EP1540844B1

Methods for transmitting a waveform having a controllable attenuation and propagation velocity

Abstract

This record has no abstract on file.

EP1540844B1, drawing sheet 1 — Sheet 1 of 175

Term

Term ended

Expired 19 August 2023, 3.1 years ago.

48 claims: 5 independent, 43 dependent

1
A method, comprising:applying an input waveform to a transmission medium, wherein the input waveform has a leading edge characterized by the expression De αt , where D is a non-zero constant, α is a non-zero constant, and t is time;recording the input waveform;recording an output waveform, wherein the output waveform is produced by the transmission medium in response to the input waveform;determining a sequence of time-of-flight values between the recorded input waveform and the recorded output waveform, wherein each of the time-of-flight values is determined by subtracting a time t 1 (Y) from a time t 2 (Y), wherein the time t 1 (Y) is the time when the leading edge of the input waveform crosses a threshold signal level, wherein the time t 2 (Y) is the time when a leading edge of the output waveform crosses the threshold signal level,wherein each of the time-of-flight values is determined using a different value of the threshold signal level Y;and computing an average of the time-of-flight values to determine an average time-of-flight value.
19
The method of claim, 18, further comprising reducing a clock skew.
31
A method comprising:applying an input pulse to a transmission medium;recording the input pulse;recording a response pulse produced by the transmission medium in response to said applying the input pulse;computing a transfer function of the transmission medium based on the recorded input pulse and the recorded response pulse;computing a simulated response signal using the transfer function and a simulated input signal, wherein the simulated input signal has a leading-edge of the form De αt , wherein D is a non-zero constant, wherein α is a non-zero constant, wherein t is time;computing a sequence of time-of-flight values between the simulated input signal and the simulated response signal, wherein each of the time-of-flight values is computed by subtracting a time t 1 (Y) from a time t 2 (Y), wherein the time t 1 (Y) is the time when the leading edge of the simulated input signal crosses a threshold signal level Y, wherein the time t 2 (Y) is the time when a leading edge of the simulated response signal crosses the threshold signal level Y, wherein each of the time-of-flight values is computed using a different value of the threshold signal level Y;and computing an average of the time-of-flight values.
39
A method comprising:applying an input pulse to a first end of a transmission medium;recording a first signal occurring at the first end of the transmission medium, wherein the first signal includes a combination of the input pulse and a reflected pulse, wherein the reflected pulse is produced by the transmission medium in response to said applying the input pulse;repeatedly applying the input pulse to a variable resistor load and recording a second signal occurring at the variable resistor load;varying the resistance of the variable resistor load until the recorded second signal approximates the recorded first signal;computing a transfer function of the transmission medium based on the recorded first signal and the recorded second signal that approximates the recorded first signal;computing a simulated response signal using the transfer function and a simulated input signal, wherein the simulated input signal has a leading-edge of the form De αt , wherein D is a non-zero constant, wherein α is a non-zero constant, wherein t is time;computing a sequence of time-of-flight values between the simulated input signal and the simulated response signal, wherein each of the time-of-flight values is computed by subtracting a time t 1 (Y) from a time t 2 (Y), wherein the time t 1 (Y) is the time when the leading edge of the simulated input signal crosses a threshold signal level Y, wherein the time t 2 (Y) is the time when a leading edge of the simulated response signal crosses the threshold signal level Y, wherein each of the time-of-flight values is computed using a different value of the threshold signal level Y;and computing an average of the time-of-flight values.
42
An apparatus, comprising:a waveform generator configured to generate an input waveform having a leading edge characterized by the expression De αt , wherein D is a non-zero constant, a is a non-zero constant and t is time;an input recorder configured to record the input waveform;an output recorder configured to record an output waveform that is produced by a transmission medium in response to the input waveform being applied to the transmission medium;and a computer configured to compute a sequence of time-of-flight values between the recorded input waveform and the recorded output waveform and to compute an average of the time-of-flight values, wherein each of the time-of-flight values is computed by subtracting a time t 1 (Y) from a time t 2 (Y), wherein t 1 (Y) is the time when the leading edge of the input waveform crosses a threshold signal level Y, wherein t 2 (Y) is the time when a leading edge of the output waveform crosses the threshold signal level Y, wherein each of the time-of-flight values is computed using a different value of the threshold signal level Y.

Independent claims5

152 paragraphs, as filed

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to the field of signal transmission. More particularly, a representative embodiment of the invention relates to transmission line testing and communication.

2. Discussion of the Related Art

Transmission lines, with their characteristic loss of signal as well as inherent time delay, may create problems in designing systems that employ a plurality of signals, which may be subject to delay and distortion. Typical signals used to generate inputs to transmission lines generally exhibit delay or propagation times that are not easily determinable. The propagation velocity of these waves is also variable with displacement along the transmission line.

An important measurement in TDR test technology is the time-of-flight (TOF) of a test pulse generated by the instrument and applied to the line. The time-of-flight may be measured by timing the passage of the pulse detected at two locations along the line. Along with a value of the propagation speed of the pulse, time-of-flight measurements can allow one to obtain the distance between measurement points or, in the case of a reflected wave, the distance from the pulse launch point to the location of the impedance change causing the pulse to be reflected and returned.

A fundamental limitation in TDR technology is the poor accuracy of TOF measurements in lossy, dispersive transmission lines. The relatively high TDR accuracy of TOF values obtainable in short low loss, low dispersion transmissions lines is possible only because the propagating test pulses keep their shape and amplitude in tact over the distances they travel during TOF measurements. By contrast, in dispersive, lossy long transmission lines the test pulses used in the art change shape, amplitude, and speed as they travel. An example can be found in <patcit id="pcit0001" dnum="WO167628A"><text>WO167628</text></patcit>.

Further, it is difficult to provide high-speed communications in a lossy, frequency dependent transmission media. It would be advantageous to have a method to increase data transmission rates in such transmission lines.

Until now, the requirements of providing a method and/or apparatus for accurately measuring times-of-flight, estimating line lengths and other parameters characterizing lossy, dispersive transmission lines, and providing high-speed communications via such transmission media have not been met. What is needed is a solution that addresses these requirements.

SUMMARY OF THE INVENTION

There is a need for the following embodiments. Of course, the invention is not limited to these embodiments.

According to an aspect of the invention, a method for transmitting a waveform having an essentially constant propagation velocity along a transmission line comprises: generating an exponential waveform, the exponential waveform (a) being characterized by the equation Vin = De-ASD[x-vSD'] where D is a magnitude, Vin is a voltage, t is time, ASD is an attenuation coefficient, and vSD is a propagation velocity; and (b) being truncated at a maximum value. According to another aspect of the invention, an apparatus for determining an attenuation coefficient and a propagation velocity comprises: an exponential waveform generator; an input recorder coupled to an output of the exponential waveform generator; a transmission line under test coupled to the output of the exponential waveform generator; an output recorder coupled to the transmission line under test; an additional transmission line coupled to the transmission line under test; and a termination impedance coupled to the additional transmission line and to a ground.

These, and other, embodiments of the invention will be better appreciated and understood when considered in conjunction with the following description and the accompanying drawings. It should be understood, however, that the following description, while indicating various embodiments of the invention and numerous specific details thereof, is given by way of illustration and not of limitation. Many substitutions, modifications, additions and/or rearrangements may be made within the scope of the invention without departing from the spirit thereof, and the invention includes all such substitutions, modifications, additions and/or rearrangements.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings accompanying and forming part of this specification are included to depict certain aspects of the invention. A clearer conception of the invention, and of the components and operation of systems provided with the invention, will become more readily apparent by referring to the exemplary, and therefore nonlimiting, embodiments illustrated in the drawings, wherein like reference numerals (if they occur in more than one view) designate the same or similar elements. The invention may be better understood by reference to one or more of these drawings in combination with the description presented herein. It should be noted that the features illustrated in the drawings are not necessarily drawn to scale. <ul id="ul0001" list-style="none" compact="compact"><li><figref idref="f0001">FIG. 1A</figref> is a graph of an SD waveform for (x = 0,t), representing an embodiment of the invention.</li><li><figref idref="f0001">FIG. 1B</figref> is a graph of an SD waveform in moving frame at x'= ℓ (x = ℓ, t'), representing an embodiment of the invention.</li><li><figref idref="f0002">FIG. 2</figref> is block diagram of a setup to measure SD parameters, representing an embodiment of the invention.</li><li><figref idref="f0002">FIG. 3</figref> is a graph of voltage traces at x = 0 and x = 1002 ft of a 2004 ft transmission line (T1 cable) with α = 3.0 x 1061/sec indicating time-of-flight measured at 4V threshold, representing an embodiment of the invention.</li><li><figref idref="f0003">FIG. 4A</figref> is a graph of SD threshold overlap regions for voltages at x = 0 and at x = d (1002 ft), representing an embodiment of the invention.</li><li><figref idref="f0003">FIG. 4B</figref> is a graph of measured constant threshold times-of-flight for 1002 ft T1 line with α = 3.0 x 106 1/sec, representing an embodiment of the invention.</li><li><figref idref="f0004">FIG. 5A</figref> is a graph of SD threshold overlap regions for voltages at x = 0 and 1/2 voltage across Rt = ∞, representing an embodiment of the invention.</li><li><figref idref="f0004">FIG. 5B</figref> is a graph of measured constant threshold times-of-flight for 2004 ft T1 line with α = 3.0 x 106 1/sec, representing an embodiment of the invention.</li><li><figref idref="f0005">FIG. 6A</figref> is a graph of an input pulse applied to a T1 cable to empirically determine a transfer function, representing an embodiment of the invention.</li><li><figref idref="f0005">FIG. 6B</figref> is a graph of a pulse measured at 1002 for a T1 cable, representing an embodiment of the invention.</li><li><figref idref="f0006">FIG. 7A</figref> is a graph of a Power Spectral Density of an input pulse applied to a T1 cable, representing an embodiment of the invention.</li><li><figref idref="f0006">FIG. 7B</figref> is a graph of a Power Spectral Density of a pulse measured at 1002 for a T1 cable, representing an embodiment of the invention.</li><li><figref idref="f0007">FIG. 8</figref> is a graph of an empirically derived transfer function for a 1002 ft T1 cable, representing an embodiment of the invention.</li><li><figref idref="f0008">FIG. 9</figref> is a graph of a measured initial waveform with simulated 1002 ft waveform for T1 cable and constant threshold times-of-flight, representing an embodiment of the invention.</li><li><figref idref="f0008">FIG. 10</figref> is a graph of a difference between measured and simulated 1002 ft voltage traces, representing an embodiment of the invention.</li><li><figref idref="f0009">FIG. 11</figref> is a graph of a simulated initial waveform with 1002 ft waveform for T1 cable, α = 1x105 sec-1, and constant threshold times-of-flight, representing an embodiment of the invention.</li><li><figref idref="f0009">FIG. 12</figref> is a graph of a simulated initial waveform with 1002 ft waveform for T1 cable, α = 1x107 sec-1, and constant threshold times-of-flight, representing an embodiment of the invention.</li><li><figref idref="f0010">FIG. 13A</figref> is a graph of SD parameters VSD determined by simulation with empirical transfer function, representing an embodiment of the invention.</li><li><figref idref="f0010">FIG. 13B</figref> is a graph of SD parameters ASD determined by simulation with empirical transfer function, representing an embodiment of the invention.</li><li><figref idref="f0011">FIG. 14</figref> is block diagram of a setup to determine ZSD(α), representing an embodiment of the invention.</li><li><figref idref="f0011">FIG. 15A</figref> is a graph of voltage traces measured at d = 1002 ft with varying termination resistances, α = 1.5x 106 1/sec, representing an embodiment of the invention.</li><li><figref idref="f0012">FIG. 15B</figref> is a graph of ratios of output signal amplitudes with known termination traces and one-half of line output signal amplitude for open termination trace, α = 1.5 x 106 1/sec, representing an embodiment of the invention.</li><li><figref idref="f0013">FIG. 16</figref> is a graph of experimentally determined SD impedances for a T1 cable as a function of α, representing an embodiment of the invention.</li><li><figref idref="f0014">FIG. 17</figref> is a graph of SD voltage traces and the natural logarithm of SD voltage traces measured at input, ∼6kft, ∼12kft and ∼18kft, along 24 AWG twisted wire pairs connected in series, representing an embodiment of the invention.</li><li><figref idref="f0015">FIG. 18</figref> is a graph used to detect end of SD region demonstrating sensitivity to noise and speed of response for the filters at 12-kft., representing an embodiment of the invention.</li><li><figref idref="f0016">FIG. 19</figref> is a graph of a variation of <maths id="math0001"><math display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msqrt><mi>LC</mi></msqrt></mrow></msub></mrow><mprescripts /><none /><mn>1</mn></mmultiscripts></mrow></math><img file="EP1540844B1_D0001.tif" /></maths> as a fraction of the speed of light in a vacuum with α, representing an embodiment of the invention.</li><li><figref idref="f0017">FIG. 20</figref> is a graph of an input waveform attenuated and time shifted for 11,573 ft (Top), 12,219 ft (Middle) and 12,859 ft (Bottom) shown with waveform measured at ∼12 kft, representing an embodiment of the invention.</li><li><figref idref="f0018">FIG. 21</figref> is a graph of a standard deviation of VPredicted -VMeasured in the SD region vs. estimated distance in region of correlation, representing an embodiment of the invention.</li><li><figref idref="f0019">FIG. 22A</figref> is a graph of VPredicted -VMeasured and minimum variance of VPredicted -VMeasured at estimated distance of 12,216 ft, representing an embodiment of the invention.</li><li><figref idref="f0019">FIG. 22B</figref> is a graph of VPredicted -VMeasured and variance of VPredicted -VMeasured at estimated distance of 12,254 ft, representing an embodiment of the invention.</li><li><figref idref="f0020">FIG. 23</figref> is a graph of voltage traces measured a ∼6, ∼12, and ∼18 kft of 24 AWG BKMA 50 cable and applied waveform attenuated and shifted in time according to SD prediction, representing an embodiment of the invention.</li><li><figref idref="f0021">FIG. 24</figref> is a graph of voltage trace measured at input showing applied truncated SD pulse and wave reflected from open termination at 1002 ft, representing an embodiment of the invention.</li><li><figref idref="f0021">FIG. 25</figref> is a graph of voltage trace measured at input showing wave reflected from open termination at 1002 ft and applied wave attenuated and shifted forward in time for best fit resulting in 2004 ft estimated total distance traveled, representing an embodiment of the invention.</li><li><figref idref="f0022">FIG. 26A</figref> is a graph of SD TDR threshold overlap regions for voltages at x = 0 for 1002 ft open terminated T1 line, representing an embodiment of the invention.</li><li><figref idref="f0022">FIG. 26B</figref> is a graph of measured constant threshold TDR times-of-flight for 1002 ft T1 line with α = 6.7 x 106 1/sec resulting in total estimated distance traveled by the reflected wave of 2010 ft., representing an embodiment of the invention.</li><li><figref idref="f0023">FIG. 27</figref> is a graph of voltage trace measured at input showing applied truncated SD pulse and wave reflected from short circuit termination at 1002 ft., representing an embodiment of the invention.</li><li><figref idref="f0023">FIG. 28</figref> is a graph of a voltage trace measured at input inverted, showing wave reflected from short circuit termination at 1002 ft inverted and the non-inverted applied input wave attenuated and shifted forward in time for best fit resulting in 2004 ft estimated total distance traveled, representing an embodiment of the invention.</li><li><figref idref="f0024">FIG. 29A</figref> is a graph of SD TDR threshold overlap regions for voltages at x = 0 for 1002 ft short circuit terminated T1 line, representing an embodiment of the invention.</li><li><figref idref="f0024">FIG. 29B</figref> is a graph of measured constant threshold TDR times-of-flight for 1002 ft T1 line with α = 6.7 x 106 1/sec resulting in total estimated distance traveled of 2008 ft., representing an embodiment of the invention.</li><li><figref idref="f0025">FIG. 30</figref> is a graph of a voltage pulse applied to 1002 ft T1 cable (dotted line), and to resistor with value Zo, representing an embodiment of the invention.</li><li><figref idref="f0025">FIG. 31</figref> is another graph of the voltage pulse applied to 1002 ft T1 cable (dotted line), and to resistor with value Zo, representing an embodiment of the invention.</li><li><figref idref="f0026">FIG. 32</figref> is a graph of a transfer function between the difference of waveform applied to the twisted wire pair and the waveform applied to Zo, representing an embodiment of the invention.</li><li><figref idref="f0027">FIG. 33</figref> is a graph of simulated applied SD waveform with α = 6.7 x 106 1/sec, and reflected waveform calculated with transfer function, representing an embodiment of the invention.</li><li><figref idref="f0027">FIG. 34</figref> is a graph of simulated voltage traces showing wave reflected from open termination at 1002 ft and applied wave attenuated and shifted forward in time for best fit resulting in 2004 ft estimated total distance traveled, representing an embodiment of the invention.</li><li><figref idref="f0028">FIG. 35A</figref> is a graph of simulated SD TDR threshold overlap regions for voltages at x = 0 for 1002 ft open terminated T1 line, representing an embodiment of the invention.</li><li><figref idref="f0028">FIG. 35B</figref> is a graph of simulated constant threshold TDR times-of-flight for 1002 ft T1 line with α = 6.7 x 106 1/sec resulting in total estimated distance traveled of 2005 ft, representing an embodiment of the invention.</li><li><figref idref="f0029">FIG. 36A</figref> is a graph of an uncompensated SD waveform applied to a 2004-ft twisted wire pair transmission line, representing an embodiment of the invention.</li><li><figref idref="f0029">FIG. 36B</figref> is a graph of an uncompensated SD waveform measured 100-Ohm termination on a 2004-ft twisted wire pair transmission line, representing an embodiment of the invention.</li><li><figref idref="f0029">FIG. 37A</figref> is a graph of a compensated SD waveform applied to a 2004-ft twisted wire pair transmission line, representing an embodiment of the invention.</li><li><figref idref="f0029">FIG. 37B</figref> is a graph of a compensated SD waveform measured 100-Ohm termination on a 2004-ft twisted wire pair transmission line, representing an embodiment of the invention.</li><li><figref idref="f0030">FIG. 38</figref> is a graph of a series of four symbols measured at input to a 2004-ft twisted wire pair transmission line, representing an embodiment of the invention.</li><li><figref idref="f0030">FIG. 39</figref> is a graph of a series of four symbols measured at 100-Ohm termination at 2004-ft on twisted wire pair transmission line, representing an embodiment of the invention.</li><li><figref idref="f0031">FIG. 40</figref> is a natural logarithmic graph of voltages measured at 100-ohm termination at 2004-ft for each symbol period showing a least mean square fit of linear region, representing an embodiment of the invention.</li><li><figref idref="f0032">FIG. 41</figref> is a graph of Xi = e(α<sub2>o</sub2> +(i-1)·Δα)·t, representing an embodiment of the invention.</li><li><figref idref="f0032">FIG. 42</figref> is a graph of an orthonormal vector set Y =[Y1, Y2, Y3, Y4, Y5] , representing an embodiment of the invention.</li><li><figref idref="f0033">FIG. 43</figref> is a graph of non-orthogonal waveforms Y' =[Y1, Y2, Y3, Y4, <maths id="math0002"><math display="inline"><mrow><mrow><msub><mi mathvariant="normal">Y</mi><mn>5</mn></msub><mo>]</mo></mrow><mo>⋅</mo><mi>sin</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mrow><mn>7</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></mfrac><mo>⋅</mo><mi mathvariant="normal">t</mi></mfenced><mo>,</mo></mrow></math><img file="EP1540844B1_D0002.tif" /></maths> representing an embodiment of the invention.</li><li><figref idref="f0034">FIG. 44</figref> is a graph of dispersed waveforms Y' at the receiver at 8000 ft, representing an embodiment of the invention.</li><li><figref idref="f0035">FIG. 45</figref> is a graph of a compensating Pulse C, at Input and at 8000 ft, representing an embodiment of the invention.</li><li><figref idref="f0035">FIG. 46</figref> is a graph of component functions at input with compensating pulses, Z = Y'+b · C, representing an embodiment of the invention.</li><li><figref idref="f0036">FIG. 47</figref> is a graph of component functions Z at 8 kft, representing an embodiment of the invention.</li><li><figref idref="f0037">FIG. 48</figref> is a graph of linear independent waveforms, S, transmitted to generate orthogonal waveforms at 8 kft, representing an embodiment of the invention.</li><li><figref idref="f0038">FIG. 49</figref> is a graph of orthogonal waveforms, S, received at 8 kft, representing an embodiment of the invention.</li><li><figref idref="f0039">FIG. 50</figref> is a flow diagram of an orthonormal set, S, generation algorithm, representing an embodiment of the invention.</li><li><figref idref="f0040">FIG. 51</figref> is a graph of three successive transmitted symbols, Q, with five bits encoded on each of four orthonormal pulses of S, representing an embodiment of the invention.</li><li><figref idref="f0040">FIG. 52</figref> is a graph of three successive symbols, Q, at the input to the receiver at 8kft with five bits encoded on each of four orthonormal pulses of S, representing an embodiment of the invention.</li><li><figref idref="f0041">FIG. 53</figref> is a graph of histogram of signal error, aexpected - adetected, for one second, or 1.67x106 bits, of data transmitted with -140 dBm AWGN present, representing an embodiment of the invention.</li></ul>

