Filter for impulse response shortening, with additional spectral constraints, for multicarrier transmission

Background

0001The invention relates to time-domain equalization in a discrete multi-tone (DMT) receiver.

0002Conventional single carrier modulation techniques translate data bits for transmission through a communication channel by varying the amplitude and/or phase of a single sinusoidal carrier. By contrast, DMT, which is also referred to as Orthogonal Frequency Division Multiplexing (OFDM) or Multicarrier Modulation (MCM), employs a large number of sinusoidal subcarriers, e.g., 128 or 256 subcarriers. The available bandwidth of the communication channel is divided into subchannels and each subchannel communicates a part of the data. A DMT system may employ quadrature amplitude modulation (QAM) for each of the subcarriers.

0003OFDM-based systems transmit blocks of information bits. The time required to transmit one such block is called the symbol period. The time domain waveform that corresponds to one such block of bits is called a symbol.

0004Intersymbol interference (ISI) arises from the characteristics of practical communication channels and limits the rate at which information can be transmitted through them. Specifically, communication channels typically have an Effective Discrete-Time Impulse Response (EDIR) that is greater than one sample time in length, which causes ISI. ISI is a well-known phenomenon in single-carrier communication systems and there are many techniques for reducing it. The process of such ISI reduction is called equalization. ISI is discussed, for example, in <nplcit id="ncit0001" npl-type="b"><text>Proakis, Digital Communications, McGraw Hill, 2nd Edition, 1989</text></nplcit>.

0005Equalization in OFDM-based systems is achieved by a two stage process. First, at the transmitter, a Cyclic Prefix (CP) is employed by affixing an end-portion of each symbol to the beginning of the symbol. A cyclic prefix that is greater than the EDIR of the channel prevents one symbol from interfering with another. Furthermore, it also facilitates a simple method of neutralizing the time domain spread of each symbol forced by the channel. This is achieved through a simple frequency domain process in the receiver which requires one multiplication operation for each subcarrier used. The use of a Cyclic Prefix to reduce ISI is discussed, for example, in: <nplcit id="ncit0002" npl-type="s"><text>Cimini, "Analysis and Simulation of a Digital Mobile Channel using Orthogonal Frequency Division Multiplexing," IEEE Transactions on communications, pp 665-675, July 1985</text></nplcit>; <nplcit id="ncit0003" npl-type="s"><text>Chow, "A Discrete Multi-Tone Transceiver System for HDSL applications," IEEE Journal on Selected Areas of Communications, 9 (6): 895-908, August 1991</text></nplcit>; and "<nplcit id="ncit0004" npl-type="s"><text>DMT Group VDSL PMD Draft Standard Proposal, "Technical Report, T1E1.4/96-329R2, ANSI 1997</text></nplcit>.

0006Another problem arising in conventional DMT systems is noise bleeding, which occurs when noise from one frequency band interferes with a signal whose subcarrier is in another frequency band. Noise bleeding is caused, in general, by a discrete Fourier transform (DFT) operation at the receiver. Noise bleeding is discussed in, for example, <nplcit id="ncit0005" npl-type="s"><text>Worthen et. al., "Simulation of VDSL Test Loops, "Technical ReportT1E1.4/97-288, ANSI 1997</text></nplcit>.

0007In a perfectly synchronized DMT system, a signal in one frequency band does not interfere with a signal whose subcarrier is in another frequency band. However, noise from one band may interfere with other less noisy bands and render them unusable. Techniques for dealing with noise-bleeding include wavelet-based solutions. However, wavelet-based solutions are, in general, computationally intensive.

0008Other references dealing with time domain equalization include: <nplcit id="ncit0006" npl-type="s"><text>Chow, J. S. and Cioffi, J. M., "A Cost-effective Maximum Likelihood Receiver for Multicarrier Systems", Proceedings of the ICC, 1992</text></nplcit>;<nplcit id="ncit0007" npl-type="s"><text> Melsa, Peter J.W., Younce, Richard C., and Rohrs, Charles E., "Optimal Impulse Response Shortening", Proceedings of the thirtv-third Annual Allerton Conference on Communication. Control and Computing, 1995, pp. 431-438</text></nplcit>; <nplcit id="ncit0008" npl-type="s"><text>Harikumar, Gopal and Marchok, Daniel, "Shortening the Channel Impulse Response of VDSL Loops for Multicarrier Applications", Technical report T1E1.4197-289, ANSI, 1997</text></nplcit>.

0009<patcit id="pcit0001" dnum="EP0768778A1"><text>EP 0 768 778 A1</text></patcit> describes an arrangement in which an energy constraint is used in order to ensure that a solution for an optimal target response vector is not found which places all the received energy in unused frequency bands. <patcit id="pcit0002" dnum="WO9326096A"><text>WO 93/26096</text></patcit> describes a system in which a set of parameters is optimized for equalizing a multi-carrier data signal using an encoded, predetermined signal.

Summary

0010The present invention provides a method for equalizing a communication channel according to claim 1. There is further provided an impulse response shortening filter according to claim 3 and a corresponding computer program according to claim 6.

0011The invention provides a spectrally constrained impulse shortening filter (SCISF) a filter according to claims 3-5 being called in the following, which could be used, for example, in DMT systems. The SCISF serves two primary functions.

0012First, the SCISF reduces intersymbol interference (ISI) by reducing the length of the effective discrete-time impulse response (EDIR) of the communication channel. Conventional impulse shortening filters may have deep nulls in their frequency response. By contrast, the SCISF has a filter characteristic that is essentially free from undesired nulls that may attenuate or completely eliminate certain subcarriers.

0013Second, the SCISF reduces noise bleeding between subchannels by attenuating noisy channels in a manner that does not reduce the signal to noise ratio (SNR) in these channels, but reduces the noise power that may appear in the sidelobes of adjacent subchannels. The SCISF accomplishes these functions by applying a frequency constraint to the signal based on a desired spectral response.

0014In one general aspect, the invention features equalizing a channel according to claim 1.

0015In another general aspect, the invention features an impulse response shortening filter according to claim 3.

0016In another aspect, the invention features a computer program according to claim 6. a computer program according to claim 6.

0017The techniques described here are not limited to any particular hardware or software configuration. They may find applicability in any computing or processing environment that may be used for a communication system. The techniques may be implemented in hardware or software, or a combination of the two. Preferably, the techniques are implemented in computer programs executing on a digital signal processor that includes a processor and a storage medium readable by the processor (including volatile and non-volatile memory).

0018Further details of the method of claim 1 are provided by claim 2.

0019Further details of the filter of claim 3 are provided by claims 4 and 5.

0020Further details of the computer program of claim 6 are provided in claim 7.

Brief Description of the Drawings

0021<ul id="ul0001" list-style="none" compact="compact"><li><figref idref="f0001">Fig. 1</figref> is a block diagram of a discrete multi-tone communication system.</li><li><figref idref="f0002">Fig. 2</figref> is a plot of effective discrete-time impulse response (EDIR) of a communication channel including transmit and receive filters.</li><li><figref idref="f0002">Fig. 3</figref> is a plot of the shortened EDIR due to a spectrally constrained impulse shortening filter (SCISF).</li><li><figref idref="f0003">Fig. 4</figref> is a block diagram of a SCISF.</li><li><figref idref="f0004">Fig. 5</figref> is a plot of transmit signal power, signal power at SCISF input and noise power at SCISF input.</li><li><figref idref="f0004">Fig. 6</figref> is a plot of signal and noise power at the output of the SCISF.</li><li><figref idref="f0005">Fig. 7</figref> shows the filter response of a discrete Fourier transform for one frequency band.</li><li><figref idref="f0005">Fig. 8</figref> is a plot of the desired spectral response of the SCISF, G_d(ω), versus the actual frequency response, G(ω).</li><li><figref idref="f0006">Fig. 9</figref> is a plot of signal-to-noise ratio at the output of the SCISF, the output of the DFT with the SCISF and the output of the DFT without the SCISF.</li><li><figref idref="f0007">Fig. 10</figref> is a block diagram of a system for adapting a spectrally constrained impulse shortening filter in an initial training process.</li><li><figref idref="f0008">Figs. 11</figref> and <figref idref="f0009">12</figref> are block diagrams of a system for adapting a spectrally constrained impulse shortening filter in an periodic or continuous training process.</li><li><figref idref="f0010">Fig. 13</figref> is a block diagram of a generalized adaptation system that employs frequency scaling in the feedback loop.</li><li><figref idref="f0011">Fig. 14</figref> is a block diagram of a system for adapting a spectrally constrained impulse shortening filter in an initial training process that includes frequency scaling in the feedback loop.</li><li><figref idref="f0012">Figs. 15</figref> and <figref idref="f0013">16</figref> are block diagrams of a system for adapting a spectrally constrained impulse shortening filter in an periodic or continuous training process that includes frequency scaling in the feedback loop.</li><li><figref idref="f0014 f0015">Figs. 17-20</figref> are plots of simulation results for a discrete multi-tone system.</li><li><figref idref="f0016 f0017">Figs. 21-24</figref> are plots of simulation results for a discrete multi-tone system.</li></ul>

Description

0022As shown in <figref idref="f0001">Fig. 1</figref>, a discrete multi-tone (DMT) communication system 10 has a transmitter 12 and a receiver 14. The transmitter 12 accepts an input data bit stream which passes through a constellation encoder 20. The encoder 20 divides the serial input bit stream into blocks of data. These blocks of data are further subdivided into smaller blocks corresponding to subchannels. Each of these smaller blocks are used to compute a complex value representing a constellation point. Each constellation point corresponds to a subsymbol. The subsymbols are then output by the encoder. Taken together, the subsymbols constitute a symbol.