DESCRIPTION OF PREFERRED EMBODIMENT

The invention and the various features and advantageous details thereof are explained more fully with reference to the nonlimiting embodiments that are illustrated in the accompanying drawings and detailed in the following description. Descriptions of well known starting materials, processing techniques, components and equipment are omitted so as not to unnecessarily obscure the invention in detail. It should be understood, however, that the detailed description and the specific examples, while indicating specific embodiments of the invention, are given by way of illustration only and not by way of limitation. Various substitutions, modifications, additions and/or rearrangements within the spirit and/or scope of the underlying inventive concept will become apparent to those skilled in the art from this disclosure.

Within this application several publications are referenced by Arabic numerals within parentheses or brackets. Full citations for these, and other, publications may be found at the end of the specification immediately preceding the claims after the section heading References. The disclosures of all these publications in their entireties are hereby expressly incorporated by reference herein for the purpose of indicating the background of the invention and illustrating the state of the art.

The below-referenced allowed U.S. Patent Application discloses embodiments that were satisfactory for the purposes for which they are intended. The entire contents of <patcit id="pcit0002" dnum="US51992200A" dnum-type="L"><text>U.S. Patent Application 09/519,922, filed March 7th, 2000</text></patcit>, is hereby expressly incorporated by reference herein for all purposes.

In general, the context of the invention can include signal transmission. The context of the invention can include a method and/or apparatus for measuring and estimating transmission line parameters. The context of the invention can also include a method and/or apparatus for performing high-speed communication over a lossy, dispersive transmission line.

A practical application of the invention that has value within the technological arts is in time domain reflectometry. Further, the invention is useful in conjunction with lossy transmission lines. The invention is also useful in conjunction with frequency dependent transmission lines. Another practical application of the invention that has value within the technological arts is high-speed communications. There are virtually imnumerable other uses for the invention. In fact, any application involving the transmission of a signal may benefit.

A method for transmitting a waveform having a controllable attenuation and propagation velocity, representing an embodiment of the invention, can be cost effective and advantageous. The invention improves quality and/or reduces costs compared to previous approaches.

The invention includes a method and/or apparatus for utilizing a truncated waveform, coined a Speedy Delivery (SD) waveform by the inventors. An analysis of SD propagation in coupled lossy transmission lines is presented and practical considerations associated with truncating the SD waveforms are addressed. Parameters used to describe the propagation of the SD waveform are defined and techniques for determining their values are presented. These parameters include the Speedy Delivery propagation velocity, vSD, the Speedy Delivery attenuation coefficient, ASD, and the Speedy Delivery impedance, ZSD. These parameters may depend on properties of a transmission media as well as an exponential coefficient α (SD input signal parameter).

The invention describes SD signal propagation in coupled transmission lines. An embodiment involving two transmission lines is illustrated. More complex configurations that include a larger number of coupled lines may be readily developed by one of ordinary skill in the art in light of this disclosure. Additionally, the behavior of the propagation of a slowly varying envelope of an optical pulse containing a truncated SD signal in a single mode communication fiber is described.

The invention teaches a method for using SD test pulses to obtain empirical estimates of the SD signal propagation parameters, ASD and vSD, in lossy transmission lines, including those with frequency dependent parameters. Once the value of vSD is determined for a calibration cable length, constant threshold time of flight measurements can be used to measure distances to discontinuities in similar cables. The calibration procedure can be repeated experimentally for each exponential coefficient α.

The invention provides a method for utilizing a wideband, non-SD waveform in order to develop an empirical transfer function for a calibration length of a given transmission media. This transfer function may be used to simulate the transmission of SD waveforms with a wide range of values for the exponential coefficient α. The SD parameters ASD(α) and vSD(α) can be determined from these simulated waveforms. The inclusion of a simulation step in the calibration process can significantly reduce the experimental measurements needed to determine the values of ASD(α) and vSD(α) for a variety of values for α. The SD behavior of a media can then be determined by simulation, without the need for an SD waveform generator.

The invention teaches a method for determining the SD impedance of a transmission line. It also illustrates the predicable variation of ZSD as a function of α. This predictability can allow a designer to control the transmission line SD impedance by appropriately selecting the parameter α.

The invention also describes an alternate precision distance measurement method using truncated SD test pulses with high loss, long lines. As the test pulses travel down these lines, the attenuation can be so extensive that their amplitudes are too small for common threshold crossing time of flight measurements to be feasible. The invention includes a method for predicting an attenuation and overcoming this difficulty, extracting accurate length measurements for long lossy lines.

The invention demonstrates the utility of the SD test waveform in accurately determining distances to impedance discontinuities. Examples show how to use SD wave propagation as the basis for an accurate time domain reflectometry (TDR) unit. This unit may include an SD waveform generator, a means of coupling with the transmission media, a means of measuring the applied and reflected waveforms, a display, memory storage, and computational ability to analyze and interpret the SD waveforms. The invention can also include a computer. The invention can include a method which can be applied to modify currently available TDR units to increase their accuracy. This modification may be a TDR software modification resulting in a nonrecurring unit manufacturing cost. A standard TDR waveform may be used to determine an empirical transfer function for the line under test, and this transfer function may then be used to simulate the propagation of the SD waveform. This process allows all the advantages of SD to be incorporated into currently available units with only a modification of existing computational algorithms, creating a virtual SD TDR unit.

The truncated SD test pulse approach can also be applied to the propagation of acoustic waves in sonar and geophysical applications. Reflected acoustic signals can be used to accurately determine the location of underwater objects and to determine locations of geophysical strata boundaries. Empirical transfer functions can be determined and used to simulate SD pulse propagation eliminating the need to physically generate complex acoustic pulses.

The invention provides an SD waveform that has several propagation properties in lossy, frequency dependent transmission media, which make it suitable for enhancing a data transmission scheme.

The invention provides a method to increase the number of bits transmitted per pulse by varying the value of α to encode data on truncated SD waveforms. The example provided utilizes SD pulses with four different α's sequentially transmitted on a 2kft cable. These SD pulses are received at the end of the cable with the exponential constants undistorted. The four pulses may be transmitted as positive or negative signals resulting in eight possible states. Therefore, each pulse represents three bits of transmitted information in each symbol period. This strategy can be used with short, low loss transmission lines.

The invention also teaches the use of SD propagation properties to generate a set of pulses consisting of linear combinations of truncated SD waveforms that are orthogonal at the receiver. Data can be encoded on these pulses by varying their amplitude, and these amplitude-modulated pulses may be transmitted simultaneously on a transmission line. The individual pulse amplitudes are computed at the receiver taking advantage of their orthogonality. This method provides improved noise immunity, and can be used with longer, higher loss transmission lines.

The invention also teaches the use of SD waveforms in digital circuits, specifically, on-chip clock circuits. It provides a method for utilizing truncated SD clock waveforms to reduce clock skew.

The propagation of a Speedy Delivery (SD) waveform in a transmission line can be described by the equation V(x,t) = De-A<sub2>SD</sub2>[x-v<sub2>SD</sub2>t] The associated boundary condition at the input to the line is V(x = 0, t) = Deαt, wherein ASD is the exponential coefficient describing the attenuation of the SD signal as a function of distance along the line at a fixed time instant, and vSD is the SD signal propagation velocity. These quantities are also related by the simple equation, ASD(α)·vSD(α)= a. The parameters ASD and vSD of the propagating wave and the line boundary condition are all functions of α, and may be determined experimentally for a transmission line.

For the case of two balanced twisted wire pair transmission lines, the differential voltage can be equally divided between the two wires and the current on each wire in the pair is equal in magnitude but opposite in direction. In this case, the telegrapher's equations can be written: <maths id="math0003"><math display="block"><mrow><mrow><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac></mrow><mi mathvariant="bold">V</mi><mo>=</mo><mi mathvariant="bold">Z</mi><mo>⋅</mo><mi mathvariant="bold">I</mi></mrow></math><img file="EP1540844B1_D0003.tif" /></maths><maths id="math0004"><math display="block"><mrow><mrow><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac></mrow><mi mathvariant="bold">I</mi><mo>=</mo><mi mathvariant="bold">Y</mi><mo>⋅</mo><mi mathvariant="bold">V</mi></mrow></math><img file="EP1540844B1_D0004.tif" /></maths><maths id="math0005"><math display="block"><mrow><mrow><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr></mtable></mfenced></mrow><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">R</mi><mn mathvariant="normal">1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">L</mi><mn mathvariant="normal">1</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd><mtd><msub><mi mathvariant="normal">M</mi><mn mathvariant="normal">12</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">M</mi><mn mathvariant="normal">21</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd><mtd><msub><mi mathvariant="normal">R</mi><mn mathvariant="normal">2</mn></msub><mo>+</mo><msub><mi mathvariant="normal">L</mi><mn mathvariant="normal">2</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr></mtable></mfenced></mrow></math><img file="EP1540844B1_D0005.tif" /></maths><maths id="math0006"><math display="block"><mrow><mrow><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr></mtable></mfenced></mrow><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">G</mi><mn mathvariant="normal">1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">C</mi><mn mathvariant="normal">1</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd><mtd><msub><mi mathvariant="normal">C</mi><mn mathvariant="normal">12</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">C</mi><mn mathvariant="normal">21</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd><mtd><msub><mi mathvariant="normal">G</mi><mn mathvariant="normal">2</mn></msub><mo>+</mo><msub><mi mathvariant="normal">C</mi><mn mathvariant="normal">2</mn></msub><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd></mtr></mtable></mfenced></mrow></math><img file="EP1540844B1_D0006.tif" /></maths> This analysis includes the net cross capacitance, Cij, and the net cross inductances, Mij. The cross resistance and conductance terms are neglected. These equations are then Laplace transformed into the complex frequency domain. <maths id="math0007"><math display="block"><mrow><mrow><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mrow><mover><mi mathvariant="normal">R</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">L</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">M</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>12</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">M</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">21</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><msub><mrow><mover><mi mathvariant="normal">R</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">L</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow></math><img file="EP1540844B1_D0007.tif" /></maths><maths id="math0008"><math display="block"><mrow><mrow><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mrow><mover><mi mathvariant="normal">G</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>12</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">21</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><msub><mrow><mover><mi mathvariant="normal">G</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow></math><img file="EP1540844B1_D0008.tif" /></maths> Taking the partial derivative of the top equation with respect to x, and substituting the bottom equation for the <maths id="math0009"><math display="inline"><mrow><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn>1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">I</mi><mn>2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow></math><img file="EP1540844B1_D0009.tif" /></maths> term, gives the result: <maths id="math0010"><math display="block"><mrow><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mn>2</mn></msup></mrow><mrow><mo>∂</mo><msup><mi mathvariant="normal">x</mi><mn mathvariant="normal">2</mn></msup></mrow></mfrac></mrow><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced><mo>=</mo><mover><mi mathvariant="bold">Z</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">s</mi></mfenced><mover><mi mathvariant="bold">Y</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>⋅</mo><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mrow></math><img file="EP1540844B1_D0010.tif" /></maths><maths id="math0011"><math display="block"><mrow><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow><mn>2</mn></msup></mrow></mfrac><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mrow><mover><mi mathvariant="normal">R</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">L</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">M</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>12</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">M</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">21</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><msub><mrow><mover><mi mathvariant="normal">R</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">L</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mrow><mover><mi mathvariant="normal">G</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>12</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">21</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd><mtd><msub><mrow><mover><mi mathvariant="normal">G</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow></mrow></math><img file="EP1540844B1_D0011.tif" /></maths> or, <maths id="math0012"><math display="block"><mrow><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mn>2</mn></msup></mrow><mrow><mo>∂</mo><msup><mi mathvariant="normal">x</mi><mn mathvariant="normal">2</mn></msup></mrow></mfrac></mrow><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced><mo>=</mo><mover><mi mathvariant="normal">Γ</mi><mrow><mo>‾</mo></mrow></mover><msup><mfenced><mi mathvariant="normal">s</mi></mfenced><mn>2</mn></msup><mo>⋅</mo><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mrow></math><img file="EP1540844B1_D0012.tif" /></maths><maths id="math0013"><math display="block"><mrow><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mn>2</mn></msup></mrow><mrow><msup><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow><mn>2</mn></msup></mrow></mfrac><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mfenced><mrow><msub><mrow><mover><mi mathvariant="normal">R</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">L</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mrow></mfenced><mfenced><msub><mrow><mover><mi mathvariant="normal">G</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>1</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mfenced></mtd><mtd><msup><mi mathvariant="normal">s</mi><mn>2</mn></msup><msub><mrow><mover><mi mathvariant="normal">M</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>12</mn></msub><mrow><mfenced><mi mathvariant="normal">s</mi></mfenced><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>21</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mrow></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">s</mi><mn>2</mn></msup><msub><mrow><mover><mi mathvariant="normal">M</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn mathvariant="normal">21</mn></msub><mrow><mfenced><mi mathvariant="normal">s</mi></mfenced><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>12</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mrow></mtd><mtd><mfenced><mrow><msub><mrow><mover><mi mathvariant="normal">R</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">L</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mrow></mfenced><mfenced><msub><mrow><mover><mi mathvariant="normal">G</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mi mathvariant="normal">s</mi><msub><mrow><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover></mrow><mn>2</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced></mtd></mtr></mtable></mfenced></mrow></mrow></math><img file="EP1540844B1_D0013.tif" /></maths> This system of equations has the solution: <maths id="math0014"><math display="block"><mrow><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced><mo>=</mo><msup><mi mathvariant="bold">e</mi><mrow><mo>−</mo><mover><mi mathvariant="normal">Γ</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>⋅</mo><mi mathvariant="bold">A</mi><mo>+</mo><msup><mi mathvariant="bold">e</mi><mrow><mo>+</mo><mover><mi mathvariant="normal">Γ</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>⋅</mo><mi mathvariant="bold">B</mi></mrow></math><img file="EP1540844B1_D0014.tif" /></maths>

Applying the boundary conditions for a semi-infinite pair of transmission lines, V(x = 0,s) at x = 0, and V(x → ∞ ,s) = 0 gives: <maths id="math0015"><math display="block"><mrow><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced><mo>=</mo><msup><mi mathvariant="bold">e</mi><mrow><mo>−</mo><mover><mi mathvariant="normal">Γ</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>⋅</mo><mi mathvariant="bold">V</mi><mfenced><mi mathvariant="normal">x</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">s</mi></mfenced><mn>.</mn></mrow></math><img file="EP1540844B1_D0015.tif" /></maths>

Because <o ostyle="single">Γ</o> is a complex symmetric matrix, it can be written <o ostyle="single">Γ</o> = (<o ostyle="single">ZY</o>)1/2 = <o ostyle="single">PγP</o>-1, where <o ostyle="single">P</o> is the eigenvector matrix of the <o ostyle="single">ZY</o> product and <o ostyle="single">γ</o> is the diagonal eigenvalue matrix of <o ostyle="single">Γ</o>. Therefore, we can write e-<o ostyle="single">Γ</o>(s)·x = P(s)·e-<o ostyle="single">γ</o>(s)·x · P(s)-1. This gives the result: <maths id="math0016"><math display="block"><mrow><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">s</mi></mfenced><mo>=</mo><mi mathvariant="bold">P</mi><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>⋅</mo><msup><mi mathvariant="bold">e</mi><mrow><mo>−</mo><mover><mi mathvariant="normal">γ</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>⋅</mo><mi mathvariant="bold">P</mi><msup><mfenced><mi mathvariant="normal">s</mi></mfenced><mrow><mo>−</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⋅</mo><mi mathvariant="bold">V</mi><mfenced><mi mathvariant="normal">x</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">s</mi></mfenced></mrow></math><img file="EP1540844B1_D0016.tif" /></maths>