0023The subsymbols are supplied to an inverse discrete Fourier transform (IDFT) 30, which may be implemented, for example, in a digital signal processor. The IDFT 30 outputs N time samples of a symbol. The time samples are processed by a parallel to serial converter 40 to form a single stream of time samples.

0024Following the parallel to serial converter 40, a prefix adder 50 adds a cyclic prefix to the beginning of each symbol to reduce intersymbol interference (ISI). Alternatively, the cyclic prefix may be added in the parallel to serial converter. After the cyclic prefix is added, the resulting signal passes through a digital-to-analog (D/A) converter 60 for transmission to the receiver 14 through a communication channel 70. An analog transmit filter 65 may be included following the D/A converter to band limit the transmitted signal.

0025At the receiver 14, the signal passes through an analog-to-digital (A/D) converter 80 and then through a spectrally constrained impulse shortening filter (SCISF) 90. A prefix stripper 100 strips the cyclic prefixes from the resulting symbols and a serial to parallel converter 110 divides the stream of time samples into parallel signal paths that form the inputs to a discrete Fourier transform (DFT) 120. The DFT 120 converts the time samples into subsymbols. A frequency domain equalization filter 130 equalizes the subsymbols. A decoder 140 converts the subsymbols into a data bits and outputs the resulting data. An analog receive filter 75 may be included prior to the A/D converter in order to band limit the received signal.

0026As discussed above, a cyclic prefix is added to each symbol prior to transmission through the communication channel to reduce the effects of ISI. The cyclic prefix is added by copying the last v time samples from the end of a symbol and placing them at the beginning of the symbol. To eliminate ISI, the length of the cyclic prefix, v, is chosen to be longer than the effective discrete-time impulse response (EDIR) of the channel. However, because the cyclic prefix constitutes redundant data, increasing the length of the cyclic prefix reduces the efficiency of the communication system. For example, in a system having N time samples per symbol and a cyclic prefix of v time samples, the efficiency of the system will be reduced by a factor of N/(N+v). Efficiency may be maximized either by minimizing v or maximizing N. However, increasing N increases the complexity, latency and computational requirements of the system and at some point becomes impractical. Accordingly, it is desirable to minimize v.

0027A spectrally constrained impulse shortening filter having an impulse response, g(n), may be employed in the receiver to minimize the length of the cyclic prefix by decreasing the EDIR of the effective communication channel, which includes the transmit and receive filters, the impulse shortening filter, and the physical transmission channel. The use of an impulse shortening filter is referred to as time domain equalization. Decreasing the EDIR allows a shorter cyclic prefix to be used without increasing ISI.

0028<figref idref="f0002">Fig. 2</figref> is a plot of the EDIR for a DMT test configuration having a communication channel that is 4500 feet in length and operates at a sampling frequency of 11.04 MHz (test loop 4, as described in "Very-high Speed Digital Subscriber Lines: System Requirements," <u>Technical Report T1E1.4/97-131R1</u>, ANSI 1998). The EDIR includes the effects of a transmit filter, the communication channel and a receive filter. <figref idref="f0002">Fig. 3</figref> shows the impulse response as shortened by the addition of an impulse shortening filter.

0029The impulse shortening filter is selected so that a significant part of the energy of the joint impulse response of the filter and the effective communication channel, <i>g</i>(<i>n</i>)*<i>h</i>(<i>n</i>), is confined to a region that is shorter in length than the length of the cyclic prefix. Some prior algorithms for computing <i>g</i>(<i>n</i>) considered only shortening the EDIR and did not consider the spectral characteristics of the resulting impulse shortening filter. Such filters often had deep nulls in some frequency bands, which rendered some of the corresponding subchannels useless.

0030Since increasing the length of the cyclic prefix reduces system efficiency, the receiver may dynamically compute an optimal length for the cyclic prefix and may send that information to the transmitter. For example, the receiver may compute a set of impulse responses for the impulse shortening filter based on a set of predetermined cyclic prefix lengths. The receiver then computes the system throughput for each particular cyclic prefix length. The length that maximizes system throughput is selected and the result of that selection is communicated to the transmitter. The transmitter then operates using the selected cyclic prefix length.

0031To avoid the possible attenuation of frequency bands, the spectral response of the impulse shortening filter is further required to have a spectral response that |<i>G</i>(ω)|, meets a specified spectral constraint. A spectral constraint of the form |<i>G</i>(ω)<i>H</i>(ω)|><i>τ</i>, where <i>τ</i> is a threshold, is sufficient to avoid nulls in the frequency response of the impulse shortening filter. However, it is possible to compute a spectral constraint or desired spectral response, |<i>G_d</i>(ω)|, that provides additional performance improvements, such as reducing noise bleeding between subchannels. A spectrally constrained impulse filter is configured to have a spectral response that approximates the desired spectral response.

0032As shown in <figref idref="f0003">Fig. 4</figref>, the spectrally constrained impulse shortening filter (SCISF) 90 may be implemented as a time domain digital filter, which has a digital filter structure 92 with a number of taps 94 or filter coefficient inputs for adjusting the filter response. The coefficients may be computed and supplied to the taps by a digital signal processor (DSP). Alternatively, the SCISF may be implemented entirely in software, i.e., within a DSP.

0033A desired spectral response may be applied to a received signal using a filter that is separate from the impulse shortening or time domain equalization (TEQ) filter. For example, an analog filter may be placed prior to the A/D converter. However, the adjustability of such a filter would be limited. As a further example, a digital filter could be added prior to the TEQ filter. Both of these configurations suffer the disadvantage that the TEQ filter may distort the desired spectral characteristics of the added filter. A filter also might be positioned after the TEQ filter, which would reduce noise bleeding, but might reduce the impulse shortening provided by the TEQ filter. Accordingly, the SCISF integrates the TEQ (i.e., impulse shortening) function with the desired spectral response in a single filter.

0034In summary, the filter characteristic, <i>g</i>(<i>n</i>), of the SCISF satisfies two conditions. First, the effective length of the convolution of the filter characteristic with the impulse response of the communication channel, <i>g</i>(<i>n</i>)*<i>h</i>(<i>n</i>), is less than a target length. Second, the cost function (error function) between the desired spectral response, <i>G_d</i>(ω), and the actual filter spectral response, <i>G</i>(ω), is minimized.

0035The desired spectral response is an ideal filter characteristic that is selected to maximize the data bit throughput in the subchannels of a DMT system by reducing the impact of noise. There are many sources for noise in a DMT communication system, such as near-end cross-talk (NEXT), radio frequency interference (RFI) and noise generated in the communication channel (white noise). As shown in <figref idref="f0004">Figs. 5 and 6</figref>, the noise spectral density is generally not uniform across the frequency band of the communication system. This non-uniformity contributes to the problem of noise bleeding, in which noise in one frequency band interfering with a signal in another frequency band.

0036In general, noise bleeding is caused by sidelobes of filters in a DFT. <figref idref="f0005">Fig. 7</figref> shows the filter response of a DFT for one frequency band or bin (i.e., bin 128). The first sidelobes (96) are only 13 dB below the main lobe (98). Therefore, noise located outside of bin 128, but within the first sidelobe of bin 128, i.e., approximately mid-way between bins 126 and 127, would appear in bin 128 with an attenuation of only 13 dB. Consequently, noisy subchannels in a DMT system may degrade the performance of non-noisy subchannels.

0037The desired spectral response is essentially a spectral constraint that attenuates noisy channels more than non-noisy channels. The signal and noise in the noisy channels are reduced equally, so the attenuation does not affect the signal-to-noise ratio in these channels. However, because the absolute noise level in the noisy channels is reduced, there is less noise available to enter the sidelobes of the non-noisy channels. Hence, the noise bleeding problem is minimized.

0038To determine the desired spectral response, the noise power spectral density (noise PSD) at the receiver must be known. The noise PSD may be determined, for example, by performing a periodigram on received data. This measurement is more complicated if a transmitter is transmitting, since the transmitted signal must be separated from the noise measurement. The noise PSD is determined by: (i) slicing the received constellation of subcarriers after the DFT to determine the nearest valid constellation point; (ii) determining an error signal based on the difference between the received constellation point and the valid constellation point; (iii) performing an IDFT on the error signal; and (iv) generating a periodigram (with windowing) from the error signals. The noise PSD may then be determined from the periodigram.

0039An example of a noise PSD characteristic for a DMT communication system is shown in <figref idref="f0004">Fig. 5</figref> (test loop 4, as described in "Very-high Speed Digital Subscriber Lines: System Requirements," <u>Technical Report T1E1.4/97-131R1</u>, ANSI 1998). The transmit signal power is measured at the output of the transmit filter in the transmitter. The signal and noise PSD plots shown in <figref idref="f0004">Fig. 5</figref> are measured at the input of the A/D converter in the receiver, which is prior to the SCISF.

0040Measured noise PSD is used by a digital signal processor (DSP) to compute a desired spectral response, <i>G_d</i>(ω), using the algorithm described below. Alternatively, the inverse of the noise PSD may be used as an approximation for the desired spectral response. A spectral response, <i>G</i>(ω), is then determined for the SCISF that minimizes the error between the spectral response of the SCISF and the desired spectral response. A set of filter coefficients may then be generated to configure the SCISF to the determined characteristic. These calculations may be done periodically to adjust the performance of the communication system. Frequency domain equalization coefficients and symbol synchronization may also be adjusted based on these calculations.

0041<figref idref="f0005">Fig. 8</figref> is a plot of the desired spectral response of the SCISF, G_d(ω), versus the actual frequency response, G(ω). The difference between the responses is only a few dB. <figref idref="f0004">Fig. 6</figref> shows the signal and noise PSD at the output of the SCISF.

0042<figref idref="f0006">Fig. 9</figref> shows the dramatic effect of the SCISF on system performance. Without the SCISF (i.e., using a filter that provides only impulse shortening), the signal-to-noise ratio (SNR) decreases significantly at the output of the Fourier transform (i.e., FFT or DFT) to less than about 7 dB. This decrease is due, in large part, to noise bleeding caused by the sidelobes of the Fourier transform. By contrast, with the SCISF, the SNR at the output of the Fourier transform tracks the SNR at the output of the SCISF within a few dB. Overall, the SCISF provides an improvement in SNR.

0043The desired spectral response of the SCISF, <i>G_d</i>(ω) and the actual frequency response, <i>G</i>(ω), are derived from an energy constraint for the SCISF, i.e., the desired spectral response must localize the energy of the effective impulse response within a desired frequency band. The energy constraint is combined with a desired spectral response based on the measured noise power spectral density. The resulting cost function (or error function) is then minimized to obtain a practical filter characteristic for the SCISF. This process is presented in detail below.

0044In the following derivation, all vectors are column vectors by default. Vectors are denoted by bold lower case letters (e.g., <b>t</b>). The size <i>m_t</i> of a vector <b>t</b>, is written <b>t</b>(<i>m_t</i>). The components of a vector are denoted by lower case letters, e.g., <b>t</b><i>(m_t)</i> = [<i>t</i>₀···<i>t</i>_{<i>m_t</i>-1}]<i>^T</i>. The convolution of length <i>m_t</i> + <i>m_h</i> -1 of vectors <i>t</i>(<i>m_t</i>) and <i>h</i>(<i>m_h</i>) is denoted <b>t*h.</b> The Toeplitz matrix of a vector <maths id="math0001"><math display="inline"><mi>t</mi><mfenced><msub><mi>m</mi><mi>t</mi></msub></mfenced><mo>,</mo><msub><mrow><mspace width="1em" /><mi>t</mi></mrow><mfenced><mi>m</mi><mo>:</mo><mi>n</mi></mfenced></msub><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msup><mfenced open="[" close="]"><msub><mi>t</mi><mi>m</mi></msub><mo>⋯</mo><msub><mi>t</mi><mi>n</mi></msub></mfenced><mi>T</mi></msup><mo>,</mo></math><img file="EP1068704B1_D0001.tif" /></maths> is written <maths id="math0002"><math display="inline"><msubsup><mi mathvariant="bold">T</mi><mfenced><mi>m</mi><mo>:</mo><mi>n</mi></mfenced><msub><mi>m</mi><mi>x</mi></msub></msubsup><mo>,</mo></math><img file="EP1068704B1_D0002.tif" /></maths> and <maths id="math0003"><math display="inline"><msubsup><mi mathvariant="bold">T</mi><mfenced><mi>m</mi><mo>:</mo><mi>n</mi></mfenced><msub><mi>m</mi><mi>x</mi></msub></msubsup><mo>⁢</mo><mi mathvariant="bold">x</mi><mo>=</mo><msub><mfenced><mi mathvariant="bold">t</mi><mo mathvariant="bold">*</mo><mi mathvariant="bold">x</mi></mfenced><mfenced><mi>m</mi><mo>:</mo><mi>n</mi></mfenced></msub><mn>.</mn></math><img file="EP1068704B1_D0003.tif" /></maths> The Discrete Time Fourier Transform (DTFT) of a vector t may be represented as: <maths id="math0004" num="(1)"><math display="block"><mi>T</mi><mfenced><mi>ω</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><msub><mi>m</mi><mi>t</mi></msub><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><msub><mi>t</mi><mi>n</mi></msub><mo>⁢</mo><msup><mi>e</mi><mrow><mo>-</mo><mi mathvariant="italic">jωn</mi></mrow></msup><mn>.</mn></math><img file="EP1068704B1_D0004.tif" /></maths>

0045The EDIR of the channel and the impulse response of the SCISF are expressed as <b>h</b>(<i>m_h</i>) and g(<i>m_g</i>), respectively. The cyclic prefix has a length of <i>m_c</i> per symbol. The energy of the effective impulse response must be localized to a contiguous region of target length <i>m_l</i>, where <i>m_t</i>&lt;<i>m_c</i>, while satisfying a spectral constraint. The energy criterion to be satisfied by the SCISF may be written as: <maths id="math0005" num="(2)"><math display="block"><mfrac><msup><mrow><mo>‖</mo><msub><mfenced><mi mathvariant="bold">g</mi><mo>*</mo><mi mathvariant="bold">h</mi></mfenced><mrow><mi mathvariant="italic">m</mi><mo mathvariant="italic">:</mo><mi mathvariant="italic">m</mi><mo>+</mo><msub><mi>m</mi><mi>l</mi></msub><mo>-</mo><mn>1</mn></mrow></msub><mo>‖</mo></mrow><mn>2</mn></msup><msup><mrow><mo>‖</mo><mfenced><mi mathvariant="bold">g</mi><mo>*</mo><mi mathvariant="bold">h</mi></mfenced><mo>‖</mo></mrow><mn>2</mn></msup></mfrac><mo>≥</mo><mi>α</mi></math><img file="EP1068704B1_D0005.tif" /></maths> for some 0 ≤ <i>m ≤ m_g</i> + <i>m_h -</i> 2 and some 0 &lt; a &lt; 1. Defining the set <i>S^m</i> as <maths id="math0006" num="(3)"><math display="block"><msup><mi>s</mi><mi>m</mi></msup><mo>=</mo><mfenced open="{" close="}"><mi mathvariant="bold">g</mi><mo>∈</mo><msup><mi>R</mi><msub><mi>m</mi><mi>g</mi></msub></msup><mo>:</mo><mfrac><msup><mrow><mo>‖</mo><msub><mfenced><mi mathvariant="bold">g</mi><mo>*</mo><mi mathvariant="bold">h</mi></mfenced><mrow><mi mathvariant="italic">m</mi><mo mathvariant="italic">:</mo><mi mathvariant="italic">m</mi><mo>+</mo><msub><mi>m</mi><mi>l</mi></msub><mo>-</mo><mn>1</mn></mrow></msub><mo>‖</mo></mrow><mn>2</mn></msup><msup><mrow><mo>‖</mo><mfenced><mi mathvariant="bold">g</mi><mo>*</mo><mi mathvariant="bold">h</mi></mfenced><mo>‖</mo></mrow><mn>2</mn></msup></mfrac><mo>≥</mo><mi>α</mi></mfenced></math><img file="EP1068704B1_D0006.tif" /></maths> the impulse response of the SCISF must belong to <i>S^m</i> for some <i>m</i>.

0046Let <i>ω</i>₁<i>,..., ω_N</i> be the location of the sub-carriers in the frequency domain. The spectral constraint may be applied by selecting g ∈ <i>S^m</i>, so that the cost function <maths id="math0007" num="(4)"><math display="block"><mi>J</mi><mfenced><mi mathvariant="bold">g</mi></mfenced><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><msup><mfenced><mfenced open="|" close="|"><mi>G</mi><mfenced><msub><mi>ω</mi><mi>i</mi></msub></mfenced></mfenced><mo>-</mo><msub><mi>G</mi><mi>d</mi></msub><mfenced><msub><mi>ω</mi><mi>i</mi></msub></mfenced></mfenced><mn>2</mn></msup></math><img file="EP1068704B1_D0007.tif" /></maths> is minimized for the desired spectral response <i>G_d</i>(<i>ω</i>).

0047Typically, this optimization would have to be repeated for all possible <i>m</i> to select the filter impulse response, <b>g,</b> that achieves the lowest possible value of <i>J</i>. However, as discussed below, optimization may be limited to a few well chosen values of <i>m</i>.

0048The determination of the filter impulse response may be done in two stages. First, the desired magnitude frequency response <i>G_d</i>(<i>ω</i>) of the impulse shortening filter g over the bins used by the DMT system is obtained. Second, <i>J</i> is optimized over <i>S^m</i> for a specific value of <i>m</i>.