For the case where the applied boundary conditions at x = 0 are SD waveforms, V1(x = 0,t)=D1eα<sub2>1</sub2>t, V2(x = 0,t)=D2eα<sub2>2</sub2>t, the SD solution is: <maths id="math0017"><math display="block"><mrow><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced><mo>=</mo><mi mathvariant="bold">P</mi><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mi mathvariant="normal">γ</mi><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>⋅</mo><msup><mi mathvariant="bold">P</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi mathvariant="bold">P</mi><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mi mathvariant="normal">γ</mi><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>⋅</mo><msup><mi mathvariant="bold">P</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">D</mi><mn>2</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr></mtable></mfenced><mo>,</mo></mrow></math><img file="EP1540844B1_D0017.tif" /></maths> where P(αi) is a real matrix. <maths id="math0018"><math display="block"><mrow><mtable columnalign="left" width="auto"><mtr><mtd><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd><mtd><mo>=</mo><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd></mtr></mtable></mfenced></mrow><mo>⋅</mo><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn>22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mfenced></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>+</mo><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd></mtr></mtable></mfenced></mrow></mrow><mo>⋅</mo><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn>22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">D</mi><mn>2</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr></mtable></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0018.tif" /></maths> If we designate P-1 as Q, we can write: <maths id="math0019"><math display="block"><mrow><mtable><mtr><mtd><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd><mtd><mo>=</mo><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd></mtr></mtable></mfenced></mrow><mo>⋅</mo><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">Q</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">Q</mi><mn>22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mfenced></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>+</mo><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd></mtr></mtable></mfenced></mrow></mrow><mo>⋅</mo><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">Q</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd><mtd><msub><mi mathvariant="normal">Q</mi><mn>22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">D</mi><mn>2</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr></mtable></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0019.tif" /></maths> or, <maths id="math0020"><math display="block"><mrow><mtable><mtr><mtd><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd><mtd><mo>=</mo><mi mathvariant="bold">P</mi><mfenced><msub><mi>α</mi><mn>1</mn></msub></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">11</mn></msub><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">21</mn></msub><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn>2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr></mtable></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">D</mi><mn>1</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>+</mo><mi mathvariant="bold">P</mi><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn>12</mn></msub><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn>1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn>22</mn></msub><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn>2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr></mtable></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mrow></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0020.tif" /></maths> or, <maths id="math0021"><math display="block"><mrow><mtable columnalign="left" width="auto"><mtr><mtd><mi mathvariant="bold">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd><mtd><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">11</mn></msub><mrow><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mrow><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mrow></mrow><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">11</mn></msub><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>1</mn></msub></mfenced><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn>1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr></mtable></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /><mo>+</mo><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">12</mn></msub><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">12</mn></msub><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup></mrow></mtd></mtr></mtable></mfenced><mo>⋅</mo><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mrow></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0021.tif" /></maths>

The real constants associated with the eigenvector matrix, P, its inverse, P-1 = Q, and the SD parameters D can be written: <maths id="math0022"><math display="block"><mrow><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">11</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0022.tif" /></maths><maths id="math0023"><math display="block"><mrow><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">12</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">P</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0023.tif" /></maths><maths id="math0024"><math display="block"><mrow><msub><mi mathvariant="normal">A</mi><mn>21</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn>2</mn></msub><msub><mi mathvariant="normal">P</mi><mn>11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0024.tif" /></maths><maths id="math0025"><math display="block"><mrow><msub><mi mathvariant="normal">A</mi><mn>22</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn>2</mn></msub><msub><mi mathvariant="normal">P</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn>22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0025.tif" /></maths><maths id="math0026"><math display="block"><mrow><msub><mi mathvariant="normal">B</mi><mn>11</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">P</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn>11</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0026.tif" /></maths><maths id="math0027"><math display="block"><mrow><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">12</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn mathvariant="normal">21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0027.tif" /></maths><maths id="math0028"><math display="block"><mrow><msub><mi mathvariant="normal">B</mi><mn>21</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn>2</mn></msub><msub><mi mathvariant="normal">P</mi><mn>21</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn>12</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0028.tif" /></maths><maths id="math0029"><math display="block"><mrow><msub><mi mathvariant="normal">B</mi><mn>22</mn></msub><mo>=</mo><msub><mi mathvariant="normal">D</mi><mn>2</mn></msub><msub><mi mathvariant="normal">P</mi><mn mathvariant="normal">22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced><msub><mi mathvariant="normal">Q</mi><mn>22</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn>2</mn></msub></mfenced></mrow></math><img file="EP1540844B1_D0029.tif" /></maths>

The signals on transmission lines will be: <maths id="math0030"><math display="block"><mrow><msub><mi mathvariant="normal">V</mi><mn>1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced><mo>=</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mrow></math><img file="EP1540844B1_D0030.tif" /></maths><maths id="math0031"><math display="block"><mrow><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced><mo>=</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup></mrow></math><img file="EP1540844B1_D0031.tif" /></maths> or, <maths id="math0032"><math display="block"><mrow><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd><mtd><mo>=</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">12</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><mi mathvariant="normal">SD</mi><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0032.tif" /></maths><maths id="math0033"><math display="block"><mrow><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn>2</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd><mtd><mo>=</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>+</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">12</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn>12</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">B</mi><mn mathvariant="normal">22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0033.tif" /></maths>

Where: <maths id="math0034"><math display="block"><mrow><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mo>=</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>,</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">12</mn></msub></mrow></msub><mo>=</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>,</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mo>=</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo>,</mo><mspace width="1em" /><mi mathvariant="normal">and</mi><mspace width="1em" /><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mo>=</mo><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0034.tif" /></maths><maths id="math0035"><math display="block"><mrow><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mrow><mrow><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mrow></mfrac><mo>,</mo><mspace width="1em" /><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">12</mn></msub></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mrow><mrow><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">1</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mrow></mfrac><mo>,</mo><mspace width="1em" /><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mrow><mrow><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub></mfenced></mrow></mfrac><mo>,</mo><mspace width="1em" /><mi mathvariant="normal">and</mi><mspace width="1em" /><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mrow><mrow><msub><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></msub><mfenced><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub></mfenced></mrow></mfrac><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0035.tif" /></maths>

Thus, there are SD attenuations and velocities associated with each eigenvalue, γi, of the product <o ostyle="single">ZY</o> evaluated at each α.

Because positive exponential waveforms continually increase, practical considerations may create the need to truncate or limit the waveform at some level determined by the specific application. Truncation may be accomplished by several methods, as is known in the art.

The result of this truncation of the input of a transmission media is that the propagating exponential (SD) signal is also limited in magnitude as it travels in the media, and in lossy media, this maximum amplitude decreases with distance traveled exhibiting an attenuating behavior.

The propagating (SD) signal is described in the coordinate frame, (x, t), by <maths id="math0036"><math display="block"><mrow><mi mathvariant="normal">V</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">De</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mfenced open="[" close="]"><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></mfenced></mrow></msup><mo>=</mo><msup><mi mathvariant="normal">De</mi><mrow><mi mathvariant="normal">α</mi><mo>⋅</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi></msub><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>,</mo><msub><mrow><mspace width="1em" /><mi mathvariant="normal">where A</mi></mrow><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>=</mo><mi mathvariant="normal">α</mi><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0036.tif" /></maths>

Referring to <figref idref="f0001">FIG. 1A</figref>, a graph of an SD waveform for (x = 0, t) is depicted. The truncated input waveform at x = 0 reaching a limit at t = ti is shown.

In the case of a four parameter (LCRG) transmission line, we have the expressions <maths id="math0037"><math display="block"><mrow><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>=</mo><msqrt><mrow><msup><mi mathvariant="normal">LCα</mi><mn mathvariant="normal">2</mn></msup><mo>+</mo><mfenced><mi mathvariant="normal">RC</mi><mo>+</mo><mi mathvariant="normal">LG</mi></mfenced><mi mathvariant="normal">α</mi><mo>+</mo><mi mathvariant="normal">RG</mi></mrow></msqrt><mo>,</mo><msub><mi mathvariant="normal">and v</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">α</mi><mrow><msqrt><mrow><msup><mi mathvariant="normal">LCα</mi><mn mathvariant="normal">2</mn></msup><mo>+</mo><mfenced><mi mathvariant="normal">RC</mi><mo>+</mo><mi mathvariant="normal">LG</mi></mfenced><mi mathvariant="normal">α</mi><mo>+</mo><mi mathvariant="normal">RG</mi></mrow></msqrt></mrow></mfrac><mn>.</mn></mrow></math><img file="EP1540844B1_D0037.tif" /></maths>

The attenuation of the traveling SD wave may be viewed from the perspective of a moving reference frame traveling at the speed <maths id="math0038"><math display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msqrt><mi>LC</mi></msqrt></mrow></msub></mrow><mprescripts /><none /><mn>1</mn></mmultiscripts><mn>.</mn></mrow></math><img file="EP1540844B1_D0038.tif" /></maths>

Introducing the new coordinates: <maths id="math0039"><math display="block"><mrow><mi>tʹ</mi><mo>=</mo><mi mathvariant="normal">t</mi><mo>−</mo><mi mathvariant="normal">x</mi><msqrt><mi>LC</mi></msqrt><mn>.</mn></mrow></math><img file="EP1540844B1_D0039.tif" /></maths><maths id="math0040"><math display="block"><mrow><mi>xʹ</mi><mo>=</mo><mi mathvariant="normal">x</mi></mrow></math><img file="EP1540844B1_D0040.tif" /></maths>

Then <maths id="math0041"><math display="block"><mrow><mi mathvariant="normal">V</mi><mfenced separators=","><mi>xʹ</mi><mi>tʹ</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">De</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mi>xʹ</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mfenced><mi>tʹ</mi><mo>+</mo><mi>xʹ</mi><mo>⋅</mo><msqrt><mi>LC</mi></msqrt></mfenced></mrow></mfenced><mo>,</mo></mrow></msup></mrow></math><img file="EP1540844B1_D0041.tif" /></maths> or <maths id="math0042"><math display="block"><mrow><mi mathvariant="normal">V</mi><mfenced separators=","><mi>xʹ</mi><mi>tʹ</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">De</mi><mrow><mi mathvariant="normal">α</mi><mo>⋅</mo><mi>tʹ</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><msqrt><mi mathvariant="normal">LC</mi></msqrt></mrow></mfenced><mo>⋅</mo><mi>xʹ</mi></mrow></msup><mn>.</mn></mrow></math><img file="EP1540844B1_D0042.tif" /></maths>

If we view the traveling wave after traveling a distance, x = x' = ℓ, in the transmission media, then <maths id="math0043"><math display="block"><mrow><mi mathvariant="normal">V</mi><mfenced><mo>ℓ</mo><mo>,</mo><mi>tʹ</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">De</mi><mrow><mi mathvariant="normal">α</mi><mo>⋅</mo><mi>tʹ</mi></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><msqrt><mi mathvariant="normal">LC</mi></msqrt></mrow></mfenced><mo>⋅</mo><mo>ℓ</mo></mrow></msup><mn>.</mn></mrow></math><img file="EP1540844B1_D0043.tif" /></maths>

Setting t' = ti, yields the maximum value of the truncated SD waveform at x = ℓ, and for any value 0 ≤ tj ≤ ti, the value of the signal at the associated point tj in the moving reference frame at x = ℓ is: <maths id="math0044"><math display="block"><mrow><mi mathvariant="normal">V</mi><mfenced><mo>ℓ</mo><mo>,</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">j</mi></msub></mfenced><mo>=</mo><msup><mi mathvariant="normal">De</mi><mrow><mi mathvariant="normal">α</mi><mo>⋅</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">j</mi></msub></mrow></msup><mspace width="1em" /><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><msqrt><mi mathvariant="normal">LC</mi></msqrt></mrow></mfenced><mo>⋅</mo><mo>ℓ</mo></mrow></msup></mrow></math><img file="EP1540844B1_D0044.tif" /></maths>

Referring to <figref idref="f0001">FIG. 1B</figref>, a graph of an SD waveform in moving frame at x' = ℓ (x = ℓ, t') is depicted. The attenuation of this truncated waveform is defined by the ratio: <maths id="math0045"><math display="block"><mrow><mtable><mtr><mtd><mfrac><mrow><mi mathvariant="normal">V</mi><mfenced><mo>ℓ</mo><mo>,</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">j</mi></msub></mfenced></mrow><mrow><mi mathvariant="normal">V</mi><mfenced separators=","><mn mathvariant="normal">0</mn><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">j</mi></msub></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">De</mi><mrow><mi mathvariant="normal">α</mi><mo>⋅</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">j</mi></msub></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><msqrt><mi mathvariant="normal">LC</mi></msqrt></mrow></mfenced><mo>⋅</mo><mo>ℓ</mo></mrow></msup></mrow><mrow><msup><mi mathvariant="normal">De</mi><mrow><mi mathvariant="normal">α</mi><mo>⋅</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">j</mi></msub></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><msqrt><mi mathvariant="normal">LC</mi></msqrt></mrow></mfenced><mo>⋅</mo><mo>ℓ</mo></mrow></msup></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0045.tif" /></maths>

This attenuation of the signal at a point tj of a truncated SD waveform traveling in a transmission line with frequency dependent parameters whose Laplace transforms are <o ostyle="single">L</o>(s), <o ostyle="single">C</o>(s), <o ostyle="single">R</o>(s), and <o ostyle="single">G</o>(s), is: <maths id="math0046"><math display="block"><mrow><mfrac><mrow><mi>V</mi><mfenced><mo>ℓ</mo><mo>,</mo><msub><mi>t</mi><mi>j</mi></msub></mfenced></mrow><mrow><mi>V</mi><mfenced separators=","><mn>0</mn><msub><mi>t</mi><mi>j</mi></msub></mfenced></mrow></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mrow><mover><mi>A</mi><mrow><mo>‾</mo></mrow></mover></mrow><mi mathvariant="italic">SD</mi></msub><mfenced><mi>α</mi></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mn>1</mn><mo>−</mo><msub><mrow><mover><mi>v</mi><mrow><mo>‾</mo></mrow></mover></mrow><mi mathvariant="italic">SD</mi></msub><mrow><mfenced><mi>α</mi></mfenced><mo>⋅</mo><msqrt><mrow><mover><mi>L</mi><mrow><mo>‾</mo></mrow></mover><mover><mi mathvariant="italic">C</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi>α</mi></mfenced></mrow></msqrt></mrow></mfenced><mo>⋅</mo><mo>ℓ</mo></mrow></msup></mrow></math><img file="EP1540844B1_D0046.tif" /></maths> where: <maths id="math0047"><math display="block"><mrow><msub><mrow><mover><mi mathvariant="normal">A</mi><mrow><mo>‾</mo></mrow></mover></mrow><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>=</mo><msqrt><mrow><mover><mi mathvariant="normal">LC</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msup><mo>+</mo><mfenced><mover><mi mathvariant="normal">RC</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>+</mo><mover><mi mathvariant="normal">LC</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced></mfenced><mo>⋅</mo><mi mathvariant="normal">α</mi><mo>+</mo><mover><mi mathvariant="normal">RG</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced></mrow></msqrt><mo>,</mo></mrow></math><img file="EP1540844B1_D0047.tif" /></maths> and <maths id="math0048"><math display="block"><mrow><msub><mrow><mover><mi mathvariant="normal">v</mi><mrow><mo>‾</mo></mrow></mover></mrow><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">α</mi><mrow><msqrt><mrow><mover><mi mathvariant="normal">LC</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>⋅</mo><msup><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msup><mo>+</mo><mfenced><mover><mi mathvariant="normal">RC</mi><mrow><mo>‾</mo></mrow></mover><mrow><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>+</mo><mover><mi mathvariant="normal">LG</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced></mrow></mfenced><mo>⋅</mo><mi mathvariant="normal">α</mi><mo>+</mo><mover><mi mathvariant="normal">RG</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced></mrow></msqrt></mrow></mfrac><mn>.</mn></mrow></math><img file="EP1540844B1_D0048.tif" /></maths>

Furthermore, in this transmission media with frequency dependent parameters, if the input exponential pulse is rapidly closed when the exponential amplitude truncation limit is reached, then the end of the exponential region of the leading edge of the transmitted pulse is further reduced in amplitude and rounded by the chromatic dispersion.