0049To determine the desired spectral response <i>G_d</i>(<i>ω</i>), it is necessary to use expressions for the signal to noise ratios observed in the various frequency bins at the output of the DFT in the receiver. The DMT system has <i>M</i> tones (subcarriers), <i>N</i> of which (those from <i>M</i>₁ through <i>M₂</i>) are used, and the communication channel has an analog frequency response <i>H_c</i>(<i>f</i>). The analog noise power spectral density observed at the input of the receiver A/D is <i>s_η</i>(<i>f</i>). Prior to conversion in the A/D converter, the received analog signal may be filtered by an anti-aliasing filter with transfer function <i>H_a</i>(<i>f</i>). The EDIR in the absence of the impulse shortening filter is <i>h</i>(<i>n</i>). After the A/D converter, the signal is fed into the impulse shortening filter with an impulse response of <i>g</i>(<i>n</i>). The impulse shortening filter ensures that (<i>h</i>(<i>n</i>)*<i>g</i>(<i>n</i>)) is shorter in length than the cyclic prefix. <i>G</i>(ω) is the discrete time Fourier transform of <i>g</i>(<i>n</i>). Under these conditions, the expected signal energy µ(<i>k</i>) observed in bin <i>k</i> at the output of the length-2<i>M</i> receiver DFT is given by: <maths id="math0008" num="(5)"><math display="block"><mi>μ</mi><mfenced><mi>k</mi></mfenced><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>D</mi><mi>k</mi></msub><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>H</mi><mfenced><mfrac><mi mathvariant="italic">πk</mi><mi>M</mi></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>G</mi><mfenced><mfrac><mi mathvariant="italic">πk</mi><mi>M</mi></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>;</mo><mi>H</mi><mfenced><mi>ω</mi></mfenced><mo>=</mo><msub><mi>H</mi><mi>c</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced><mo>⁢</mo><msub><mi>H</mi><mi>a</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced></math><img file="EP1068704B1_D0008.tif" /></maths> where C₁ is a constant, 1/<i>T</i> the sampling frequency and <i>D_k</i> the transmitted power in bin k. The noise power <i>η</i>(<i>k</i>) in bin <i>k</i> is: <maths id="math0009" num="(6)"><math display="block"><mi>η</mi><mfenced><mi>k</mi></mfenced><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><mfenced open="[" close="]"><msub><mi>S</mi><mi>η</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>G</mi><mfenced><mi>ω</mi></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><msup><mfenced open="|" close="|"><msub><mi>H</mi><mi>a</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced></mfenced><mn>2</mn></msup></mfenced><mo>*</mo><msub><mrow><mfenced open="[" close="]"><mfrac><mrow><msup><mi>sin</mi><mn>2</mn></msup><mfenced><mi>M</mi><mo>⁢</mo><mi>ω</mi></mfenced></mrow><mrow><msup><mi>sin</mi><mn>2</mn></msup><mfenced><mfrac><mi>ω</mi><mn>2</mn></mfrac></mfenced></mrow></mfrac></mfenced><mo>|</mo></mrow><mrow><mi>ω</mi><mo>=</mo><mfrac><mi mathvariant="italic">πk</mi><mi>M</mi></mfrac></mrow></msub></math><img file="EP1068704B1_D0009.tif" /></maths> where <i>C</i>₂ is another constant and * denotes the convolution. Assuming that the noise in the bands corresponding to unused tones is sufficiently attenuated by the anti-alias filter, <i>_η</i>(<i>k</i>) is approximately equal to: <maths id="math0010" num="(7)"><math display="block"><mi>η</mi><mfenced><mi>k</mi></mfenced><mo>≈</mo><msub><mi>C</mi><mn>3</mn></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><msub><mi>M</mi><mn>1</mn></msub></mrow><msub><mi>M</mi><mn>2</mn></msub></munderover></mstyle><msub><mi>S</mi><mi>η</mi></msub><mfenced><mfrac><mi>l</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">MT</mi></mrow></mfrac></mfenced><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>G</mi><mfenced><mfrac><mi mathvariant="italic">πl</mi><mi>M</mi></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><msup><mfenced open="|" close="|"><msub><mi>H</mi><mi>a</mi></msub><mfenced><mfrac><mi>l</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">MT</mi></mrow></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><mfenced><mi>α</mi><mo>⁢</mo><mfenced><mi>k</mi><mo>-</mo><mi>l</mi></mfenced><mo>+</mo><mi>α</mi><mo>⁢</mo><mfenced><mn>2</mn><mo>⁢</mo><mi>M</mi><mo>-</mo><mi>k</mi><mo>-</mo><mi>l</mi></mfenced></mfenced><mo>,</mo></math><img file="EP1068704B1_D0010.tif" /></maths> where <i>_α</i>(<i>n</i>) is defined as: <maths id="math0011" num="(8)"><math display="block"><mi>α</mi><mfenced><mi>n</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∫</mo><mrow><mo>-</mo><mfrac><mi>π</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">M</mi></mrow></mfrac></mrow><mfrac><mi>π</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">M</mi></mrow></mfrac></msubsup></mstyle><mfenced open="[" close="]"><mfrac><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo>⁢</mo><mfenced><mi>M</mi><mo>⁢</mo><mfenced><mfrac><mi mathvariant="italic">πn</mi><mi mathvariant="italic">M</mi></mfrac><mo>-</mo><mi>λ</mi></mfenced></mfenced></mrow><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mfenced><mfrac><mi mathvariant="italic">πn</mi><mi mathvariant="italic">M</mi></mfrac><mo>-</mo><mi>λ</mi></mfenced></mfenced></mrow></mfrac></mfenced><mo>ⅆ</mo><mi>λ</mi><mo>,</mo></math><img file="EP1068704B1_D0011.tif" /></maths><i>M</i>₁ ...<i>M</i>₂ are the used tones and <i>C</i>₃ another constant. Defining <b>x</b> to be the vector of frequency magnitudes to be solved for as: <maths id="math0012" num="(9)"><math display="block"><mi mathvariant="bold">x</mi><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mfenced open="[" close="]"><mtable><mtr><mtd><mrow><mo>|</mo><msup><mrow><mi>G</mi><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><msub><mi>M</mi><mn>1</mn></msub></mrow><mi>M</mi></mfrac></mfenced><mo>|</mo></mrow><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msup><mtable><mtr><mtd><mfenced open="|" close="|"><mtable><mtr><mtd><mi>G</mi></mtd></mtr></mtable><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><msub><mi>M</mi><mn>2</mn></msub></mrow><mi>M</mi></mfrac></mfenced></mfenced></mtd></mtr></mtable><mn>2</mn></msup></mtd></mtr></mtable></mfenced><mo>,</mo></math><img file="EP1068704B1_D0012.tif" /></maths> the SNR in bin <i>k</i> can be seen to be of the form <maths id="math0013"><math display="inline"><mfrac><mrow><msub><mi>b</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi>x</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>a</mi><mi mathvariant="normal">k</mi><mi>T</mi></msubsup><mo>⁢</mo><mi mathvariant="bold">x</mi></mrow></mfrac><mo>,</mo></math><img file="EP1068704B1_D0013.tif" /></maths> in which <i>b_k</i> are scalars and <b>a</b><i>_k</i> are vectors.

0050To determine the desired spectral response, <b>x</b> is chosen to maximize the bit throughput. Approximating the capacity of bin <i>k</i> by <maths id="math0014"><math display="inline"><msqrt><mi>SNR</mi><mfenced><mi>k</mi></mfenced></msqrt><mo>,</mo></math><img file="EP1068704B1_D0014.tif" /></maths> the optimal spectral profile is obtained by minimizing the cost function <i>F</i>, where: <maths id="math0015" num="(10)"><math display="block"><mi>F</mi><mfenced><mi mathvariant="bold">x</mi></mfenced><mo>=</mo><mo>-</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><msub><mi>M</mi><mn>1</mn></msub></mrow><msub><mi>M</mi><mn>2</mn></msub></munderover></mstyle><msqrt><mfrac><mrow><msub><mi>b</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi>x</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi mathvariant="bold">a</mi><mi>k</mi><mi>T</mi></msubsup><mo>⁢</mo><mi mathvariant="bold">x</mi></mrow></mfrac></msqrt><mn>.</mn></math><img file="EP1068704B1_D0015.tif" /></maths> This minimization is performed over: <maths id="math0016" num="(11)"><math display="block"><mi>X</mi><mo>=</mo><mrow><mo>{</mo><mi mathvariant="bold">x</mi><mo>∈</mo><msup><mi mathvariant="normal">R</mi><mi>N</mi></msup><mo>:</mo><mo>‖</mo><mi mathvariant="bold">x</mi><mo>‖</mo><mo>=</mo><mn>1</mn><mo>,</mo><mspace width="1em" /><msub><mi>x</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>N</mi></mrow></math><img file="EP1068704B1_D0016.tif" /></maths> and can be accomplished by any one of a variety of constrained optimization strategies, as discussed in Bertsekas, <u>Nonlinear Programming</u>, Athena Scientific, Belmont, MA, 1995. A median filter may be applied to the output of the optimization algorithm to smooth the resulting desired spectral response.

0051Let <maths id="math0017"><math display="inline"><mi mathvariant="bold">A</mi><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msubsup><mi mathvariant="bold">H</mi><mfenced><mn>0</mn><mo>:</mo><msub><mi>m</mi><mi>h</mi></msub><mo>+</mo><msub><mi>m</mi><mi>g</mi></msub><mo>-</mo><mn>2</mn></mfenced><msub><mi>m</mi><mi>g</mi></msub></msubsup></math><img file="EP1068704B1_D0017.tif" /></maths> and <maths id="math0018"><math display="inline"><mi mathvariant="bold">B</mi><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msubsup><mi mathvariant="bold">H</mi><mfenced><mi>m</mi><mo>:</mo><mi>m</mi><mo>+</mo><msub><mi>m</mi><mi>l</mi></msub><mo>-</mo><mn>1</mn></mfenced><msub><mi>m</mi><mi>g</mi></msub></msubsup><mn>.</mn></math><img file="EP1068704B1_D0018.tif" /></maths> The energy constraint in equation (3) can be written as: <maths id="math0019" num="(12)"><math display="block"><msup><mi>S</mi><mi>m</mi></msup><mo>=</mo><mfenced open="{" close="}"><mi mathvariant="normal">g</mi><mo>∈</mo><msup><mi>R</mi><msub><mi>m</mi><mi>g</mi></msub></msup><mo>:</mo><mfrac><mrow><msup><mi mathvariant="bold">g</mi><mi>T</mi></msup><mo>⁢</mo><msup><mi mathvariant="bold">B</mi><mi>T</mi></msup><mo>⁢</mo><mi mathvariant="bold">Bg</mi></mrow><mrow><msup><mi mathvariant="bold">g</mi><mi>T</mi></msup><mo>⁢</mo><msup><mi mathvariant="bold">A</mi><mi>T</mi></msup><mo>⁢</mo><mi mathvariant="bold">Ag</mi></mrow></mfrac><mo>≥</mo><mi>α</mi></mfenced><mn>.</mn></math><img file="EP1068704B1_D0019.tif" /></maths>