Coupled transmission lines can be treated in a similar fashion. The equations governing SD propagation on one of two coupled transmission lines is: <maths id="math0049"><math display="block"><mrow><mtable><mtr><mtd><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mtd><mtd><mo>=</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">12</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><mi mathvariant="normal">SD</mi><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mfenced><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mi mathvariant="normal">t</mi></mfenced></mrow></msup></mtd></mtr></mtable><mn>.</mn></mrow></math><img file="EP1540844B1_D0049.tif" /></maths>

Applying the substitution: <maths id="math0050"><math display="inline"><mrow><mi mathvariant="normal">t</mi><mo>′</mo><mo>=</mo><mi mathvariant="normal">t</mi><mo>−</mo><mi mathvariant="normal">x</mi><msqrt><mrow><mover><mi>LC</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt><mo>,</mo></mrow></math><img file="EP1540844B1_D0050.tif" /></maths> x' = x, gives the result: <maths id="math0051"><math display="block"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mfrac><mrow><mrow><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced><mi mathvariant="normal">xʹ</mi><mo>=</mo><mo>ℓ</mo><mo>,</mo><mi mathvariant="normal">tʹ</mi><mo>=</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mfenced></mrow></mrow><mrow><mrow><msub><mi mathvariant="normal">V</mi><mn mathvariant="normal">1</mn></msub><mfenced><mi mathvariant="normal">x</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">t</mi><mo>=</mo><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mfenced></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><msqrt><mrow><mover><mi mathvariant="normal">LC</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">21</mn></msub></mrow></msub><msqrt><mrow><mover><mi mathvariant="normal">LC</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup></mrow><mrow><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn>12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn>22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">12</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">12</mn></msub></mrow></msub><msqrt><mrow><mover><mi mathvariant="normal">LC</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mrow><msub><mi mathvariant="normal">SD</mi><mn mathvariant="normal">22</mn></msub></mrow></msub><msqrt><mrow><mover><mi mathvariant="normal">LC</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup></mrow><mrow><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup><mo>+</mo><msub><mi mathvariant="normal">A</mi><mn mathvariant="normal">22</mn></msub><mo>⋅</mo><msup><mi mathvariant="normal">e</mi><mrow><msub><mi mathvariant="normal">α</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi></msub></mrow></msup></mrow></mfrac></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0051.tif" /></maths>

Which can be written: <maths id="math0052"><math display="block"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mfrac><mrow><mrow><msub><mi>V</mi><mn mathvariant="normal">1</mn></msub><mfenced><mo>ℓ</mo><mo>,</mo><msub><mi>t</mi><mi>j</mi></msub></mfenced></mrow></mrow><mrow><mrow><msub><mi>V</mi><mn mathvariant="normal">1</mn></msub><mfenced><mn mathvariant="normal">0</mn><msub><mi>t</mi><mi>j</mi></msub></mfenced></mrow></mrow></mfrac><mo>=</mo><mfrac><mfenced><msub><mi>A</mi><mn mathvariant="normal">11</mn></msub><mo>⋅</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi mathvariant="italic">A</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="italic">v</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn mathvariant="normal">11</mn></msub></mrow></msub><msqrt><mrow><mover><mi>L</mi><mrow><mo>‾</mo></mrow></mover><mover><mi mathvariant="italic">C</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup><mo>+</mo><msub><mi mathvariant="italic">A</mi><mn mathvariant="normal">21</mn></msub><mo>⋅</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi mathvariant="italic">A</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn>21</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="italic">v</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn>21</mn></msub></mrow></msub><msqrt><mrow><mover><mi>L</mi><mrow><mo>‾</mo></mrow></mover><mover><mi mathvariant="italic">C</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup></mfenced><mrow><mfenced><msub><mi>A</mi><mn mathvariant="normal">11</mn></msub><mo>+</mo><msub><mi>A</mi><mn>21</mn></msub></mfenced><mo>+</mo><mfenced><msub><mi>A</mi><mn>12</mn></msub><mo>+</mo><msub><mi>A</mi><mn>22</mn></msub></mfenced><mo>⋅</mo><msup><mi>e</mi><mrow><mfenced><msub><mi>α</mi><mn>2</mn></msub><mo>−</mo><msub><mi>α</mi><mn>1</mn></msub></mfenced><msub><mi>t</mi><mi>i</mi></msub></mrow></msup></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mfenced><msub><mi>A</mi><mn mathvariant="normal">12</mn></msub><mo>⋅</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi mathvariant="italic">A</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn>12</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="italic">v</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn>12</mn></msub></mrow></msub><msqrt><mrow><mover><mi>L</mi><mrow><mo>‾</mo></mrow></mover><mover><mi mathvariant="italic">C</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup><mo>+</mo><msub><mi mathvariant="italic">A</mi><mn>22</mn></msub><mo>⋅</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi mathvariant="italic">A</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn>22</mn></msub></mrow></msub><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><msub><mi mathvariant="italic">v</mi><mrow><msub><mi mathvariant="italic">SD</mi><mn>22</mn></msub></mrow></msub><msqrt><mrow><mover><mi>L</mi><mrow><mo>‾</mo></mrow></mover><mover><mi mathvariant="italic">C</mi><mrow><mo>‾</mo></mrow></mover></mrow></msqrt></mfenced><mo>ℓ</mo></mrow></msup></mfenced><mrow><mfenced><msub><mi>A</mi><mn mathvariant="normal">11</mn></msub><mo>+</mo><msub><mi>A</mi><mn>21</mn></msub></mfenced><mo>⋅</mo><msup><mi>e</mi><mrow><mfenced><msub><mi>α</mi><mn>1</mn></msub><mo>−</mo><msub><mi>α</mi><mn>2</mn></msub></mfenced><msub><mi>t</mi><mi>i</mi></msub></mrow></msup><mo>+</mo><mfenced><msub><mi>A</mi><mn>12</mn></msub><mo>+</mo><msub><mi>A</mi><mn>22</mn></msub></mfenced></mrow></mfrac></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd></mtr></mtable><mn>.</mn></mrow></math><img file="EP1540844B1_D0052.tif" /></maths>

Another example includes of the evolution of a slowly varying envelope, E(x, t), of an optical pulse in a single mode communication fiber. The Schrodinger partial differential equation [3], <maths id="math0053"><math display="block"><mrow><mrow><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">E</mi></mrow><mrow><mo>∂</mo><mi mathvariant="normal">x</mi></mrow></mfrac></mrow><mo>+</mo><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">1</mn></msub><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">E</mi></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac><mo>−</mo><mfrac><mi mathvariant="normal">j</mi><mn mathvariant="normal">2</mn></mfrac><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">2</mn></msub><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mn mathvariant="normal">2</mn></msup><mi mathvariant="normal">E</mi></mrow><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow></mfrac></mrow><mo>−</mo><mfrac><mn mathvariant="normal">1</mn><mn mathvariant="normal">6</mn></mfrac><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">3</mn></msub><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mn mathvariant="normal">3</mn></msup><mi mathvariant="normal">E</mi></mrow><mrow><msup><mrow><mo>∂</mo><mi mathvariant="normal">t</mi></mrow><mn mathvariant="normal">3</mn></msup></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></mfrac><mi mathvariant="normal">E</mi><mo>,</mo></mrow></math><img file="EP1540844B1_D0053.tif" /></maths> describes the evolution of the shape of the propagating pulse envelope undergoing chromatic dispersion in fiber. Chromatic dispersion occurs because the mode-propagation constant, β(ω), is a nonlinear function of the angular frequency ω, where <maths id="math0054"><math display="block"><mrow><msub><mi mathvariant="normal">β</mi><mi mathvariant="normal">n</mi></msub><mo>=</mo><msub><mfenced open="[" close="]"><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mi mathvariant="normal">n</mi></msup><mi mathvariant="normal">β</mi></mrow><mrow><mo>∂</mo><msup><mi mathvariant="normal">ω</mi><mi mathvariant="normal">n</mi></msup></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant="normal">ω</mi><mo>=</mo><msub><mi mathvariant="normal">ω</mi><mi mathvariant="normal">o</mi></msub></mrow></msub></mrow></math><img file="EP1540844B1_D0054.tif" /></maths> and ωo is the frequency of the light being modulated in the fiber.<maths id="math0055"><math display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msub><mi mathvariant="normal">β</mi><mi mathvariant="normal">o</mi></msub></mrow></msub></mrow><mprescripts /><none /><mrow><msub><mi mathvariant="normal">ω</mi><mi mathvariant="normal">o</mi></msub></mrow></mmultiscripts></mrow></math><img file="EP1540844B1_D0055.tif" /></maths> is the phase velocity of the pulse. <maths id="math0056"><math display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">1</mn></msub></mrow></msub></mrow><mprescripts /><none /><mn mathvariant="normal">1</mn></mmultiscripts></mrow></math><img file="EP1540844B1_D0056.tif" /></maths> is the group velocity, β2 is the group velocity dispersion (GVD) parameter which causes the pulse to broaden as it propagates in the fiber. β3 is the third-order dispersion (TOD) parameter [4]

The higher order derivatives of β with respect to ω are assumed negligible. However, they may be added in these analyses by anyone familiar with the art. The fiber attenuation is represented by the parameter γ.

The SD solution of the linear Schrodinger equation with loss (γ ≠ 0) is: <maths id="math0057"><math display="block"><mrow><mi mathvariant="normal">E</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">De</mi><mfenced open="[" close="]"><mfrac><mi mathvariant="normal">t</mi><mrow><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mrow></mfrac><mo>−</mo><mfenced><mfrac><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">1</mn></msub></mrow><mrow><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">3</mn></msub></mrow><mrow><mn mathvariant="normal">6</mn><msubsup><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi><mn mathvariant="normal">3</mn></msubsup></mrow></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></mfrac></mfenced><mo>⋅</mo><mi mathvariant="normal">x</mi></mfenced></msup><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">j</mi><mfenced><mfrac><mrow><msub><mi mathvariant="normal">β</mi><mn>2</mn></msub><mo>⋅</mo><mi mathvariant="normal">x</mi></mrow><mrow><mn mathvariant="normal">2</mn><msubsup><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi><mn mathvariant="normal">2</mn></msubsup></mrow></mfrac></mfenced></mrow></msup><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0057.tif" /></maths>

The boundary condition at the input to the fiber at x = 0 is <maths id="math0058"><math display="inline"><mrow><mi mathvariant="normal">E</mi><mfenced separators=","><mn>0</mn><mi mathvariant="normal">t</mi></mfenced><mo>=</mo><msup><mi>De</mi><mrow><mfrac><mi mathvariant="normal">t</mi><mi>To</mi></mfrac></mrow></msup><mo>,</mo></mrow></math><img file="EP1540844B1_D0058.tif" /></maths> (see <figref idref="f0001">FIG. 1A</figref> with <maths id="math0059"><math display="inline"><mrow><mi mathvariant="normal">α</mi><mo>=</mo><mfrac><mn mathvariant="normal">1</mn><mrow><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mrow></mfrac></mrow></math><img file="EP1540844B1_D0059.tif" /></maths> ). The receiver detector in a fiber optic communication network responds to the square of the magnitude |E(x, t)|2 of the propagating pulse envelope. In this case <maths id="math0060"><math display="block"><mrow><msup><mfenced open="|" close="|"><mi mathvariant="normal">E</mi><mfenced separators=","><mi mathvariant="normal">x</mi><mi mathvariant="normal">t</mi></mfenced></mfenced><mn mathvariant="normal">2</mn></msup><mo>=</mo><msup><mi mathvariant="normal">D</mi><mn mathvariant="normal">2</mn></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mn mathvariant="normal">2</mn><mo>⋅</mo><mi mathvariant="normal">A</mi><mfenced><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mfenced><mfenced open="[" close="]"><mi mathvariant="normal">x</mi><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">sd</mi></msub><mfenced><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">t</mi></mfenced></mrow></msup><mo>,</mo></mrow></math><img file="EP1540844B1_D0060.tif" /></maths> where the velocity of propagation of a truncated SD leading edge of the envelope is <maths id="math0061"><math display="block"><mrow><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">sd</mi></msub><mfenced><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mfenced><mo>=</mo><mfrac><mn>1</mn><mfenced><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">1</mn></msub><mo>−</mo><mfrac><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">3</mn></msub></mrow><mrow><mn mathvariant="normal">6</mn><msubsup><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi><mn>2</mn></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub><mi mathvariant="normal">γ</mi></mrow><mn mathvariant="normal">2</mn></mfrac></mfenced></mfrac><mo>,</mo><mspace width="1em" /><mi>and A</mi><mfenced><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mfenced><mo>=</mo><mfenced><mfrac><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">1</mn></msub></mrow><mrow><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">3</mn></msub></mrow><mrow><mn mathvariant="normal">6</mn><msubsup><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi><mn mathvariant="normal">3</mn></msubsup></mrow></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></mfrac></mfenced><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0061.tif" /></maths>

Note that <maths id="math0062"><math display="inline"><mrow><msub><mi mathvariant="normal">v</mi><mi>sd</mi></msub><mfenced><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mfenced><mo>⋅</mo><mi mathvariant="normal">A</mi><mfenced><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mrow></mfrac><mn>.</mn></mrow></math><img file="EP1540844B1_D0062.tif" /></maths>

Transforming to a moving reference frame traveling at the group velocity <maths id="math0063"><math display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">1</mn></msub></mrow></msub></mrow><mprescripts /><none /><mn mathvariant="normal">1</mn></mmultiscripts><mo>,</mo></mrow></math><img file="EP1540844B1_D0063.tif" /></maths><maths id="math0064"><math display="block"><mrow><mi mathvariant="normal">tʹ</mi><mo>=</mo><mi mathvariant="normal">t</mi><mo>−</mo><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">1</mn></msub><mi mathvariant="normal">x</mi><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0064.tif" /></maths><maths id="math0065"><math display="block"><mrow><mi mathvariant="normal">xʹ</mi><mo>=</mo><mi mathvariant="normal">x</mi></mrow></math><img file="EP1540844B1_D0065.tif" /></maths>

The SD propagation speed in this moving reference frame is <maths id="math0066"><math display="block"><mrow><msub><mrow><mover><mi mathvariant="normal">v</mi><mrow><mo>‾</mo></mrow></mover></mrow><mi mathvariant="normal">sd</mi></msub><mfenced><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub></mfenced><msub><mrow><mo>|</mo></mrow><mrow><mi mathvariant="normal">in xʹ</mi><mo>,</mo><mi mathvariant="normal">tʹ frame</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi></msub><mfenced><mfrac><mi mathvariant="normal">γ</mi><mn mathvariant="normal">2</mn></mfrac><mo>−</mo><mfrac><mrow><msub><mi mathvariant="normal">β</mi><mn mathvariant="normal">3</mn></msub></mrow><mrow><mn mathvariant="normal">6</mn><msubsup><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi><mn>3</mn></msubsup></mrow></mfrac></mfenced></mrow></mfrac><mo>,</mo></mrow></math><img file="EP1540844B1_D0066.tif" /></maths>

The attenuation of the truncated SD portion of the leading edge of the pulse envelope is <maths id="math0067"><math display="block"><mrow><mfrac><mrow><msup><mfenced open="|" close="|"><mi>E</mi><mfenced><mo>ℓ</mo><mo>,</mo><msub><mi>t</mi><mi>j</mi></msub></mfenced></mfenced><mn>2</mn></msup></mrow><mrow><msup><mfenced open="|" close="|"><mi>E</mi><mfenced separators=","><mn>0</mn><msub><mi>t</mi><mi>j</mi></msub></mfenced></mfenced><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mfenced><mfrac><mi>γ</mi><mn>2</mn></mfrac><mo>−</mo><mfrac><mrow><msub><mi>β</mi><mn>3</mn></msub></mrow><mrow><mn>6</mn><msubsup><mi>T</mi><mi>o</mi><mn>3</mn></msubsup></mrow></mfrac></mfenced><mo>ℓ</mo></mrow></msup><mn>.</mn></mrow></math><img file="EP1540844B1_D0067.tif" /></maths>

This result implies that if To can be made small enough, this attenuation may be reduced. Higher order terms of β(ω), which were not included is this particular analysis, may become more significant as To becomes smaller.

EXAMPLES

Specific embodiments of the invention will now be further described by the following, nonlimiting examples which will serve to illustrate in some detail various features. The following examples are included to facilitate an understanding of ways in which the invention may be practiced. It should be appreciated that the examples which follow represent embodiments discovered to function well in the practice of the invention, and thus can be considered to constitute preferred modes for the practice of the invention. However, it should be appreciated that many changes can be made in the exemplary embodiments which are disclosed while still obtaining like or similar result without departing from the spirit and scope of the invention. Accordingly, the examples should not be construed as limiting the scope of the invention.

Example 1

Method for determining SD waveform parameters, ASD and vSD

This example teaches how SD test pulses may be used to obtain empirical estimates of the transmission line parameters, ASD and vSD, that describe the propagation of SD waveforms in lossy, dispersive lines, including those with frequency dependent parameters. The numerical values of ASD and vSD as a function of α can be determined empirically.

Referring to <figref idref="f0002">FIG. 2</figref>, an experimental set up that may be used to determine these parameters for a copper, twisted wire pair transmission line is depicted. The test length of the transmission line, d, and the value of α for the applied SD input signal are measured. An additional length of transmission line that is identical to the line under test is attached as shown and terminated with an open circuit. In this example the cables are two 24 AWG, individually shielded, twisted wire pairs in a 1002 ft coil of T1 cable.

The values of D, α and the duration of the SD signal were chosen to prevent the occurrence of reflections at the measurement point, d, during the initial time of flight of the SD signal propagating in the test line. The additional length of transmission line also ensures that all reflections occurring in the test line are delayed until well after the propagating SD waveform measurement at d is complete.

The SD signal time of flight can be directly measured by timing the propagating SD waveform crossing of constant voltage thresholds (<figref idref="f0002">FIG. 3</figref>). The value of the SD signal propagation velocity in the line, vSD , can be calculated using the relationship, TOF = (tf - to) = vsdd. The times of flight for these waveforms shown again in <figref idref="f0003">FIG. 4A</figref> at thresholds from 1.2 volts to 4.8 volts are plotted in <figref idref="f0003">FIG. 4B</figref>. The average TOF for the threshold overlap region shown in FIG. 4.3b is 1,716 ns. This TOF results in <maths id="math0068"><math display="block"><mrow><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub><mo>=</mo><mfrac><mi mathvariant="normal">d</mi><mrow><msub><mi mathvariant="normal">TOF</mi><mi mathvariant="normal">AVG</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1002</mn><mi mathvariant="normal">ft</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mn mathvariant="normal">716</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mrow><mo>−</mo><mn mathvariant="normal">9</mn></mrow></msup><mi mathvariant="normal">sec</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">5.839</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">8</mn></msup><mfrac><mi mathvariant="normal">ft</mi><mi mathvariant="normal">sec</mi></mfrac><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0068.tif" /></maths>

The SD attenuation coefficient, ASD, can be calculated from α and vsd using the relationship ASD · vSD = α. The calculated ASD obtained from the waveforms shown in FIG. <maths id="math0069"><math display="block"><mrow><msub><mi mathvariant="normal">A</mi><mi mathvariant="normal">SD</mi></msub><mo>=</mo><mfrac><mi mathvariant="normal">α</mi><mrow><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">SD</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">3.0</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">6</mn></msup><mspace width="1em" /><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mi mathvariant="normal">sec</mi></msub></mrow><mprescripts /><none /><mn mathvariant="normal">1</mn></mmultiscripts></mrow><mrow><mn mathvariant="normal">5.839</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">8</mn></msup><mspace width="1em" /><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mi mathvariant="normal">sec</mi></msub></mrow><mprescripts /><none /><mi mathvariant="normal">ft</mi></mmultiscripts></mrow></mfrac><mo>=</mo><mn mathvariant="normal">5.138</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mrow><mo>−</mo><mn mathvariant="normal">3</mn></mrow></msup><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mi mathvariant="normal">ft</mi></msub></mrow><mprescripts /><none /><mn mathvariant="normal">1</mn></mmultiscripts><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0069.tif" /></maths>

Once this velocity is known for a calibrated length of the cable, this type of threshold crossing TOF measurement can be used to determine the unknown length of another sample of the same transmission line.