0052Matrix A has full column rank, since it corresponds to a full convolution, so <maths id="math0020"><math display="inline"><msub><mi mathvariant="bold">R</mi><mi>A</mi></msub><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msup><mi mathvariant="bold">A</mi><mi>T</mi></msup><mo>⁢</mo><mi mathvariant="bold">A</mi></math><img file="EP1068704B1_D0020.tif" /></maths> is invertible. Let <maths id="math0021"><math display="inline"><msub><mi mathvariant="bold">R</mi><mi>B</mi></msub><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msup><mi mathvariant="bold">B</mi><mi>T</mi></msup><mo>⁢</mo><mi mathvariant="bold">B</mi><mn>.</mn></math><img file="EP1068704B1_D0021.tif" /></maths> Defining <maths id="math0022"><math display="inline"><mi mathvariant="bold">q</mi><mo>=</mo><msubsup><mi mathvariant="bold">R</mi><mi>A</mi><mn>0.5</mn></msubsup><mo>⁢</mo><mi mathvariant="bold">h</mi><mo>,</mo></math><img file="EP1068704B1_D0022.tif" /></maths> where <maths id="math0023"><math display="inline"><msubsup><mi mathvariant="bold">R</mi><mi>A</mi><mn>0.5</mn></msubsup></math><img file="EP1068704B1_D0023.tif" /></maths> is the square root of the positive definite matrix <b>R</b><i>_A</i>, the energy constraint set can be written in terms of <b>q</b> as: <maths id="math0024" num="(13)"><math display="block"><mi>Q</mi><mo>⁢</mo><munder><mi mathvariant="normal">Δ</mi><mo>=</mo></munder><mfenced open="{" close="}"><mi mathvariant="bold">q</mi><mo>∈</mo><msup><mi>R</mi><msub><mi>m</mi><mi>g</mi></msub></msup><mo>:</mo><mfrac><mrow><msup><mi mathvariant="bold">q</mi><mi>T</mi></msup><mo>⁢</mo><msubsup><mi mathvariant="bold">R</mi><mi>A</mi><mrow><mo>-</mo><mn>0.5</mn></mrow></msubsup><mo>⁢</mo><msub><mi mathvariant="bold">R</mi><mi>B</mi></msub><mo>⁢</mo><msubsup><mi mathvariant="bold">R</mi><mi>A</mi><mrow><mo>-</mo><mn>0.5</mn></mrow></msubsup><mo>⁢</mo><mi mathvariant="bold">q</mi></mrow><mrow><msup><mi mathvariant="bold">q</mi><mi>T</mi></msup><mo>⁢</mo><mi mathvariant="bold">q</mi></mrow></mfrac><mo>≥</mo><mi>α</mi></mfenced><mn>.</mn></math><img file="EP1068704B1_D0024.tif" /></maths>

0053The next step is to reduce the dimensionality, i.e., the number of variables to search over in the optimization process. For example, in a video digital subscriber line(VDSL) application, an impulse shortening filter having a few hundred taps may be required. Searching over a variable space so large is difficult and impractical. Instead, the optimization is performed by searching over a cleverly chosen lower-dimensional subset of the variable space. This simplification in the optimization process may be done without significant reduction in the performance of the communication system.

0054The reduction in dimensionality is accomplished by a transformation of variables. Let <maths id="math0025"><math display="inline"><mi>C</mi><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msubsup><mi mathvariant="bold">R</mi><mi>A</mi><mrow><mo>-</mo><mn>0.5</mn></mrow></msubsup><mo>⁢</mo><msub><mi mathvariant="bold">R</mi><mi>B</mi></msub><mo>⁢</mo><msubsup><mi mathvariant="bold">R</mi><mi>A</mi><mrow><mo>-</mo><mn>0.5</mn></mrow></msubsup><mn>.</mn></math><img file="EP1068704B1_D0025.tif" /></maths> Let <i>S</i> = U∑U<i>^T</i> be the singular-value decomposition of <b>C</b>, where ∑ is a diagonal matrix whose (<i>i</i>, <i>i</i>)th element is σ<i>_i</i>, and where the σ<i>_i</i> are arranged in descending order. If <i>σ</i>₁ &lt; <i>α</i>, there is no feasible solution corresponding to delay <b><i>m.</i></b> If <i>σ</i>₁ ≥ <i>α</i>, let <maths id="math0026" num="(14)"><math display="block"><mi mathvariant="normal">U</mi><mo mathvariant="normal">=</mo><mfenced open="[" close="]"><msub><mi mathvariant="normal">U</mi><mn mathvariant="normal">1</mn></msub><mo>⁢</mo><msub><mi mathvariant="normal">U</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>;</mo><mspace width="1em" /><mi mathvariant="normal">Σ</mi><mo mathvariant="normal">=</mo><mfenced open="[" close="]"><mstyle displaystyle="false"><mtable><mtr><mtd><munder><mo mathvariant="normal">∑</mo><mn mathvariant="normal">1</mn></munder></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mstyle displaystyle="false"><munder><mo mathvariant="normal">∑</mo><mn mathvariant="normal">2</mn></munder><mspace width="1em" /></mstyle></mtd></mtr></mtable></mstyle></mfenced><mo mathvariant="normal">,</mo></math><img file="EP1068704B1_D0026.tif" /></maths> where U₁ has size (<i>m_g</i>, <i>m_d</i>) and ∑₁ has size (<i>m_d</i>, <i>m_d</i>) for some <i>m_d</i>. These equations define the dimension of the range space of the matrix <b>U₁</b>, over which the search is confined. The dimension <i>m_d</i> may be chosen either to include all <i>σ_i</i> greater than some threshold <i>β, β</i> &lt; <i>α</i>, or <i>m_d</i> may be a fixed number. Simulations indicate that <i>m_d</i> can be less than <i>m_g</i> by more than an order of magnitude without significantly affecting communication system performance.

0055The dimensionality reduction is achieved by a further transformation of variables: q = U₁v. The energy constraint set now becomes: <maths id="math0027" num="(15)"><math display="block"><mi>V</mi><mo>⁢</mo><munder><mi>Δ</mi><mo>=</mo></munder><mfenced open="{" close="}"><mi mathvariant="bold">v</mi><mo>∈</mo><msup><mi mathvariant="normal">R</mi><msub><mi>m</mi><mi>d</mi></msub></msup><mo>:</mo><mfrac><mrow><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mstyle displaystyle="false"><munder><mo>∑</mo><mn>1</mn></munder><mo>⁢</mo><mi mathvariant="bold">v</mi></mstyle></mrow><mrow><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><mi mathvariant="bold">v</mi></mrow></mfrac><mo>≥</mo><mi>α</mi></mfenced><mo>=</mo><mrow><mo>{</mo><mi mathvariant="bold">v</mi><mo>∈</mo><msup><mi mathvariant="normal">R</mi><msub><mi>m</mi><mi>d</mi></msub></msup><mo>:</mo><mi mathvariant="bold">v</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>⁢</mo><msup><mrow><mo>‖</mo><mi mathvariant="bold">v</mi><mo>‖</mo></mrow><mn>2</mn></msup><mo>-</mo><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mstyle displaystyle="false"><munder><mo>∑</mo><mn>1</mn></munder><mo>⁢</mo><mi mathvariant="bold">v</mi><mo>≤</mo><mn>0</mn></mstyle></mrow></math><img file="EP1068704B1_D0027.tif" /></maths> using the identity <maths id="math0028"><math display="inline"><msubsup><mi mathvariant="normal">U</mi><mn mathvariant="normal">1</mn><mi>T</mi></msubsup><mo>⁢</mo><msub><mi>CU</mi><mn mathvariant="normal">1</mn></msub><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">Σ</mi><mn mathvariant="normal">1</mn></msub><mn mathvariant="normal">.</mn></math><img file="EP1068704B1_D0028.tif" /></maths>

0056Next, the cost function is expressed in terms of <b>v</b>. For a particular <b>v,</b> the corresponding <b>g</b> is given by <maths id="math0029"><math display="inline"><mi mathvariant="normal">g</mi><mo mathvariant="normal">=</mo><msubsup><mi mathvariant="normal">R</mi><mi>A</mi><mrow><mo mathvariant="normal">-</mo><mn mathvariant="normal">0.5</mn></mrow></msubsup><mo>⁢</mo><msub><mi mathvariant="normal">U</mi><mn mathvariant="normal">1</mn></msub><mo>⁢</mo><mi mathvariant="normal">v</mi><mo mathvariant="normal">,</mo></math><img file="EP1068704B1_D0029.tif" /></maths>, which leads to: <maths id="math0030" num="(16)"><math display="block"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>G</mi><mfenced><msub><mi>ω</mi><mn>1</mn></msub></mfenced></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mi>G</mi><mfenced><msub><mi>ω</mi><mi>N</mi></msub></mfenced></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold">Fg</mi><mo mathvariant="bold">;</mo><mspace width="1em" /><mi mathvariant="bold">F</mi><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mo>⁢</mo><msub><mi>ω</mi><mn>1</mn></msub></mrow></msup></mtd><mtd><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mo>⁢</mo><mn>2</mn><mo>⁢</mo><msub><mi>ω</mi><mn>1</mn></msub></mrow></msup></mtd><mtd><mo>⋯</mo></mtd><mtd><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mo>⁢</mo><mfenced><msub><mi>m</mi><mi>g</mi></msub><mo>-</mo><mn>1</mn></mfenced><mo>⁢</mo><msub><mi>ω</mi><mn>1</mn></msub></mrow></msup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mo>⁢</mo><msub><mi>ω</mi><mn>1</mn></msub></mrow></msup></mtd><mtd><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mo>⁢</mo><mn>2</mn><mo>⁢</mo><msub><mi>ω</mi><mi>N</mi></msub></mrow></msup></mtd><mtd><mo>…</mo></mtd><mtd><msup><mi>e</mi><mrow><mo>-</mo><mi>j</mi><mo>⁢</mo><mfenced><msub><mi>m</mi><mi>g</mi></msub><mo>-</mo><mn>1</mn></mfenced><mo>⁢</mo><msub><mi>ω</mi><mi>N</mi></msub></mrow></msup></mtd></mtr></mtable></mfenced></math><img file="EP1068704B1_D0030.tif" /></maths> Let <maths id="math0031"><math display="inline"><mi mathvariant="bold">D</mi><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msubsup><mi>FAR</mi><mi>A</mi><mrow><mo>-</mo><mn>0.5</mn></mrow></msubsup><mo>⁢</mo><msub><mi mathvariant="normal">U</mi><mn>1</mn></msub><mn>.</mn></math><img file="EP1068704B1_D0031.tif" /></maths> Let <maths id="math0032"><math display="inline"><msup><mi mathvariant="normal">D</mi><mi>R</mi></msup><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><mi>real</mi><mfenced><mi>D</mi></mfenced></math><img file="EP1068704B1_D0032.tif" /></maths> real (<i>D</i>) and <maths id="math0033"><math display="inline"><msup><mi mathvariant="normal">D</mi><mi>I</mi></msup><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><mi>imag</mi><mfenced><mi>D</mi></mfenced><mn>.</mn></math><img file="EP1068704B1_D0033.tif" /></maths>. D<i>^R</i> and D<i>^I</i> are real matrices of size (<i>N</i>, <i>m_d</i>). Let <maths id="math0034"><math display="inline"><msubsup><mi mathvariant="normal">d</mi><mrow><mi>R</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup><mo>,</mo><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>N</mi></math><img file="EP1068704B1_D0034.tif" /></maths> and <maths id="math0035"><math display="inline"><msubsup><mi mathvariant="normal">d</mi><mrow><mi>I</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup><mo>,</mo><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>N</mi></math><img file="EP1068704B1_D0035.tif" /></maths> be the rows of <i>D^R</i> and D<i>^I</i>, respectively. Then: <maths id="math0036"><math display="block"><mtable columnalign="left"><mtr><mtd><mfenced open="|" close="|"><mi>G</mi><mfenced><msub><mi>ω</mi><mi>n</mi></msub></mfenced></mfenced></mtd><mtd><mo>=</mo><msqrt><msup><mfenced><msubsup><mi mathvariant="bold">d</mi><mrow><mi>R</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup><mo>⁢</mo><mi mathvariant="bold">v</mi></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msubsup><mi mathvariant="bold">d</mi><mrow><mi>I</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup><mo>⁢</mo><mi mathvariant="bold">v</mi></mfenced><mn>2</mn></msup></msqrt></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><msqrt><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><mfenced><msub><mi mathvariant="bold">d</mi><mrow><mi>R</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>⁢</mo><msubsup><mi mathvariant="bold">d</mi><mrow><mi>R</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup><mo>+</mo><msub><mi mathvariant="bold">d</mi><mrow><mi>I</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>⁢</mo><msubsup><mi mathvariant="bold">d</mi><mrow><mi>I</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup></mfenced><mo>⁢</mo><mi mathvariant="bold">v</mi></msqrt><mo>=</mo><msqrt><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><msub><mi mathvariant="bold">T</mi><mi>n</mi></msub><mo>⁢</mo><mi mathvariant="bold">v</mi></msqrt></mtd></mtr></mtable></math><img file="EP1068704B1_D0036.tif" /></maths> where <maths id="math0037"><math display="inline"><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><msub><mi mathvariant="normal">d</mi><mrow><mi>R</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>⁢</mo><msubsup><mi mathvariant="normal">d</mi><mrow><mi>R</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup><mo>+</mo><msub><mi mathvariant="normal">d</mi><mrow><mi>I</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>⁢</mo><msubsup><mi mathvariant="normal">d</mi><mrow><mi>I</mi><mo>,</mo><mi>n</mi></mrow><mi>T</mi></msubsup><mn>.</mn></math><img file="EP1068704B1_D0037.tif" /></maths> These definitions result in: <maths id="math0038" num="(17)"><math display="block"><msub><mi>J</mi><mi>o</mi></msub><mfenced><mi mathvariant="bold">v</mi></mfenced><mo>⁢</mo><mover><mo>=</mo><mi>Δ</mi></mover><mo>⁢</mo><mi>j</mi><mfenced><mi>g</mi><mfenced><mi mathvariant="bold">v</mi></mfenced></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><msup><mfenced><msqrt><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>⁢</mo><mi mathvariant="bold">v</mi></msqrt><mo>-</mo><msub><mi>G</mi><mi>d</mi></msub><mfenced><msub><mi>ω</mi><mi>n</mi></msub></mfenced></mfenced><mn>2</mn></msup></math><img file="EP1068704B1_D0038.tif" /></maths> The gradient and Hessian of <i>J_o</i> are: <maths id="math0039" num="(18)"><math display="block"><msub><mo>∇</mo><msub><mi>J</mi><mi>o</mi></msub></msub><mfenced><mi mathvariant="bold">v</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><mn>2</mn><mrow><mo>[</mo><mfenced><mn>1</mn><mo>-</mo><mfrac><mrow><msub><mi>G</mi><mi>d</mi></msub><mfenced><msub><mi>ω</mi><mi>n</mi></msub></mfenced></mrow><msqrt><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>⁢</mo><mi mathvariant="bold">v</mi></msqrt></mfrac></mfenced><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>⁢</mo><mi mathvariant="bold">v</mi></mrow></math><img file="EP1068704B1_D0039.tif" /></maths><maths id="math0040" num="(19)"><math display="block"><msub><mi>H</mi><msub><mi>J</mi><mi>o</mi></msub></msub><mfenced><mi mathvariant="bold">v</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><mn>2</mn><mo>⁢</mo><mfenced open="[" close="]"><mfenced><mn>1</mn><mo>-</mo><mfrac><mrow><msub><mi>G</mi><mi>d</mi></msub><mfenced><msub><mi>ω</mi><mi>n</mi></msub></mfenced></mrow><msqrt><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>⁢</mo><mi mathvariant="bold">v</mi></msqrt></mfrac></mfenced><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>G</mi><mi>d</mi></msub><mfenced><msub><mi>ω</mi><mi>n</mi></msub></mfenced></mrow><mfenced><msqrt><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>⁢</mo><mi mathvariant="bold">v</mi></msqrt></mfenced></mfrac><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub><mo>⁢</mo><msup><mi mathvariant="bold">vv</mi><mi>T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">Γ</mi><mi>n</mi></msub></mfenced></math><img file="EP1068704B1_D0040.tif" /></maths>

0057The projection <b>P<i>_v</i>(y)</b> of any y ∈ R<i>^m</i>d on to <i>V</i> is defined to be <maths id="math0041" num="(20)"><math display="block"><msub><mi>P</mi><mi>V</mi></msub><mfenced><mi>y</mi></mfenced><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="normal">Δ</mi></mover><mo>⁢</mo><mi>arg</mi><mo>⁢</mo><munder><mi>min</mi><mrow><mi mathvariant="bold-italic">ν</mi><mo>∈</mo><mi>V</mi></mrow></munder><mo>⁢</mo><msup><mrow><mo>‖</mo><mi mathvariant="bold">y</mi><mo mathvariant="bold">-</mo><mi mathvariant="bold">v</mi><mo>‖</mo></mrow><mn>2</mn></msup></math><img file="EP1068704B1_D0041.tif" /></maths>