Referring to <figref idref="f0004">FIG. 5A</figref>, the measured input voltage of the two twisted wire pairs in a T1 line connected in series is depicted (see <figref idref="f0002">FIG. 2</figref>) and one-half of the voltage measured at the end of the open terminated second twisted wire pair in the T1 line (Rt = ∞ in <figref idref="f0002">FIG. 2</figref>). The average TOF for the threshold overlap region shown in <figref idref="f0004">FIG. 5B</figref> is 3,418 ns. This time of flight together with the previously calibrated vSD, results in an estimated distance of 1,996 ft, or 99.6% of the actual length (2004 ft).

<figref idref="f0004">FIG. 5A</figref> also indicates a potential difficulty in longer lines caused by the attenuation of the truncated SD pulse. The region of SD threshold overlap grows smaller as the pulse propagates, and is attenuated over longer distances. The accuracy of this measurement can be improved, and the difficulty caused by the reduced region of threshold overlap can be overcome for very long transmission lines using the TOF measurement technique discussed in example 4.

The high accuracy of this method results from the use of an average of TOF measurements taken over a range of signal threshold amplitudes. It is feasible to use the average of these values to improve the accuracy of the measurement of TOF since all points on the leading SD edge of the waveform travel at the same speed. This is in contrast with the current time domain reflectometry art in which a measurement of the time of emergence of the dispersing test pulse is attempted. This is equivalent to attempting to accurately measure the threshold crossing time of the pulse at a zero threshold level when the slope of the pulse is also nearly zero.

Example 2

Extracting SD Parameters From Empirical Transfer Function

The method described in example 1 can be limiting in that it should be experimentally repeated for each exponential coefficient α. This may require a lengthy laboratory calibration time for each cable type. Simulation can be included in the calibration process to significantly reduce the experimental measurements needed to determine the values of ASD and vSD. The use of simulation in this process requires a transfer function that describes the response of the specific type of cable. The α's analyzed for the T1 cable range from 1x105 sec-1 to 1x107 sec-1. This range of α's corresponds to the frequency band of 16 kHz to 1.6 MHz, so the transfer function must be accurate for this range. The transfer function of the cable may available from the manufacturer or can be determined empirically by applying a known pulse and measuring the applied cable input waveform and the response waveform at a known (calibration) distance along the cable. The transfer function is then calculated from the ratio of the fast Fourier transforms (FFT) of the input-output pulses: <maths id="math0070"><math display="inline"><mrow><msub><mi mathvariant="normal">H</mi><mi>est</mi></msub><mfenced><mi>jω</mi></mfenced><mo>=</mo><mfrac><mrow><mi>FFT</mi><mfenced><msub><mi mathvariant="normal">V</mi><mrow><mi>Measured at</mi><mspace width="1em" /><mo>ℓ</mo></mrow></msub></mfenced></mrow><mrow><mi>FFT</mi><mfenced><msub><mi mathvariant="normal">V</mi><mi>Applied</mi></msub></mfenced></mrow></mfrac><mn>.</mn></mrow></math><img file="EP1540844B1_D0070.tif" /></maths> Once the transfer function is determined, the propagation of SD waveforms with a series of different values for α can be simulated and the times of flight and SD parameters, ASD and vSD, determined.

This method can be demonstrated for the T1 cable used in example 1. The end of the total 2004 length has been terminated with a 100-Ohm resistor (Rt in FIG. 4.1) to minimize reflections at the 1002 ft measurement point. <figref idref="f0005">FIG. 6A</figref> shows a measured input pulse applied to the T1. <figref idref="f0005">FIG. 6B</figref> shows the output pulse measured at 1002 ft. <figref idref="f0006">FIGS. 7A-B</figref> show the Power Spectral Density of the measured pulses. -The range of α's is indicated to show that there is sufficient energy in the frequency band of interest to empirically estimate an accurate transfer function. The magnitude and the phase of the transfer function determined from the measured signals are shown in <figref idref="f0007">FIG. 8</figref>.

Once the transfer function is known, a waveform with any α in the indicated range of interest can be simulated and the value of ASD and vSD determined. The values of ASD and vSD of the measured SD waveforms of section three provide a test of the transfer function for an α of 3x106 sec-1. <figref idref="f0008">FIG. 9</figref> shows the time of flight plots determined using the input signal measured in section two, and the 1002 ft waveform predicted by simulation using the empirical transfer function. The simulation TOF result (<figref idref="f0008">FIG. 9</figref>) of 1,714 nsec is almost identical to the experimentally measured (<figref idref="f0004">FIG. 5B</figref>) TOF of 1,716 nsec. <figref idref="f0008">FIG. 10</figref> shows the difference between the measured waveform at 1002 ft and the simulated waveform at 1002 ft (<figref idref="f0008">FIG. 9</figref>, Top). During the SD propagation region the difference is on the order of ± 20 mV.

This process is performed for the range of α's by simulating the initial SD waveforms and using the transfer function to predict the waveforms at 1002 ft. In all cases, the simulated initial waveforms were closed with a 1 µsec ramp to reduce their high frequency components. <figref idref="f0009">FIGS. 11-12</figref> show the simulated input and 1002 ft curves for the endpoint α's of 1x105 sec-1 and 1x107 sec-1 along with their respective constant threshold times of flight curves. The values of vSD and ASD from simulation are presented in <figref idref="f0010">FIGS 13A-B</figref>, illustrated in Table I for the range of α's, as set forth below (* values in parenthesis obtained using methods of example 1, with measured input and output SD signals along actual T1 Line): <tables id="tabl0001" num="0001"><table frame="all"><title>Table I</title><tgroup cols="4"><colspec colnum="1" colname="col1" colwidth="24mm" /><colspec colnum="2" colname="col2" colwidth="45mm" /><colspec colnum="3" colname="col3" colwidth="42mm" /><colspec colnum="4" colname="col4" colwidth="42mm" /><thead><row><entry namest="col1" nameend="col4" align="center" valign="top">vSD and ASD Determined By Simulation with Empirical Transfer Function</entry></row><row><entry align="center" valign="top">Alpha (sec-1)</entry><entry align="center" valign="top">1002 Simulation TOF (µsec)</entry><entry align="center" valign="top">Simulation vSD (x108 ft/sec)</entry><entry align="center" valign="top">Simulation ASD (x10-31/ft)</entry></row></thead><tbody><row><entry align="center">1x105</entry><entry align="center">4.487</entry><entry align="center">2.233</entry><entry align="center">0.448</entry></row><row><entry align="center">2x105</entry><entry align="center">3.068</entry><entry align="center">3.266</entry><entry align="center">0.612</entry></row><row><entry align="center">3x105</entry><entry align="center">2.608</entry><entry align="center">3.842</entry><entry align="center">0.781</entry></row><row><entry align="center">4x105</entry><entry align="center">2.382</entry><entry align="center">4.207</entry><entry align="center">0.951</entry></row><row><entry align="center">5x105</entry><entry align="center">2.271</entry><entry align="center">4.412</entry><entry align="center">1.133</entry></row><row><entry align="center">6x105</entry><entry align="center">2.172</entry><entry align="center">4.613</entry><entry align="center">1.301</entry></row><row><entry align="center">7x105</entry><entry align="center">2.103</entry><entry align="center">4.765</entry><entry align="center">1.469</entry></row><row><entry align="center">8x105</entry><entry align="center">2.040</entry><entry align="center">4.912</entry><entry align="center">1.629</entry></row><row><entry align="center">9x105</entry><entry align="center">1.999</entry><entry align="center">5.013</entry><entry align="center">1.796</entry></row><row><entry align="center">1x106</entry><entry align="center">1.973</entry><entry align="center">5.079</entry><entry align="center">1.969</entry></row><row><entry align="center">2x106</entry><entry align="center">1.792</entry><entry align="center">5.592</entry><entry align="center">3.577</entry></row><row><entry align="center">3x106</entry><entry align="center">1.714 (1.716*)</entry><entry align="center">5.849 (5.839*)</entry><entry align="center">5.129 (5.138*)</entry></row><row><entry align="center">4x106</entry><entry align="center">1.666</entry><entry align="center">6.014</entry><entry align="center">6.65.1</entry></row><row><entry align="center">5x106</entry><entry align="center">1.634</entry><entry align="center">6.132</entry><entry align="center">8.154</entry></row><row><entry align="center">6x106</entry><entry align="center">1.610</entry><entry align="center">6.224</entry><entry align="center">9.641</entry></row><row><entry align="center">7x106</entry><entry align="center">1.592</entry><entry align="center">6.294</entry><entry align="center">11.122</entry></row><row><entry align="center">8x106</entry><entry align="center">1.577</entry><entry align="center">6.354</entry><entry align="center">12.591</entry></row><row><entry align="center">9x106</entry><entry align="center">1.565</entry><entry align="center">6.403</entry><entry align="center">14.057</entry></row><row><entry align="center">1x107</entry><entry align="center">1.555</entry><entry align="center">6.444</entry><entry align="center">15.519</entry></row></tbody></tgroup></table></tables>

Example 3

Measurement of SD Transmission Line Impedance

The SD line impedance, ZSD (α), is a real number which may be determined for the α's of interest. If the Laplace transform of the frequency dependent transmission line parameters <o ostyle="single">R</o>(s), <o ostyle="single">L</o>(s), C(s), and <o ostyle="single">G</o>(s), are known, then ZSD(α) may be computed from, <maths id="math0071"><math display="inline"><mrow><msub><mi mathvariant="normal">Z</mi><mi mathvariant="normal">SD</mi></msub><mo>=</mo><msqrt><mrow><mfrac><mrow><mi mathvariant="normal">α</mi><mover><mi mathvariant="normal">L</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>+</mo><mover><mi mathvariant="normal">R</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced></mrow><mrow><mi mathvariant="normal">α</mi><mover><mi mathvariant="normal">C</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced><mo>+</mo><mover><mi mathvariant="normal">G</mi><mrow><mo>‾</mo></mrow></mover><mfenced><mi mathvariant="normal">α</mi></mfenced></mrow></mfrac></mrow></msqrt><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0071.tif" /></maths>

The SD line impedance, ZSD (α); may also be experimentally determined for a test transmission line by measuring the SD portion of the voltage waveform across various known, termination impedances and computing the value ZSD(α) from the measurements. The measured SD waveform at a termination, VSD(Rt), consists of the sum of the incident SD waveform, <maths id="math0072"><math display="inline"><mrow><msubsup><mi mathvariant="normal">V</mi><mi>SD</mi><mrow><mo>+</mo></mrow></msubsup><mo>,</mo></mrow></math><img file="EP1540844B1_D0072.tif" /></maths> and a reflected SD wave. The reflected wave is the product of the SD reflection coefficient, r, and the incident wave. The measured SD signal at the termination will be <maths id="math0073"><math display="inline"><mrow><msub><mi mathvariant="normal">V</mi><mi>SD</mi></msub><mfenced><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub></mfenced><mo>=</mo><mfenced><mn>1</mn><mo>+</mo><msub><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">t</mi></msub></mfenced><mo>⋅</mo><msubsup><mi mathvariant="normal">V</mi><mi>SD</mi><mrow><mo>+</mo></mrow></msubsup><mo>=</mo><mfrac><mrow><mn>2</mn><mo>⋅</mo><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub></mrow><mrow><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub><mo>+</mo><msub><mi mathvariant="normal">Z</mi><mi>SD</mi></msub></mrow></mfrac><mo>⋅</mo><msubsup><mi mathvariant="normal">V</mi><mi>SD</mi><mrow><mo>+</mo></mrow></msubsup><mn>.</mn></mrow></math><img file="EP1540844B1_D0073.tif" /></maths>Note that all lumped and distributed impedance values are real for SD signals.

The experimental setup used to determine ZSD from line measurements is shown in <figref idref="f0011">FIG. 14</figref>. The measured input and termination waveforms of the voltages for a 1002 ft of T1 line, are shown in <figref idref="f0011">FIG. 15A</figref>. These measurements were made with an α of 1.5x 106 1/sec and termination resistances of 49.5 ohms, 98.6 ohms and an open circuit. The magnitude of the waveform with no reflection, (the incident waveform <maths id="math0074"><math display="inline"><mrow><msubsup><mi mathvariant="normal">V</mi><mi>SD</mi><mrow><mo>+</mo></mrow></msubsup></mrow></math><img file="EP1540844B1_D0074.tif" /></maths>), is determined by dividing the signal measured with an open termination, (Rt = ∞, Γ∞ = 1), in half. The ratio of the waveforms <maths id="math0075"><math display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msubsup><mi mathvariant="normal">V</mi><mi>SD</mi><mrow><mo>+</mo></mrow></msubsup></mrow></msub></mrow><mprescripts /><none /><mrow><msub><mi mathvariant="normal">V</mi><mi>SD</mi></msub><mfenced><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub></mfenced></mrow></mmultiscripts><mo>,</mo></mrow></math><img file="EP1540844B1_D0075.tif" /></maths> with known terminations and the incident waveform gives a direct measurement of the reflection coefficient, Γt, for each termination resistance, <maths id="math0076"><math display="inline"><mrow><msub><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">t</mi></msub><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">SD</mi></msub><mfenced><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub></mfenced></mrow><mrow><msubsup><mi mathvariant="normal">V</mi><mi mathvariant="normal">SD</mi><mrow><mo>+</mo></mrow></msubsup></mrow></mfrac><mo>−</mo><mn mathvariant="normal">1.</mn></mrow></math><img file="EP1540844B1_D0076.tif" /></maths><figref idref="f0012">FIG. 15B</figref> shows the measured ratios for this cable and α. Using the relationship for the transmission coefficient, <maths id="math0077"><math display="inline"><mrow><mn>1</mn><mo>+</mo><msub><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">t</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mo>⋅</mo><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub></mrow><mrow><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub><mo>+</mo><msub><mi mathvariant="normal">Z</mi><mi>SD</mi></msub></mrow></mfrac><mo>,</mo></mrow></math><img file="EP1540844B1_D0077.tif" /></maths> gives <maths id="math0078"><math display="inline"><mrow><msub><mi mathvariant="normal">Z</mi><mi>SD</mi></msub><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">t</mi></msub></mrow><mrow><mn>1</mn><mo>+</mo><msub><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">t</mi></msub></mrow></mfrac><mfenced><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">Γ</mi><mi mathvariant="normal">t</mi></msub></mfenced><mn>.</mn></mrow></math><img file="EP1540844B1_D0078.tif" /></maths>

Using the two finite values for Rt, 98.6 Ω and 49.5 Ω, yields two estimates for Zsd: <maths id="math0079"><math display="block"><mrow><msub><mi mathvariant="normal">Z</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi><mo>=</mo><mn mathvariant="normal">1.5</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">6</mn></msup><mfrac><mn mathvariant="normal">1</mn><mi mathvariant="normal">sec</mi></mfrac></mfenced><mo>=</mo><mfrac><mn mathvariant="normal">98.6</mn><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfenced><mn mathvariant="normal">0.919</mn><mo>−</mo><mn mathvariant="normal">1</mn></mfenced></mrow></mfrac><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><mfenced><mn mathvariant="normal">0.919</mn><mo>−</mo><mn mathvariant="normal">1</mn></mfenced></mfenced><mo>=</mo><mn mathvariant="normal">116</mn><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></math><img file="EP1540844B1_D0079.tif" /></maths> or <maths id="math0080"><math display="block"><mrow><msub><mi mathvariant="normal">Z</mi><mi mathvariant="normal">SD</mi></msub><mfenced><mi mathvariant="normal">α</mi><mo>=</mo><mn mathvariant="normal">1.5</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">6</mn></msup><mfrac><mn mathvariant="normal">1</mn><mi mathvariant="normal">sec</mi></mfrac></mfenced><mo>=</mo><mfrac><mn>49.5</mn><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mfenced><mn mathvariant="normal">0.598</mn><mo>−</mo><mn mathvariant="normal">1</mn></mfenced></mrow></mfrac><mfenced><mn mathvariant="normal">1</mn><mo>−</mo><mfenced><mn mathvariant="normal">0.598</mn><mo>−</mo><mn mathvariant="normal">1</mn></mfenced></mfenced><mo>=</mo><mn mathvariant="normal">116</mn><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></math><img file="EP1540844B1_D0080.tif" /></maths> for this T1 cable.

This process was repeated for α's from 5x105 sec-1 to 10x106 sec-1 and the results are shown in <figref idref="f0013">FIG 16</figref>.

Example 4

Precision measurement of long transmission line length or distance to an impedance change using a reflected wave

This example describes an alternate precision TOF measurement method of SD test pulses for higher loss, longer lines whose attenuation of the test pulses is so extensive that their amplitudes become too small as they travel down the line for a common threshold crossing measurement to be feasible.

The time of flight measured at a constant voltage threshold becomes more difficult to estimate as the waveform is attenuated. This can be seen by examining the voltage traces along an approximately 6-kft, 24 AWG, fifty twisted wire pair, telephone cable (See <figref idref="f0014">FIG. 17</figref>). Three pairs of the fifty have been connected in series to generate an approximately 18-kft, cable with three equal length sections. The applied SD input waveform is Deαt, with a D of 0.19 volts and an α of <maths id="math0081"><math display="inline"><mrow><mn>3.5</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mspace width="1em" /><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mi>sec</mi></msub></mrow><mprescripts /><none /><mn>1</mn></mmultiscripts><mn>.</mn></mrow></math><img file="EP1540844B1_D0081.tif" /></maths> This α is approximately one tenth of the α used for the 2-kft T1 cable. The smaller α is required here to reduce the attenuation of the truncated SD leading edge of the propagating pulse over these longer distances as discussed in section two. This allows signal detection over longer distances. The value of ASD and vSD for this cable and α were determined using the procedure of example 1. These measured values are based on assuming a 6000 ft calibration length for each twisted pair length inside the telephone cable. The results are vSD= 4.348x108 ft/sec and ASD=8.279x10-4 1/ft.

Even though the signal was detectable at much greater distances than 6 kft, the constant threshold time of flight could not be measured. <figref idref="f0014">FIG. 17</figref> shows the voltage traces measured at the input, ∼6 kft, ∼12 kft, and ∼18 kft distances, and it also shows the preservation of shape of the SD region of the highly attenuated propagating waveform. The ∼18 kft trace is corrected for the reflection from the open termination by dividing the measured waveform in half. The last two traces show that even with this smaller α, beyond 12,000 ft there is little or no SD signal above 0.19 volts. This is the value of the input SD waveform parameter D, thus there is little or no SD voltage threshold common to the input SD waveform and the SD waveforms at these longer distances.

The first step in precisely measuring highly attenuated SD pulse time of flight is to detect the SD region of the propagating pulse, V(t), measured and recorded at the distance ℓ along the cable. This is done by computing the <maths id="math0082"><math display="inline"><mrow><mfrac><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mi>dt</mi></msub></mrow><mprescripts /><none /><mrow><mi>dV</mi><mfenced><mi mathvariant="normal">t</mi></mfenced></mrow></mmultiscripts></mrow><mrow><mi mathvariant="normal">V</mi><mfenced><mi mathvariant="normal">t</mi></mfenced></mrow></mfrac></mrow></math><img file="EP1540844B1_D0082.tif" /></maths> of the measured pulse waveform. In the. SD cable. This is done by computing the ratio <maths id="math0083"><math display="inline"><mrow><mfrac><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mi>dt</mi></msub></mrow><mprescripts /><none /><mrow><mi>dV</mi><mfenced><mi mathvariant="normal">t</mi></mfenced></mrow></mmultiscripts></mrow><mrow><mi mathvariant="normal">V</mi><mfenced><mi mathvariant="normal">t</mi></mfenced></mrow></mfrac></mrow></math><img file="EP1540844B1_D0083.tif" /></maths> of the measured pulse waveform. In the SD region of the pulse, this ratio is α. The end of the SD region is then found by detecting the time that this ratio diverges from α. The ratio <maths id="math0084"><math display="inline"><mrow><mfrac><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msup><mi>dt</mi><mn>2</mn></msup></mrow></msub></mrow><mprescripts /><none /><mrow><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup><mi mathvariant="normal">V</mi><mfenced><mi mathvariant="normal">t</mi></mfenced></mrow></mmultiscripts></mrow><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mi>dt</mi></msub></mrow><mprescripts /><none /><mrow><mi>dV</mi><mfenced><mi mathvariant="normal">t</mi></mfenced></mrow></mmultiscripts></mrow></mfrac></mrow></math><img file="EP1540844B1_D0084.tif" /></maths> can also be used to detect the end of the SD region. In the SD region, this ratio is also α. The later ratio of the second and first derivative responds more quickly, but is also more susceptible to noise than the ratio of the first derivative and the signal. These ratios as a function of time are plotted in <figref idref="f0015">FIG. 18</figref> for the SD waveform measured at 12-kft. In general, the ratio of <maths id="math0085"><math display="inline"><mrow><mfrac><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mi>n</mi></msup><mi>V</mi><mfenced separators=","><mi>x</mi><mi>t</mi></mfenced></mrow><mrow><msup><mrow><mo>∂</mo></mrow><mi>n</mi></msup><mi>t</mi></mrow></mfrac></mrow><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>V</mi><mfenced separators=","><mi>x</mi><mi>t</mi></mfenced></mrow><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>t</mi></mrow></mfrac></mrow></mfrac></mrow></math><img file="EP1540844B1_D0085.tif" /></maths> for any positive n will have this property. Additionally, the ratio of the signal and its integral can be used to locate the SD region. In the SD region, we have the relationship: <maths id="math0086"><math display="inline"><mrow><mfrac><mrow><msup><mi mathvariant="normal">De</mi><mi mathvariant="normal">αt</mi></msup></mrow><mrow><mstyle displaystyle="true"><mrow><msubsup><mrow><mo>∫</mo></mrow><mn mathvariant="normal">0</mn><mi mathvariant="normal">t</mi></msubsup></mrow></mstyle><mrow><msup><mi mathvariant="normal">De</mi><mi mathvariant="normal">ατ</mi></msup><mi mathvariant="normal">dτ</mi></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">De</mi><mi mathvariant="normal">αt</mi></msup></mrow><mrow><mfrac><mn mathvariant="normal">1</mn><mi mathvariant="normal">α</mi></mfrac><mfenced><msup><mi mathvariant="normal">De</mi><mi mathvariant="normal">αt</mi></msup><mo>−</mo><mi mathvariant="normal">D</mi></mfenced></mrow></mfrac><mo>=</mo><mi mathvariant="normal">α</mi><mfrac><mn mathvariant="normal">1</mn><mrow><mn mathvariant="normal">1</mn><mo>−</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mi mathvariant="normal">αt</mi></mrow></msup></mrow></mfrac><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0086.tif" /></maths> Thus, the ratio will converge on α as time progresses. This ratio diverges at the end of the SD region. In a lossy transmission medium with constant transmission line parameters, the end of the estimated SD region can provide a good marker to determine the speed of light in the medium. The truncated SD leading edge of the pulse does not disperse, even with frequency dependent parameters. The high frequency components of the closing pulse, however, do undergo dispersion. These fast, high frequency components tend to erode the end of the truncated SD region as the pulse propagates. At long distances, these high frequency components are also more highly attenuated so the amount of SD region erosion is not proportional to the distance traveled. In this case, the detected end of the SD region serves as an initial estimate of the pulse time of flight or the distance the wave has traveled and serves to define the region examined for more precise time of flight measurements in the next step.

The attenuation of the truncated SD signal propagating a distance d was shown to be <maths id="math0087"><math display="inline"><mrow><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi mathvariant="normal">A</mi><mi>sd</mi></msub><mo>⋅</mo><mfenced><mn>1</mn><mo>−</mo><msub><mi mathvariant="normal">v</mi><mi>sd</mi></msub><mo>⋅</mo><msqrt><mi>LC</mi></msqrt></mfenced><mi mathvariant="normal">d</mi></mrow></msup><mn>.</mn></mrow></math><img file="EP1540844B1_D0087.tif" /></maths> in the application of time domain reflectometry (TDR) to these lines, the parameter <maths id="math0088"><math display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mo>/</mo></mrow><mrow><msqrt><mi mathvariant="italic">LC</mi></msqrt></mrow></msub></mrow><mprescripts /><none /><mn>1</mn></mmultiscripts></mrow></math><img file="EP1540844B1_D0088.tif" /></maths> is commonly referred to as Vp and is given as a fraction of the speed of light in a vacuum. The standard value of Vp used in TDR for this air core poly 24 AWG cable is 0.67. However, the value of the ratio used in this analysis is 0.59. The reason for this can be seen in <figref idref="f0016">FIG. 19</figref>. The value of 0.67 is appropriate for the high frequencies that are found in a standard TDR pulse used to test these telephone lines, but the α selected here for low truncated SD signal attenuation in these long lines is equivalent to a low frequency and propagates at a slower speed.

Once the initial TOF estimate of end of the SD region in the leading edge of the propagating pulse is obtained at the distance ℓ, the input waveform is attenuated and time shifted assuming the end of the SD region of the traveling wave corresponds to the point of truncation of the initial SD waveform, (<figref idref="f0017">FIG. 20</figref>, top). The input waveform is then incrementally translated in time and attenuated based on each new estimated distance traveled. At each distance estimate, this shifted and attenuated input waveform is subtracted from the measured waveform. If there is no dc voltage offset and no noise, this difference will be zero in the SD region when the estimated distance is equal to the actual distance. Because there is always noise and there is frequently a varying differential, dc offset at different points along the cable, the variance of this difference is evaluated at each estimated distance. This correlation has a minimum when the input waveform has been translated to the actual distance traveled, (<figref idref="f0017">FIG. 20</figref>, middle). If there is a dc differential offset between the lines, this minimum will be non-zero. At this estimated distance, the erosion of the end of the truncated SD region due to the dispersion from the closing pulse is obvious. If the waveform is translated too far, this deviation error measurement increases, (<figref idref="f0017">FIG. 20</figref>, bottom). <figref idref="f0018">FIG. 21</figref> shows the calculated standard deviation of the difference of the shifted and attenuated input waveform and the SD waveform measured at the splice between the second and third twisted wire pairs as a function of estimated distance. The minimum occurs at an estimated distance of 12,216 ft. <figref idref="f0019">FIG. 22A</figref> shows the difference of the two waveforms and the variance of the difference of the waveforms at the minimum. <figref idref="f0019">FIG. 22A</figref> indicates that there is a five to ten millivolt differential dc offset between the two measuring points. <figref idref="f0019">FIG. 22B</figref> is presented to show how the variance has increased when the estimated distance is 38 ft longer than the distance associated with the minimum.

This process was repeated eight times giving average distances of 6,113 ft, 12,216 ft, and 18,327 ft for the best-fit shift locations of the attenuated input waveform to the measured waveforms. The standard deviations of the three lengths were 1 ft, 1ft, and 5 ft respectively. The fits associated with one of these data sets are shown in <figref idref="f0020">FIG. 23</figref>. The actual length of the cable was not exactly known, however, each pair in the cable is parallel and the only differences in length are small and due to different twist rates. This process yields the distance of two lengths within 10 ft of two times the distance of one length. The distance of three lengths is within 11 ft of three times the distance of one length. Using the ∼6 kft distance as a standard gives a discrepancy of -0.08 % for the ∼12 kft length and -0.06 % for the ∼18 kft length.

Example 5

Time Domain Reflectometer (TDR): Precision measurement of transmission line distance to impedance change using a reflected wave

A Time Domain Reflectometer (TDR) is a test instrument used to find faults in transmission lines and to empirically estimate transmission line lengths and other parameters characterizing the line such as inductance per unit length, capacitance per unit length, resistance per unit length and conductance per unit length. A fundamental measurement in TDR test technology is the time of flight (TOF) of a test pulse generated by the instrument and applied to the line. This time of flight may be measured by timing the passage of the pulse detected at two locations along the line, referred to as Time Domain Transmission measurements (TDT). Or by Time Domain Reflection measurements which estimate the launch time of a pulse at one position and the subsequent return time of the pulse back to the same location after the pulse has been reflected from a fault or other impedance change some distance along the line. These measured TOF values, along with a value of the propagation speed of the pulse, allows one to obtain the distance between measurement points or in the case of the reflected wave, the distance from the pulse launch point to the location of the impedance change causing the pulse to be reflected and returned to the launch point.

A fundamental limitation in TDR technology is the poor accuracy of these TOF measurements in lossy, dispersive transmission lines. The relatively high TDR accuracy of TOF values obtainable in short low loss, low dispersion transmissions lines is possible only because the propagating test pulses keep their shape and amplitude in tact over the distances they travel during TOF measurements. By contrast, in dispersive, lossy long transmission lines the test pulses used in the art change shape, change amplitude and speed as they travel. The TOF measurements used in the art under these circumstances focus on estimating the emergence time of the leading profile of the test pulses. This part of the signature of test pulses used in the art has characteristically a low signal level and low signal slope making an accurate pulse emergence time measurement difficult to obtain.

Several advantages can be obtained in TDR technology for lossy, dispersive transmission lines by using a test pulse that contains a truncated SD signal in its leading edge. This truncated SD leading edge travels at a constant speed along these transmission lines without changing shape. The speed of propagation of this SD edge is a function of the line parameters and the SD signal parameter alpha and is controllable by changing alpha. The truncated SD leading edge of the test pulse will be attenuated as it travels along a lossy line. However, this rate of attenuation is also a function of the line parameters and the SD signal parameter alpha and is controllable by changing alpha.

The same principles used for TDR can be applied to waveforms other than electromagnetic waves in a transmission line. The reflections of acoustic waves in SONAR or geophysical applications can be analyzed using the techniques discussed in this section to provide accurate time of flight and distance estimates as well as be used to characterize the transmission media.

The accuracy of the SD test waveform in a TDR application can be demonstrated by an example using the T1 cable, as discussed in examples 1 and 2. The two twisted wire pairs inside the T1 cable are connected in series to form a 2004 ft cable and the truncated SD signal is applied to the input. The voltage trace is measured at the splice of the two lines at 1002 ft. This can eliminate the need to correct for any line impedance mismatch at the point of measurement. The exponential coefficient α of the applied wave is 6.7x106 sec-1 The SD parameters, ASD and vSD, are obtained from the empirical transfer function analysis presented in example 2. Interpolating from the data of table I gives an ASD of 10.703x10-3 ft-1 and a vSD of 6.276x108 ft/sec for this α.

The first demonstration terminates the cable with an open circuit. The reflected coefficient of an open circuit is +1, which results in a positive reflection. <figref idref="f0021">FIG. 24</figref> shows the voltage waveform measured at the 1002 ft point showing the forward traveling applied wave and the wave reflected from an open termination at the end of the cable. <figref idref="f0021">FIG. 25</figref> shows the result of the measured reflected wave together with the applied wave time shifted and attenuated according to the procedure of example 4. The total distance traveled by the reflected wave is measured to be 2004 ft by this process, or exactly two times the 1002 ft length of this twisted wire pair section. <figref idref="f0022">FIGS. 26A-B</figref> show the constant threshold timing method result for TOF discussed in example 1. This method results in a time of flight of 3,203 nsec, which gives an estimated total distance traveled of 2010 ft. This results in an estimated twisted wire pair section length of 1005 ft, or 3 ft in error (+0.3%). This is an illustration of reduced accuracy using the constant threshold TOF measurements when the pulse travels far enough to experience significant attenuation. The accuracy of this method may decline more in cases where the test pulse experiences even more attenuation. This level of attenuation may not generally occur in digital circuits. The maximum attenuation of digital signals in interconnects is generally less than one-half of the driving signal. In such cases, the constant threshold TOF measurements with the SD wave should be preferably more accurate than current TDR methods using estimated time of emergence of the reflected pulses.

The next demonstration terminates the second twisted wire pair with a short circuit. The reflected coefficient at a short circuit is -1, which results in a negative reflection. <figref idref="f0023">FIG. 27</figref> shows the voltage waveform measured at the 1002 ft point showing the forward traveling applied wave and the wave reflected from a short circuit termination at the end of the second twisted wire pair. <figref idref="f0023">FIG. 28</figref> shows the result of the inverted reflected wave and the applied wave time shifted and attenuated according to the procedure of example 4. The total distance traveled is measured to be 2004 ft by this process, or exactly two times the 1002 ft twisted wire pair length. <figref idref="f0024">FIGS. 29A-B</figref> show the constant threshold timing method of estimating TOF discussed in example 1. This method results in a time of flight of 3,200 nsec, which gives an estimated total distance traveled of 2008 ft. This results in an estimated twisted wire pair section length of 1004 ft, or 2 ft in error (+0.2%).

The utility of the SD waveform to the TDR process can be increased by using an empirical transfer function obtained with the use of a non-SD pulse applied to the line in a way similar to the process discussed in example 2. As an example with this cable, an experimental voltage pulse is generated by an arbitrary function generator connected to a single unterminated twisted wire pair from the T1 cable. This waveform is shown in <figref idref="f0025">FIGS. 30-31</figref> (dotted). The same waveform can be applied by the signal generator to a variable resistor load substituted-for the 1002 ft twisted wire pair. The resistor is adjusted until the waveform approximates the shape and magnitude of the waveform applied by the arbitrary waveform generator to the twisted wire pair, see <figref idref="f0025">FIGS. 30-31</figref> (solid). The waveform match is not exact because the resistor does not have the exact same frequency response as that of the twisted wire pair. A load resistance equal to the characteristic impedance, Zo, of the twisted wire pair results in a waveform that is closely matched to the waveform applied to the cable and has similar frequency content. The difference between the signal measured with the resistive load and the signal measured with the transmission line consists of the response of the transmission line to the applied signal and contains any reflected waves from impedance mismatches on the transmission line. The transfer function is then calculated from the ratio of the fast Fourier transforms (FFT) of the difference of the two signals, and the signal of the resistor alone: <maths id="math0089"><math display="block"><mrow><msub><mi mathvariant="normal">H</mi><mi mathvariant="normal">est</mi></msub><mfenced><mi mathvariant="normal">jω</mi></mfenced><mo>=</mo><mfrac><mrow><mi mathvariant="normal">FFT</mi><mfenced><msub><mi mathvariant="normal">V</mi><mi mathvariant="normal">Measured with Cable</mi></msub><mo>−</mo><msub><mi mathvariant="normal">V</mi><mrow><msub><mi mathvariant="normal">Measured with Load Re sis tance Z</mi><mi mathvariant="normal">o</mi></msub></mrow></msub></mfenced></mrow><mrow><mi mathvariant="normal">FFT</mi><mfenced><msub><mi mathvariant="normal">V</mi><mrow><msub><mi mathvariant="normal">Measured with Load Re sis tan ce Z</mi><mi mathvariant="normal">o</mi></msub></mrow></msub></mfenced></mrow></mfrac><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0089.tif" /></maths> This function, shown in <figref idref="f0026">FIG. 32</figref>, can be used to evaluate the reflected transmission line response to a SD pulse.

As a test of this process, a SD pulse with an α equal to the α used in the previous two experiments was simulated. The transfer function determined for the open terminated T1 cable, (<figref idref="f0026">FIG. 32</figref>), was used to calculate the expected response, see <figref idref="f0027">FIG. 33</figref>. There is a small pulse at the time of the applied pulse due to the resistor and cable having different high frequency impedance values. The influence of this pulse was eliminated with appropriate time windowing. The simulated response was then used to estimate the distance to the open termination using the methods of example 4 (<figref idref="f0027">FIG. 34</figref>), and example 1 (<figref idref="f0028">FIGS. 35A-B</figref>). The result of the slide and attenuate method resulted in an estimated total round trip distance the pulse traveled of 2004, or double the exact distance of 1002 ft to the open. The constant threshold method resulted in time of flight of 3,196 nsec for a estimated distance traveled of 2006 ft, or two times an estimated distance of 1003 ft to the open (0.1% Error).

No actual SD signal was experimentally applied to the twisted wire pair to obtain these results. The test signal applied to the line (<figref idref="f0025">FIG. 30</figref>) was similar to the type obtained form a commercial TDR instrument for testing communication cables with lengths ranging up to 5-kft. The measurements resulting from this test line voltage were used to obtain an empirical line transfer function. The SD TOF and the line distance estimates were all obtained computationally by simulating the SD propagation along the line utilizing this empirical line transfer function estimated with non-SD signals.