0058To develop an algorithm to optimize the cost function <i>J_o</i> over <i>V</i>, an expression must be derived for <b><i>P_v</i>(y).</b> There is no closed-form expression for this projection operator, however there is a very efficient algorithm to compute it. If y ∈ V, <i>P_v</i>(y) = y. If not, the projection onto <i>V</i> is the same as the projection on to its boundary, defined by: <maths id="math0042" num="(21)"><math display="block"><mi>boundary</mi><mspace width="1em" /><mfenced><mi>V</mi></mfenced><mo>⁢</mo><munder><mi>Δ</mi><mo>=</mo></munder><mrow><mo>{</mo><mi mathvariant="bold">v</mi><mo>∈</mo><msup><mi mathvariant="normal">R</mi><mi>m</mi></msup><mo>⁢</mo><mi>d</mi><mo>:</mo><mi mathvariant="bold">v</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>⁢</mo><msup><mrow><mo>‖</mo><mi mathvariant="bold">v</mi><mo>‖</mo></mrow><mn>2</mn></msup><mo>-</mo><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mstyle displaystyle="false"><munder><mo>∑</mo><mn>1</mn></munder><mo>⁢</mo><mi mathvariant="bold">v</mi></mstyle><mo>=</mo><mn>0</mn></mrow></math><img file="EP1068704B1_D0042.tif" /></maths> The latter may be accomplished using LaGrange multipliers as follows. Defining a modified cost function S as: <maths id="math0043" num="(22)"><math display="block"><mi>S</mi><mfenced><mi mathvariant="bold">v</mi></mfenced><mo>⁢</mo><mover><mo>=</mo><mi mathvariant="italic">Δ</mi></mover><mo>⁢</mo><msup><mrow><mo>‖</mo><mi mathvariant="bold">y</mi><mo mathvariant="bold">-</mo><mi mathvariant="bold">v</mi><mo>‖</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>μ</mi><mo>⁢</mo><mfenced><mi>α</mi><mo>⁢</mo><msup><mrow><mo>‖</mo><mi mathvariant="bold">v</mi><mo>‖</mo></mrow><mn>2</mn></msup><mo>-</mo><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mo>⁢</mo><msub><mi mathvariant="normal">Σ</mi><mn>1</mn></msub><mo>⁢</mo><mi mathvariant="bold">v</mi></mfenced></math><img file="EP1068704B1_D0043.tif" /></maths> The equation ∇<b><i>_S</i>(v)</b>= 0 must be solved on the boundary of <i>V</i>. This reduces to the following simultaneous equations: <maths id="math0044" num="(23)"><math display="block"><mfenced open="[" close="]"><mfenced><mn>1</mn><mo>+</mo><mi>μ</mi><mo>⁢</mo><mi>α</mi></mfenced><mo>⁢</mo><mi mathvariant="bold">I</mi><mo>-</mo><mi>μ</mi><mstyle displaystyle="false"><munder><mo mathvariant="normal">∑</mo><mn>1</mn></munder></mstyle></mfenced><mo>⁢</mo><mi mathvariant="bold">v</mi><mo>=</mo><mn>0</mn></math><img file="EP1068704B1_D0044.tif" /></maths><maths id="math0045" num="(24)"><math display="block"><mi>α</mi><mo>⁢</mo><msup><mrow><mo>‖</mo><mi mathvariant="bold">v</mi><mo>‖</mo></mrow><mn>2</mn></msup><mo>-</mo><msup><mi mathvariant="bold">v</mi><mi>T</mi></msup><mstyle displaystyle="false"><munder><mo>∑</mo><mn>1</mn></munder><mo>⁢</mo><mi mathvariant="bold">v</mi></mstyle><mo>=</mo><mn>0</mn></math><img file="EP1068704B1_D0045.tif" /></maths> Solving for v in terms of <i>µ</i> from (23) and substituting in (24), the following equation is obtained for <i>µ</i>: <maths id="math0046" num="(25)"><math display="block"><msup><mi mathvariant="bold">y</mi><mi>T</mi></msup><mo>⁢</mo><mi mathvariant="italic">diag</mi><mo>⁢</mo><mfenced><mfrac><mrow><mi>α</mi><mo>-</mo><msub><mi>σ</mi><mi>i</mi></msub></mrow><msup><mfenced open="[" close="]"><mn>1</mn><mo>+</mo><mi>μ</mi><mo>⁢</mo><mfenced><mi>α</mi><mo>-</mo><msub><mi>σ</mi><mi>i</mi></msub></mfenced></mfenced><mn>2</mn></msup></mfrac></mfenced><mo>⁢</mo><mi mathvariant="bold">y</mi><mo>=</mo><mn>0</mn></math><img file="EP1068704B1_D0046.tif" /></maths> Equation (25) may be written as a polynomial equation of order 2<i>m_d</i> in <i>_µ</i>. The polynomial equation must be solved for the real and positive roots, which may be done using one of the variety of efficient root-finding algorithms in the literature. One of the real, positive roots must be chosen by enumeration so that, when substituted into the expression for v, it leads to the smallest ∥<b>y</b> - <b>v</b>∥².

0059Since the gradient and Hessian of the cost function <i>J_o</i> and the projection operator onto <i>V</i> are available, the optimization can be done very efficiently. For example, the penalty method may be used, as described in Bertsekas, <u>Nonlinear Programming</u>, Athena Scientific, Belmont, MA, 1995, or an iterative strategy consisting of gradient descent followed by projection onto <i>V</i>. Both techniques have been tested and perform well.

0060The SCISF may be configured, by determining the filter coefficients during a training or adaptation period. The SCISF filters the output {<i>y_k</i>} of the receiver A/D 80. The coefficients are selected using an algorithm that minimizes the squared error between a reference sequence {<i>u_k</i>} generated by the receiver and the output of the SCISF {û<i>_k</i>}. The SCISF may be a finite impulse response (FIR) filter or an infinite impulse response (IIR) filter. The SCISF may be trained following activation of the communication system or periodically during operation of the system to compensate for variations in the channel noise profile.

0061The training may be performed using a variation of one of the classical adaptive algorithms, such as least mean squares (LMS), normalized LMS, or recursive least squares (RLS). For example, the following algorithm is a version of the normalized LMS algorithm in which <i>b</i>₀,...,<i>b</i>_N and <i>a</i>₁..., <i>a_N</i> are the FIR and IIR parts of a SCISF having an impulse response, g. The z-transform G(z) is: <maths id="math0047" num="(26)"><math display="block"><mi>G</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><mo>⁢</mo><msup><mi>z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>b</mi><mi>N</mi></msub><mo>⁢</mo><msup><mi>z</mi><mrow><mo>-</mo><mi>N</mi></mrow></msup></mrow><mrow><mn>1</mn><mo>-</mo><msub><mi>a</mi><mn>1</mn></msub><mo>⁢</mo><msup><mi>z</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mo>⋯</mo><mo>-</mo><msub><mi>a</mi><mi>N</mi></msub><mo>⁢</mo><msup><mi>z</mi><mrow><mo>-</mo><mi>N</mi></mrow></msup></mrow></mfrac></math><img file="EP1068704B1_D0047.tif" /></maths> The adaptation of coefficients <i>a_i</i> and <i>b_i</i> is defined in the following equations, in which <i>a_i</i>(<i>k</i>) and <i>b_i</i>(<i>k</i>) are the values of these coefficients during the <i>k</i>^th iteration. The parameter <i>_µ</i> is a predetermined constant with a value of 0.4. <maths id="math0048" num="(27)"><math display="block"><msub><mover><mi>u</mi><mo>^</mo></mover><mi>k</mi></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></munderover></mstyle><msub><mi>b</mi><mi>i</mi></msub><mfenced><mi>k</mi></mfenced><mo>⁢</mo><msub><mi>y</mi><mrow><mi>k</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><msub><mi>a</mi><mi>i</mi></msub><mfenced><mi>k</mi></mfenced><mo>⁢</mo><msub><mover><mi>u</mi><mo>^</mo></mover><mrow><mi>k</mi><mo>-</mo><mi>i</mi></mrow></msub></math><img file="EP1068704B1_D0048.tif" /></maths><maths id="math0049" num="(28)"><math display="block"><msub><mi>α</mi><mi>k</mi></msub><mo>=</mo><mfenced open="[" close="]"><msub><mi>b</mi><mn>0</mn></msub><mfenced><mi>k</mi></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>b</mi><mi>N</mi></msub><mfenced><mi>k</mi></mfenced></mfenced></math><img file="EP1068704B1_D0049.tif" /></maths><maths id="math0050" num="(29)"><math display="block"><msub><mi>β</mi><mi>k</mi></msub><mo>=</mo><mfenced open="[" close="]"><msub><mi>a</mi><mn>1</mn></msub><mfenced><mi>k</mi></mfenced><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>N</mi></msub><mfenced><mi>k</mi></mfenced></mfenced></math><img file="EP1068704B1_D0050.tif" /></maths><maths id="math0051" num="(30)"><math display="block"><msub><mi mathvariant="normal">d</mi><mi>k</mi></msub><mo>=</mo><mfenced open="[" close="]"><msub><mi>y</mi><mi>k</mi></msub><msub><mi>y</mi><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>…</mo><msub><mi>y</mi><mrow><mi>k</mi><mo>-</mo><mi>N</mi></mrow></msub></mfenced></math><img file="EP1068704B1_D0051.tif" /></maths><maths id="math0052" num="(31)"><math display="block"><msub><mi mathvariant="normal">c</mi><mi>k</mi></msub><mo>=</mo><mfenced open="[" close="]"><msub><mover><mi>u</mi><mo>^</mo></mover><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msub><msub><mover><mi>u</mi><mo>^</mo></mover><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></msub><mo>…</mo><msub><mover><mi>u</mi><mo>^</mo></mover><mrow><mi>k</mi><mo>-</mo><mi>N</mi></mrow></msub></mfenced></math><img file="EP1068704B1_D0052.tif" /></maths><maths id="math0053" num="(32)"><math display="block"><msub><mi>e</mi><mi>k</mi></msub><mo>⁢</mo><munder><mi mathvariant="normal">Δ</mi><mo>=</mo></munder><mo>⁢</mo><msub><mi>u</mi><mi>k</mi></msub><mo>-</mo><msub><mover><mi>u</mi><mo>^</mo></mover><mi>k</mi></msub></math><img file="EP1068704B1_D0053.tif" /></maths><maths id="math0054" num="(33)"><math display="block"><msub><mi>α</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>α</mi><mi>k</mi></msub><mo>+</mo><mi>μ</mi><mo>⁢</mo><mfrac><mrow><msub><mi>e</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi mathvariant="normal">d</mi><mi>k</mi></msub></mrow><msup><mrow><mo>‖</mo><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">k</mi></msub><mo>‖</mo></mrow><mn>2</mn></msup></mfrac></math><img file="EP1068704B1_D0054.tif" /></maths><maths id="math0055" num="(34)"><math display="block"><msub><mi>β</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>β</mi><mi>k</mi></msub><mo>+</mo><mi>μ</mi><mo>⁢</mo><mfrac><mrow><msub><mi>e</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi mathvariant="normal">c</mi><mi>k</mi></msub></mrow><msup><mrow><mo>‖</mo><msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">k</mi></msub><mo>‖</mo></mrow><mn>2</mn></msup></mfrac></math><img file="EP1068704B1_D0055.tif" /></maths>