Example 6

Communication by high speed data transmission using speedy delivery waveforms

The SD waveform has several properties that make it suitable as the basis for a data transmission scheme. For example: the exponential shape is maintained as it propagates at constant speed in a uniform cable; the attenuation of the SD waveform is adjustable by changing the exponential waveform parameter α; the propagating speed of the SD waveform is adjustable by changing the exponential waveform parameter α; and the SD waveforms with different α's are linearly independent.

This example teaches using the first property as the basis for a method of transmitting data using SD waveforms. The signals are transmitted 2004 ft along a transmission line consisting of the two twisted wire pairs in a 1002 ft T1 cable connected in series. The second twisted wire pair is terminated with a 100-ohm resistor. The data is encoded on four SD waveforms, each having a distinct α. Each waveform can be transmitted with a positive or a negative D, resulting in eight total states for three bits per symbol. One symbol is transmitted every three microseconds, giving a data rate of 1 Mbps. As one of ordinary skill in the art will recognize with the benefit of this disclosure, several other communication schemes can be readily derived from the teachings contained herein.

The symbols can consist of a two-microsecond truncated SD leading edge period followed by a one-microsecond recovery period. This recovery period includes a variable width half-sine compensating pulse adjusted to shorten the time required for the transmission line to return to a zero voltage. The need for this recovery period and compensating pulse is illustrated in <figref idref="f0029">FIGS. 36A-B and FIGS. 37A-B. FIG. 36A</figref> shows the transmitted pulse without the compensating pulse, and <figref idref="f0029">FIG. 36B</figref> shows the received pulse without the compensating pulse. The truncated SD waveform in the leading edge of the input pulse is closed with a simple ramp signal, which is rounded due to the line response. Although the SD portion of the propagating pulse does not undergo deformation, the closing ramp does disperse into the region beyond the three-microsecond symbol period. <figref idref="f0029">FIGS. 37A-B</figref> show the effect of the compensating pulse in eliminating this inter-symbol interference. Each truncated SD waveform may requires a different amount of compensation so the width of the half-sine pulse can be varied based on observing the pulse at the terminated end of the second twisted wire pair.

<figref idref="f0030">FIG. 38</figref> shows a series of four symbols applied to the twisted wire pair input. Each symbol has a unique α, and a unique D. The α's are equally spaced 1.1x106 sec-1 apart at 2.2x106 sec-1, 3.3x106 sec-1, 4.4x106 sec-1, and 5.5x106 sec-1. The order α's used for this illustration of four successive symbols in FIG. 9.3 is 4.4x106 sec-1, 2.2x106 sec-1, 3.3x106 sec-1, and 5.5x106 sec-1. The D's were chosen to maintain an approximately equal maximum and minimum symbol voltage. For this demonstration, the fourth waveform is inverted. The signal measured at the 100-ohm termination at a distance of 2004-ft is shown in <figref idref="f0030">FIG. 39</figref>.<figref idref="f0031">FIG. 40</figref> shows the natural log of the signal within each symbol period for the signal of <figref idref="f0030">FIG. 39</figref>. A set of matched filters could be used to detect the individual α's. The slopes of the least square linear fits of the linear regions of these waveforms are shown in Table II (set forth below). The decision level for this demonstration would be one half of the step size in α (1.1x106 sec-1) of 0.55x106 sec- 1 All errors are well below this threshold. <tables id="tabl0002" num="0002"><table frame="all"><title>Table II</title><tgroup cols="4"><colspec colnum="1" colname="col1" colwidth="24mm" /><colspec colnum="2" colname="col2" colwidth="35mm" /><colspec colnum="3" colname="col3" colwidth="37mm" /><colspec colnum="4" colname="col4" colwidth="29mm" /><thead><row><entry namest="col1" nameend="col4" align="center" valign="top">Encoded and decoded α's from data transmission demonstration</entry></row><row><entry align="center" valign="top">Symbol</entry><entry align="center" valign="top">Applied α (x106 sec-1)</entry><entry align="center" valign="top">Detected α (x106 sec-1)</entry><entry align="center" valign="top">Error (x106 sec-1)</entry></row></thead><tbody><row><entry align="center">1</entry><entry align="center">4.4</entry><entry align="center">4.38</entry><entry align="center">-0.02</entry></row><row><entry align="center">2</entry><entry align="center">2.2</entry><entry align="center">2.23</entry><entry align="center">+0.03</entry></row><row><entry align="center">3</entry><entry align="center">3.3</entry><entry align="center">3.28</entry><entry align="center">-0.02</entry></row><row><entry align="center">4</entry><entry align="center">5.5</entry><entry align="center">5.53</entry><entry align="center">+0.03</entry></row></tbody></tgroup></table></tables>

Example 7

Communication By High Speed Data Transmission Using Orthogonal Linear Combinations of Speedy Delivery Waveforms

This example discusses using the fourth SD waveform property, that SD waveforms with different α's are linearly independent. This property can be utilized to generate an orthogonal set of pulses. Data can be encoded on these pulses by varying their amplitude. These amplitude-modulated orthogonal pulses are transmitted simultaneously on the transmission line. At the receiver, the individual pulse amplitudes are computed by taking advantage of their orthogonality. The basic pulses derived from the SD waveforms must be orthogonal at the receiver for this scheme to work.

The repetitive transmission of data may require closed pulses. As shown in example 6, that the SD portion of the leading edge of a closed propagating pulse maintains its shape, while the shape used to close the pulse will disperse as it propagates. The attenuation and propagation speed of the truncated SD leading region of the pulse is a function of the SD parameter α. Therefore, the orthogonality of a set of these pulses with a variety of α values will be reduced as the pulses propagate. Fortunately, the pulses only have to be orthogonal at the receiver. Techniques can be used to generate a set of orthogonal pulses from linearly independent component pulses. The effect of the transmission line on these component pulses can be determined empirically. These transmitted component waveforms, measured at the receiver, will be linearly independent. Linear combinations of these measured, linearly independent component pulses are used to generate a set of pulses that are orthogonal at the receiver. The constants determined by the orthogonalization process are supplied to the transmitter and used to generate transmitted pulses that will be orthogonal at the receiver. An example of this procedure follows. The process of generating and transmitting the orthogonal pulses is simulated for an 8 kft, 26 AWG twisted wire pair transmission line terminated with 100 Ohms resistance.

The example orthogonalization process can begin by using a set of SD waveforms Xm(t)=e(α<sub2>o</sub2> +(m-1)·Δα)·t for m =1...5, <maths id="math0090"><math display="inline"><mrow><msub><mi>α</mi><mi mathvariant="normal">o</mi></msub><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mfrac><mn>1</mn><mi>sec</mi></mfrac><mo>,</mo><mspace width="1em" /><mi>Δα</mi><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mfrac><mn>1</mn><mi>sec</mi></mfrac><mo>,</mo></mrow></math><img file="EP1540844B1_D0090.tif" /></maths> and sec sec t = 0... 7x10-6 sec. The five waveforms are shown for a seven-microsecond interval in <figref idref="f0032">FIG. 41</figref>. The waveforms are linearly independent on this time interval. Additionally, these SD waveforms have a polynomial structure, i.e. each waveform is equal to e(Δα)·t times the previous waveform. This allows the use of a simple recursive algorithm for polynomials [4] to create orthonormal linear combinations of these linearly independent waveforms. The set of five orthonormal Y's resulting from this procedure is shown in <figref idref="f0032">FIG. 42</figref>. In this case, X and Y are related by: <maths id="math0091"><math display="block"><mrow><mi mathvariant="bold">Y</mi><mo>=</mo><mover><mi mathvariant="bold">A</mi><mrow><mo>‾</mo></mrow></mover><mo>⋅</mo><mi mathvariant="bold">X</mi></mrow></math><img file="EP1540844B1_D0091.tif" /></maths><maths id="math0092"><math display="block"><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mrow><mi mathvariant="normal">Y</mi></mrow><mn mathvariant="normal">1</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Y</mi></mrow><mn mathvariant="normal">2</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Y</mi></mrow><mn mathvariant="normal">3</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Y</mi></mrow><mn mathvariant="normal">4</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Y</mi></mrow><mn mathvariant="normal">5</mn></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn mathvariant="normal">0.026</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">0.129</mn></mtd><mtd><mn mathvariant="normal">0.088</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.717</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">1.003</mn></mtd><mtd><mn mathvariant="normal">0.337</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">4.074</mn></mtd><mtd><mn mathvariant="normal">8.578</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">5.874</mn></mtd><mtd><mn mathvariant="normal">1.310</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">23.304</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">65.519</mn></mtd><mtd><mn mathvariant="normal">67.862</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">30.704</mn></mtd><mtd><mn mathvariant="normal">5.124</mn></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi mathvariant="normal">X</mi><mn mathvariant="normal">1</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">X</mi><mn mathvariant="normal">2</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">X</mi><mn mathvariant="normal">3</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">X</mi><mn mathvariant="normal">4</mn></msub></mtd></mtr><mtr><mtd><msub><mi mathvariant="normal">X</mi><mn mathvariant="normal">5</mn></msub></mtd></mtr></mtable></mfenced><mn mathvariant="normal">.</mn></mrow></math><img file="EP1540844B1_D0092.tif" /></maths>

The set, Y, is orthonormal in that the inner product <maths id="math0093"><math display="inline"><mrow><mrow><mo>〈</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">i</mi></msub><mo>,</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">j</mi></msub><mo>〉</mo></mrow><mo>≡</mo><mrow><mstyle displaystyle="true"><mrow><msubsup><mrow><mo>∫</mo></mrow><mn>0</mn><mrow><mi mathvariant="normal">T</mi><mo>=</mo><mn>12</mn><mi>μsec</mi></mrow></msubsup></mrow></mstyle><mrow><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">i</mi></msub><mfenced><mi mathvariant="normal">t</mi></mfenced></mrow></mrow><mo>⋅</mo><msub><mi mathvariant="normal">Y</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">t</mi></mfenced><mi>dt</mi><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mi mathvariant="normal">i</mi><mo>≠</mo><mi mathvariant="normal">j</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi mathvariant="normal">i</mi><mo>=</mo><mi mathvariant="normal">j</mi></mtd></mtr></mtable><mn>.</mn></mrow></mrow></math><img file="EP1540844B1_D0093.tif" /></maths>

The sharp waveform transitions at zero and seven microseconds have high frequency content that would disperse during transmission and contribute to inter-symbol interference. This effect is reduced by multiplying this orthonormal set by sin(ω·t), with <maths id="math0094"><math display="inline"><mrow><mi mathvariant="normal">ω</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mrow><mn>7</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></mfrac><mn>.</mn></mrow></math><img file="EP1540844B1_D0094.tif" /></maths>This has the effect of changing the SD waveform to X' , where <maths id="math0095"><math display="inline"><mrow><msubsup><mi mathvariant="normal">X</mi><mi mathvariant="normal">m</mi><mrow><mo>′</mo></mrow></msubsup><mfenced><mi mathvariant="normal">t</mi></mfenced><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">o</mi></msub><mo>+</mo><mfenced><mi mathvariant="normal">m</mi><mo>−</mo><mn mathvariant="normal">1</mn></mfenced><mi mathvariant="normal">Δα</mi></mfenced><mo>⋅</mo><mi mathvariant="normal">t</mi></mrow></msup><mo>⋅</mo><mi mathvariant="normal">sin</mi><mfenced><mi mathvariant="normal">ωt</mi></mfenced><mo>=</mo><mfrac><mn mathvariant="normal">1</mn><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">j</mi></mrow></mfrac><mfenced open="[" close="]"><msup><mi mathvariant="normal">e</mi><mfenced open="[" close="]"><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">o</mi></msub><mo>+</mo><mfenced><mi mathvariant="normal">m</mi><mo>−</mo><mn mathvariant="normal">1</mn></mfenced><mi mathvariant="normal">Δα</mi><mo>+</mo><mi mathvariant="normal">jω</mi></mfenced></msup><mo>−</mo><msup><mi mathvariant="normal">e</mi><mfenced open="[" close="]"><mrow><mfenced><msub><mi mathvariant="normal">α</mi><mi mathvariant="normal">o</mi></msub><mo>+</mo><mfenced><mi mathvariant="normal">m</mi><mo>−</mo><mn mathvariant="normal">1</mn></mfenced><mi mathvariant="normal">Δα</mi></mfenced><mo>−</mo><mi mathvariant="normal">jω</mi></mrow></mfenced></msup></mfenced></mrow></math><img file="EP1540844B1_D0095.tif" /></maths>for m = 1...5.

Although the SD waveforms, X', are still linearly independent, the previous linear combination of these waveforms, Y'= <o ostyle="single">A·</o>X', is no longer orthogonal. The waveforms, Y', are shown in <figref idref="f0033">FIG. 43</figref>. These waveforms may be transmitted through the channel to the receiver.

At the receiver, the waveforms show the trailing effects of dispersion beyond the 12 µsec symbol period, see <figref idref="f0034">FIG. 44</figref>. If the spread of the pulses into the following symbol periods is not removed, this inter-symbol interference at the receiver will render data encoded at the transmitter unreadable. A simple method of eliminating this interference is by adding a pulse with a compensating tail, Z = Y'+b· C. In this example, the compensation pulse, C(t), is one cycle of <maths id="math0096"><math display="inline"><mrow><mn>1</mn><mo>−</mo><mrow><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mrow><mn>2.9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow></mfrac><mo>⋅</mo><mi mathvariant="normal">t</mi></mfenced></mrow></mrow></math><img file="EP1540844B1_D0096.tif" /></maths> delayed one microsecond after the start of the symbol period, see <figref idref="f0035">FIG. 45. FIGS. 46</figref>,<figref idref="f0036">47</figref> show the waveforms at the input and at 8 kft with appropriately sized compensating pulses added to each signal. The duration of these compensated component pulses at the receiver are now shorter than the desired 12 µsec symbol period (see <figref idref="f0036">FIG. 47</figref>), so any linear combination of these pulses will not interfere with the next symbol.

The multiplication by the sine pulse, the adding of the compensating pulse, and the transmission in the channel, have all reduced the ortliogoriality of the initial five pulses. Initially the angles between the pulses in <figref idref="f0032">FIG. 42</figref> were all 90°. Now the angles between the pulses in <figref idref="f0036">FIG. 47</figref> range from 11° to 74°. These five linearly independent waveforms are now combined into set, S, of four orthonormal waveforms using principle component decomposition (PCD) [5]. The Gram-Schmidt (G-S) orthogonalization method [6] may also be used for this step in the procedure, however the PCD method is preferable. The G-S method expands an orthonormal vector set one vector at a time. This can lead to a poorly conditioned transformation matrix if the components are close to being linearly dependent. The PCD method is preferable because it takes into account the entire inner-product space spanned by all of the components. The process makes use of the eigenvectors of the entire inner-product space to generate the transformation matrix. Additionally, the transformation matrix can be easily manipulated to order the eigenvalues. In this example the four largest eigenvalues were selected by truncating the transformation matrix. This process yields a transformation matrix, <o ostyle="single">B</o>, that generates four waveforms that are orthonormal at 8-kft by linearly combining the five component signals. These waveforms are shown in <figref idref="f0037">FIG. 48</figref> (the transmitted signals) and <figref idref="f0038">FIG. 49</figref> (the received orthogonal signals). <maths id="math0097"><math display="block"><mrow><mi mathvariant="bold">S</mi><mo>=</mo><mover><mi mathvariant="bold">B</mi><mrow><mo>‾</mo></mrow></mover><mo>⋅</mo><mi mathvariant="bold">Z</mi></mrow></math><img file="EP1540844B1_D0097.tif" /></maths><maths id="math0098"><math display="block"><mrow><mi mathvariant="bold">S</mi><mo>=</mo><mover><mi mathvariant="bold">B</mi><mrow><mo>‾</mo></mrow></mover><mo>⋅</mo><mfenced><mi mathvariant="bold">Yʹ</mi><mo>+</mo><mi mathvariant="bold">b</mi><mo>⋅</mo><mi mathvariant="normal">C</mi></mfenced></mrow></math><img file="EP1540844B1_D0098.tif" /></maths><maths id="math0099"><math display="block"><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">3</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">4</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn mathvariant="normal">4.69</mn></mtd><mtd><mn mathvariant="normal">5.75</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">21.05</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">41.44</mn></mtd><mtd><mn mathvariant="normal">46.26</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">3.55</mn></mtd><mtd><mn mathvariant="normal">0.41</mn></mtd><mtd><mn mathvariant="normal">32.53</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">33.72</mn></mtd><mtd><mn mathvariant="normal">6.00</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">2.74</mn></mtd><mtd><mn mathvariant="normal">9.11</mn></mtd><mtd><mn mathvariant="normal">0.48</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">63.36</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">91.60</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">3.35</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">3.35</mn></mtd><mtd><mn mathvariant="normal">2.06</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">56.60</mn></mtd><mtd><mn mathvariant="normal">123.38</mn></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mrow><mi mathvariant="normal">Yʹ</mi></mrow><mn mathvariant="normal">1</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Yʹ</mi></mrow><mn mathvariant="normal">2</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Yʹ</mi></mrow><mn mathvariant="normal">3</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Yʹ</mi></mrow><mn mathvariant="normal">4</mn></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Yʹ</mi></mrow><mn mathvariant="normal">5</mn></msub></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mo>−</mo><mn mathvariant="normal">0.04034</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">0.00283</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.01971</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">0.00007</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">0.00110</mn></mtd></mtr></mtable></mfenced><mo>⋅</mo><mi mathvariant="normal">C</mi></mfenced></mrow></math><img file="EP1540844B1_D0099.tif" /></maths><maths id="math0100"><math display="block"><mrow><mfenced open="[" close="]"><mtable><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">3</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">S</mi><mn mathvariant="normal">4</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo><mn mathvariant="normal">231.1</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">3</mn><mo>,</mo><mn mathvariant="normal">365.0</mn></mtd><mtd><mn mathvariant="normal">3</mn><mo>,</mo><mn mathvariant="normal">375.8</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">1</mn><mo>,</mo><mn mathvariant="normal">474.7</mn></mtd><mtd><mn mathvariant="normal">237.0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">300.5</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">714.9</mn></mtd><mtd><mn mathvariant="normal">616.1</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">228.4</mn></mtd><mtd><mn mathvariant="normal">30.7</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn mathvariant="normal">1</mn><mo>,</mo><mn mathvariant="normal">877.3</mn></mtd><mtd><mn mathvariant="normal">5</mn><mo>,</mo><mn mathvariant="normal">458.0</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">5</mn><mo>,</mo><mn mathvariant="normal">843.5</mn></mtd><mtd><mn mathvariant="normal">2</mn><mo>,</mo><mn mathvariant="normal">729.4</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">469.3</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">3</mn><mo>,</mo><mn mathvariant="normal">107.7</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">8</mn><mo>,</mo><mn mathvariant="normal">571.5</mn></mtd><mtd><mn mathvariant="normal">8</mn><mo>,</mo><mn mathvariant="normal">705.9</mn></mtd><mtd><mo>−</mo><mn mathvariant="normal">3</mn><mo>,</mo><mn mathvariant="normal">862.4</mn></mtd><mtd><mn mathvariant="normal">632.2</mn></mtd></mtr></mtable></mfenced><mo>⋅</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mn mathvariant="normal">1</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">4</mn></msup><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mn mathvariant="normal">1.1</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">5</mn></msup><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mn mathvariant="normal">2.1</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">5</mn></msup><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mn mathvariant="normal">3.1</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">5</mn></msup><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr><mtr><mtd><msup><mi mathvariant="normal">e</mi><mrow><mn mathvariant="normal">4.1</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">5</mn></msup><mi mathvariant="normal">t</mi></mrow></msup></mtd></mtr></mtable></mfenced><mo>⋅</mo><mi mathvariant="normal">sin</mi><mfenced><mi mathvariant="normal">ωt</mi></mfenced><mo>+</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mo>−</mo><mn mathvariant="normal">0.668</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.493</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.199</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0.054</mn></mtd></mtr></mtable></mfenced><mo>⋅</mo><mi mathvariant="normal">C</mi></mrow></math><img file="EP1540844B1_D0100.tif" /></maths>

The orthonormal set S is orthogonal and normalized at the receiver. The process accounts for the effects of the channel and allows the transmitter input waveforms to be easily constructed as a linear combination of defined components prior to transmission. This set S can be used as the basis functions for an orthogonal pulse amplitude modulation data transmission scheme. This process is depicted in <figref idref="f0039">FIG. 50</figref>.

The data can be encoded by amplitude modulation with five bits on each of the four orthonormal pulses, S. Five bits requires thirty-two states. Each state corresponds to one amplitude level, ai, on each orthogonal pulse, Si. For this example the amplitude levels were ± 0.5, ± 1.5, ...± 15.5. These four modulated orthogonal pulses are summed to generate a symbol, Q=a1S1 + a2S2 +a3S3 +a4S4.

The symbol is transmitted and decoded at the receiver by using the orthogonality of the pulses. For example, the applied signal on orthogonal pulse one can be found by taking the inner product of the symbol, Q, with the pulse one signal, S1. This is done by multiplying the received symbol by the known orthogonal pulse, integrating over the symbol period, and normalizing the result. <maths id="math0101"><math display="block"><mrow><mtable columnalign="left" width="auto"><mtr><mtd><mfrac><mrow><mo>〈</mo><mi mathvariant="normal">Q</mi><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow><mrow><mo>〈</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow></mfrac></mtd><mtd><mo>=</mo><mfrac><mrow><mo>〈</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">2</mn></msub><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">3</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">3</mn></msub><mo>+</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">4</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">4</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow><mrow><mo>〈</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><mfrac><mrow><mrow><mo>〈</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">1</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow><mo>+</mo><mrow><mo>〈</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">2</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">2</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow><mo>+</mo><mrow><mo>〈</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">3</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">3</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow><mo>+</mo><mrow><mo>〈</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">4</mn></msub><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">4</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow></mrow><mrow><mo>〈</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><mfrac><mrow><mrow><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">1</mn></msub></mrow><mrow><mo>〈</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow><mo>+</mo><mn mathvariant="normal">0</mn><mo>+</mo><mn mathvariant="normal">0</mn><mo>+</mo><mn mathvariant="normal">0</mn></mrow><mrow><mo>〈</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>,</mo><msub><mi mathvariant="normal">S</mi><mn mathvariant="normal">1</mn></msub><mo>〉</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><msub><mi mathvariant="normal">a</mi><mn mathvariant="normal">1</mn></msub></mtd></mtr></mtable></mrow></math><img file="EP1540844B1_D0101.tif" /></maths>

<figref idref="f0040">FIG. 51</figref> shows three successive symbols at the output of the transmitter and <figref idref="f0040">FIG. 52</figref> shows the same three symbols at the input of the receiver. The symbols were encoded with a random selection of five bits per orthogonal pulse for a data rate of <maths id="math0102"><math display="inline"><mrow><mfrac><mrow><mn>4</mn><mfrac><mi>Pulses</mi><mi>Symbol</mi></mfrac><mo>×</mo><mn>5</mn><mspace width="1em" /><mfrac><mi>Bits</mi><mi>Pulse</mi></mfrac></mrow><mrow><mn>12</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup><mfrac><mi>sec</mi><mi>Symbol</mi></mfrac></mrow></mfrac><mo>=</mo><mn>1.67</mn><mspace width="1em" /><mi>Mbps</mi><mn>.</mn></mrow></math><img file="EP1540844B1_D0102.tif" /></maths>

<figref idref="f0041">FIG. 53</figref> is a histogram of the error in detected level, aexpected -adetected, for one second, or 1.67x106 bits, of data transmitted with -140 dBm additive white gaussian noise (AWGN) present [7]. The standard deviations of the error distributions are 0.016, 0.018, 0.020, and 0.018 with maximum detected errors of 0.074, 0.071, 0.083, and 0.078. The decoding amplitude decision level is ± 0.5.

Example 8

On-Chip Clock Circuits

This example teaches the use of SD waveforms in digital circuits, specifically, in on-chip clock circuits. On-chip clock circuits may behave like transmission lines with the increased clock rates.

A major design advantage of the SD waveforms when used as clock signals is that the delay in an RC line <maths id="math0103"><math display="inline"><mfenced><msqrt><mrow><mi mathvariant="italic">RC</mi><mo>/</mo><mi>α</mi></mrow></msqrt><mo>⋅</mo><mo>ℓ</mo></mfenced></math><img file="EP1540844B1_D0103.tif" /></maths> is linear with length, ℓ, instead of quadratic with length as it is with the commonly assumed step signal. Linear delay with length <maths id="math0104"><math display="inline"><mfenced><msqrt><mi mathvariant="italic">LC</mi></msqrt><msqrt><mrow><mn>1</mn><mo>+</mo><mfrac><mi>R</mi><mi mathvariant="italic">Lα</mi></mfrac></mrow></msqrt><mo>⋅</mo><mo>ℓ</mo></mfenced></math><img file="EP1540844B1_D0104.tif" /></maths> is also true for the SD signal in an RLC line. In contrast, the delay in RLC transmission lines excited by a conventional signal exhibit an exponentially growing increase in delay with length [8].

The fact that the SD signal delay increases as a linear function of length for all types of clock lines can simplify the clock line layout design process. This linear relationship holds for the entire range of clock line behaviors, from the very lossy clock lines that behave like RC transmission lines to low loss clock lines that behave like RLC transmission lines. This simple linear relationship between line length and SD clock signal delay provides a basis for implementing software CAD tools that may enhance accuracy while reducing design computational requirements.

Time Skew Clock control is a major issue limiting system performance. A solution to this problem includes equalizing conductor lengths to the various locations on the chip where the clock signal is needed. Several geometrical patterns are commonly used (H, etc.) to equalize the signal path lengths, thereby equalizing signal delays. This requires equalizing the path lengths along the traces delivering the clock signal to pin locations that are physically close to the master chip clock driver with those clock signal path lengths to pin locations at the longest distances from the master driver. Small adjustments to equalize the delays may be accomplished by adjusting the line widths to modify the line parameters and thereby the delay per unit length and alternatively by active means such as varying the delay of delayed-locked loops embedded in the clock lines.

The invention provides a method and/or apparatus for utilizing truncated SD clock waveforms to reduce the clock skew. An advantage of the invention is that the delay per unit length can be adjusted with the exponential coefficient, α, chosen in the clock line driver design. This controllable delay per unit length has been demonstrated for a 100-ft coaxial cable in the initial patent application [9]. The minimum coaxial cable delay measured was increased by a factor of 1.8.

The large increases in the clock signal delay per unit length achieved by varying the exponential coefficient α allows lines to the pins close to the clock driver to be shortened to more closely approximate the minimum physical distance between the driver and pin while maintaining a path delay equal to the delay of the longest clock lines on the chip. The shorter clock distribution lines may utilize less physical space on the chip and reduce the total clock line capacitance. This reduction in clock line capacitance reduces the total power consumption required by the chip for a given clock frequency. In one embodiment, an SD waveform driver may also be made to create an SD clock signal with adjustable delay by including a mechanism for adjusting the exponential coefficient α in the clock generator. This adjustable delay of the SD clock signal may provide a larger delay adjustment range than the delayed-locked loops utilized in the current technology.

The invention can include transmission media. The invention can include a method and/or apparatus for a time domain reflectometer. Further, the invention can include a method and/or apparatus for high-speed communication over a transmission line. The invention can also include a method and/or apparatus for high-speed clock signals in an integrated circuit.

The terms a or an, as used herein, are defined as one or more than one. The term plurality, as used herein, is defined as two or more than two. The term another, as used herein, is defined as at least a second or more. The terms including and/or having, as used herein, are defined as comprising (i.e., open language). The term coupled, as used herein, is defined as connected, although not necessarily directly, and not necessarily mechanically. The term approximately, as used herein, is defined as at least close to a given value (e.g., preferably within 10% of, more preferably within 1% of, and most preferably within 0.1% of). The term substantially, as used herein, is defined as at least approaching a given state (e.g., preferably within 10% of, more preferably within 1% of, and most preferably within 0.1% of). The term means, as used herein, is defined as hardware, firmware and/or software for achieving a result. The term program or phrase computer program, as used herein, is defined as a sequence of instructions designed for execution on a computer system. A program, or computer program, may include a subroutine, a function, a procedure, an object method, an object implementation, an executable application, an applet, a servlet, a source code, an object code, a shared library/dynamic load library and/or other sequence of instructions designed for execution on a computer system.

All the disclosed embodiments of the invention disclosed herein can be made and used without undue experimentation in light of the disclosure. The invention is not limited by theoretical statements recited herein. Although the best mode of carrying out the invention contemplated by the inventors is disclosed, practice of the invention is not limited thereto. Accordingly, it will be appreciated by those skilled in the art that the invention may be practiced otherwise than as specifically described herein.

Further, variation may be made in the steps or in the sequence of steps composing methods described herein.

The appended claims are not to be interpreted as including means-plus-function limitations, unless such a limitation-is explicitly recited in a given claim using the phrase(s) "means for" and/or "step for." Subgeneric embodiments of the invention are delineated by the appended independent claims and their equivalents. Specific embodiments of the invention are differentiated by the appended dependent claims and their equivalents.

REFERENCES

<ol id="ol0001" compact="compact"><li>[1] <nplcit id="ncit0001" npl-type="s"><text>A. J. Gruodis and C. S. Chang, "Coupled Lossy Transmission Characterization and Simulation," IBM J. Res. Develop. 25, 1981, pp. 25-41</text></nplcit>.</li><li>[2] <nplcit id="ncit0002" npl-type="b"><text>E.G. Sauter, Nonlinear Optics, New York: John Wiley & Sons Inc., 1996, p. 127</text></nplcit>.</li><li>[3] <nplcit id="ncit0003" npl-type="b"><text>G.P. Agrawal, Nonlinear Fiber Optics, San Diego: Academic Press. ,3rd ed., 2001, p. 127</text></nplcit>.</li><li>[4] <nplcit id="ncit0004" npl-type="b"><text>Dunham Jackson, Fourier Series and Orthogonal Polynomials, Menasha: George Banta Company, Inc., 1957, pp. 156-157</text></nplcit>.</li><li>[5] <nplcit id="ncit0005" npl-type="b"><text>Attila Szabo and Neil S. Ostlund, Modern Quantum Chemistry, Mineola: Dover Publications Inc., 1989, pp. 142-145</text></nplcit>.</li><li>[6] <nplcit id="ncit0006" npl-type="b"><text>John G. Proakis, Digital Communications, Boston: WCB McCraw-Hill, 3rd ed., 1995, pp. 167-173</text></nplcit>.</li><li>[7] <nplcit id="ncit0007" npl-type="s"><text>T1.417-2001 American National Standard-Spectrum Management for Loop Transmission Systems, Annex B: Loop Information, p. 84</text></nplcit>.</li><li>[8]<nplcit id="ncit0008" npl-type="s"><text> A. Deutsch et. al., "High-Speed Signal Propagation on Lossy Transmission Lines," IBM J. Res. Develop. Vol. 34 No 4, July 1990, p. 605</text></nplcit>.</li><li>[9] <patcit id="pcit0003" dnum="US519922A" dnum-type="L"><text>Robert H. Flake, Methods For Transmitting a Waveform Having a Controllable Attenuation and Propagation Velocity, U.S. Patent Application SN 09/519,922</text></patcit>.</li></ol>

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HK1078390A1	Hong Kong, China	A1	2006-03-10
EP1358720B1	European Patent Office (EPO)	B1	2008-01-16
AT384360T	Austria	T	2008-02-15
ATE384360T1	Austria	T1	2008-02-15
DE60132490D1	Germany	D1	2008-03-06
US7375602B2	United States of America	B2	2008-05-20
DK1358720T3	Denmark	T3	2008-05-26
US2008317166A1	United States of America	A1	2008-12-25
DE60132490T2	Germany	T2	2009-01-15
EP1540844A4	European Patent Office (EPO)	A4	2009-12-23
US7859271B2	United States of America	B2	2010-12-28
US2011035170A1	United States of America	A1	2011-02-10
JP4740505B2	Japan	B2	2011-08-03
US8093911B2	United States of America	B2	2012-01-10
JP2012156994A	Japan	A	2012-08-16
JP5068933B2	Japan	B2	2012-11-07
JP5306445B2	Japan	B2	2013-10-02
CA2402166C	Canada	C	2014-04-01
CA2496322C	Canada	C	2014-10-14
EP1540844B1This record	European Patent Office (EPO)	B1	2016-04-13
CA2833510C	Canada	C	2016-10-11

64 legal events, as 9 offices reported them to INPADOC

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Events

Legal events, newest first
	Event	Code	Office
2019-10-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2019-08-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2019-07-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2019-04-24	Gb: european patent ceased through non-payment of renewal feeCeasedGBPC	GBPC	EP
2019-03-01	Application deemed withdrawn, or ip right lapsed, due to non-payment of renewal feeWithdrawnR119	R119	DE
2018-07-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2018-06-29	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2018-05-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2018-05-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-10-31	Annual fee paid to national office [announced via postgrant information from national office to epo]GrantedPGFP	PGFP	EP
2017-10-31	Annual fee paid to national office [announced via postgrant information from national office to epo]GrantedPGFP	PGFP	EP
2017-10-31	Annual fee paid to national office [announced via postgrant information from national office to epo]GrantedPGFP	PGFP	EP
2017-08-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-08-23	Fee paymentPLFP	PLFP	FR
2017-07-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-06-09	Standard patents granted in hong kongGrantedGR	GR	HK
2017-05-31	Patent lapsedLapsedMM4A	MM4A	IE
2017-05-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-04-28	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-04-28	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-03-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-03-31	Patent ceasedCeasedPL	PL	CH
2017-03-22	No opposition filedOpposition26N	26N	EP
2017-02-17	No opposition filed within time limitOppositionORIGINAL CODE: 0009261PLBE	PLBE	EP
2017-02-17	Information on the status of an ep patent application or granted ep patentGrantedSTATUS: NO OPPOSITION FILED WITHIN TIME LIMITSTAA	STAA	EP
2017-01-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-01-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-01-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-01-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-01-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2017-01-16	No opposition filed against granted patent, or epo opposition proceedings concluded without decisionGrantedR097	R097	DE
2016-12-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-12-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-11-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-11-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-11-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-11-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-11-30	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-10-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-10-31	Lapsed in a contracting state [announced via postgrant information from national office to epo]LapsedPG25	PG25	EP
2016-09-21	Patent invalid in the netherlands as no translation has been filedMP	MP	NL
2016-09-15	Deletion acc. to par. 5 (withdrawal of the translation of the ep patent)MK05	MK05	AT
2016-08-25	Fee paymentPLFP	PLFP	FR
2016-05-25	Dpma publication of mentioned ep patent grantGrantedR096	R096	DE
2016-05-04	European patents granted designating irelandGrantedFG4D	FG4D	IE
2016-04-15	European patent takes effect as a national patent in ch/liEP	EP	CH
2016-04-15	Reference to at number (ep patent enters austrian national phase)REF	REF	AT
2016-04-13	Designated contracting statesAK	AK	EP
2016-04-13	European patent grantedGrantedFG4D	FG4D	GB
2016-03-11	(expected) grantORIGINAL CODE: 0009210GRAA	GRAA	EP
2015-12-18	Grant fee paidORIGINAL CODE: EPIDOSNIGR3GRAS	GRAS	EP
2015-09-09	Intention to grant announcedINTG	INTG	EP
2015-08-09	Despatch of communication of intention to grant a patentORIGINAL CODE: EPIDOSNIGR1GRAP	GRAP	EP
2011-03-30	First examination report despatched17Q	17Q	EP
2009-12-23	Supplementary search report drawn up and despatchedA4	A4	EP
2007-08-08	Information on inventor provided before grant (corrected)RIN1	RIN1	EP
2007-08-08	Information on inventor provided before grant (corrected)RIN1	RIN1	EP
2006-03-10	Requests to designate patent in hong kongDE	DE	HK
2006-02-01	Party data changed (applicant data changed or rights of an application transferred)RAP1	RAP1	EP
2005-12-07	Request for extension of the european patent (deleted)DAX	DAX	EP
2005-06-15	Request for examination filed17P	17P	EP
2005-06-15	Designated contracting statesAK	AK	EP
2005-06-15	Request for extension of the european patentAX	AX	EP
2005-04-29	Public reference made under article 153(3) epc to a published international application that has entered the european phaseORIGINAL CODE: 0009012PUAI	PUAI	EP

Methods for transmitting a waveform having a controllable attenuation and propagation velocity

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