0062In a first example, the coefficients of the SCISF are determined during an initial training period following the activation of the communication system using the LMS algorithm described above. A predetermined sequence of bits <i>x_k</i> is input to the transmitter 12. The sequence of bits results in a sequence of real numbers {<i>x_k</i>} at the input of the D/A 60. As shown in <figref idref="f0001">Figs. 1</figref> and <figref idref="f0007">10</figref>, the transmitted signal is filtered and noise-corrupted by the transmission channel 70, resulting in a received sequence {<i>y_k</i>} at the output of the A/D 80 in the receiver 14. The SCISF 90 filters and transforms the received sequence {<i>y_k</i>} into an output sequence {<i>x̂_k</i>}. The output sequence {<i>x̂_k</i>} is compared (in a signal comparator 205) to the predetermined sequence x_k, which is stored in memory 210 in the receiver. The comparison results in an error signal <i>e_k</i> that is input to the LMS algorithm processor 215 along with sequence {<i>x̂_k</i>}.

0063The training process determines coefficients for the SCISF so that the output {<i>x̂_k</i>} matches the predetermined sequence {<i>x_k</i>} as closely as possible in a least squares sense, i.e., the mean square error between the output and the predetermined sequence is minimized. During the training process, the coefficients of the SCISF converge to values that enable the SCISF to reduce ISI and additive noise. The resulting SCISF matches <i>P</i>₁(<i>ω</i>) in the frequency domain in a least squares sense, where: <maths id="math0056" num="(35)"><math display="block"><msub><mi>P</mi><mn>1</mn></msub><mfenced><mi>ω</mi></mfenced><mo>=</mo><mfrac><mrow><msub><mi>S</mi><mi>x</mi></msub><mfenced><mi>ω</mi></mfenced><mo>⁢</mo><mi>H</mi><mo>*</mo><mfenced><mi>ω</mi></mfenced></mrow><mrow><msub><mi>S</mi><mi>x</mi></msub><mfenced><mi>ω</mi></mfenced><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>H</mi><mfenced><mi>ω</mi></mfenced></mfenced><mn>2</mn></msup><mo>+</mo><msub><mi>S</mi><mi>η</mi></msub><mfenced><mi>ω</mi></mfenced></mrow></mfrac></math><img file="EP1068704B1_D0056.tif" /></maths> In the equation above, <i>S_x</i>(<i>ω</i>) is the power-spectral density at the input of the transmitter D/A 60, <i>S_η</i> (<i>ω</i>) is the power spectral density of the additive noise at the output of the A/D 80, and H(ω) is the frequency response of the effective discrete-time impulse response (EDIR) of the transmission channel 70, transmit filter 65, and receive filter 75 measured between the input of the transmitter D/A 60 and the output of the receiver A/D 80.

0064Upon completion of the initial training of the SCISF, the SCISF coefficients are fixed and the frequency domain equalizer (FEQ 130) of the receiver is trained using standard techniques for DMT receivers. Following the training of the FEQ, the SCISF can be periodically trained or adapted during operation of the communication system. Since it is not efficient to repeatedly transmit a predetermined bit sequence during operation of the communication system, the periodic training process uses transmitted communication data to generate the reference and output sequences, as described below.

0065During operation of the communication system, a sequence of communication data bits x_k is input to the transmitter 12 (see <figref idref="f0001">Fig. 1</figref>). The sequence of data bits results in a sequence of real numbers {<i>x_k</i>} at the input of the D/A 60. As shown in <figref idref="f0001">Figs. 1</figref> and <figref idref="f0008">11</figref>, the transmitted signal is filtered and noise-corrupted by the transmission channel 70, resulting in a received sequence {<i>y_k</i>} at the output of the A/D 80 in the receiver 14. The SCISF 90 filters and transforms the received sequence {<i>y_k</i>} into an output sequence {<i>x̂'_k</i>}.

0066The received sequence {<i>y_k</i>} is also input to a delay 305 and then to a secondary SCISF 300 that has the same coefficients as the primary SCISF 90 following the initial training. Such a configuration allows periodic training to be performed without disruption to the operation of the communication system. The secondary SCISF 300 is periodically or continuously trained using an algorithm similar to that used for the initial training. The new coefficients of the secondary SCISF 300 are periodically copied to the primary SCISF 90.

0067In an initial training process, the output sequence {<i>x̂_k</i>} of the secondary SCISF 300 would be compared (in a signal comparator 205) to a predetermined sequence x_k stored in memory. However, as discussed above, a sequence of data communication bits is used as a reference sequence for training rather than a predetermined sequence. As such, the receiver must have a way of generating a reference signal to compare to the output of the SCISF.

0068To compute the reference sequence, the receiver essentially duplicates the encoding and modulation processes of the transmitter using the output of the decoder 140. Because the initial training has already been performed, the SCISF 90 output {<i>x̂'_k</i>} matches the predetermined sequence {<i>x_k</i>} closely and ISI and additive noise are minimized. Hence, the data output of the decoder 140 closely matches the transmitted sequence of communication data bits <i>x_k</i>. The data bits are input to an encoder 320, an IDFT 330, a parallel to serial converter 340, and a prefix adder 350 similar to those in the transmitter. The output sequence {<i>x_k</i>} of this chain is input to the LMS algorithm processor 215 and used as a reference sequence in the training algorithm.

0069The output sequence {<i>x̂_k</i>} of the secondary SCISF 300 is compared (in a signal comparator 205) to the reference sequence {<i>x_k</i>}, which is output by the encoding/modulation chain (320, 330, 340 and 350). As noted above, the received sequence {<i>y_k</i>} passes through a delay 305 before being input to the secondary SCISF 300. The delay 305 compensates for the processing delay in the demodulation/decoding chain and the encoding/modulation chain. The comparison results in an error signal <i>e_k</i> that is input to the LMS algorithm processor 215. The training process determines coefficients for the secondary SCISF 300 so that the output {<i>x̂_k</i>} matches the reference sequence {<i>x_k</i>} as closely as possible in a least squares sense, i.e., the mean square error between the output and the reference sequence is minimized. Periodically, the coefficients of the secondary SCISF 300 are copied to the primary SCISF 90.

0070An another example, as shown in <figref idref="f0009">Fig. 12</figref>, the periodic training may be performed with a single SCISF 90. In this configuration, a received sequence {<i>y_k</i>} is output by the A/D 80 in the receiver 14. The SCISF 90 filters and transforms the received sequence {<i>y_k</i>} into an output sequence {<i>x̂'_k</i>}. The received sequence {<i>y_k</i>} is also input to a delay 305. After the received sequence {<i>y_k</i>} passes through the SCISF 90, a data switch 360 is changed from position A to position B, allowing the delayed received sequence to make a second pass through the SCISF 90. An output switch 370 also may be opened, so that data is not output during the training process. In addition, the SCISF coefficients are controlled by the LMS algorithm during the training process.

0071A reference sequence is computed as in the configuration of <figref idref="f0008">Fig. 11</figref>. The data bits are input to an encoder 320, an IDFT 330, a parallel to serial converter 340, and a prefix adder 350. The resulting reference sequence {<i>x_k</i>} is input to the LMS algorithm processor.

0072The output sequence {<i>x̂_k</i>} of the second pass through the SCISF 90 is compared (in signal comparator 205) to the reference sequence. As noted above, the received sequence {<i>y_k</i>} passes through a delay 305 before being input to the SCISF 90 for the second pass. The delay 305 compensates for the processing delay in the demodulation/decoding chain and the encoding/modulation chain. The comparison results in an error signal <i>e_k</i> that is input to the LMS algorithm processor 215. The training process determines coefficients for the SCISF 90 so that the output {<i>x̂_k</i>} matches the reference sequence {<i>x_k</i>} as closely as possible in a least squares sense, i.e., the mean square error between the output and the reference sequence is minimized. The coefficients of the SCISF 90 then are updated to the coefficients determined in the training process.

0073In a second examplary embodiment, the SCISF 90 coefficients are chosen so that the frequency response of the SCISF matches a desired spectral response <i>G_d</i>(<i>ω</i>) that seeks to minimize the effects of noise-bleeding and maximize system bit throughput. The desired spectral response <i>G_d</i> (<i>ω</i>) is determined based on the signal-to-noise ratios observed in the various frequency bins of the DFT 120 in the receiver.

0074For example, an OFDM system may have <i>M</i> tones, <i>N</i> of which (<i>m₁</i> through <i>m_N</i>) are used. The system operates over a channel with analog frequency response <i>H_c</i> (<i>f</i>). Referring again to <figref idref="f0001">Fig. 1</figref>, the analog noise power spectral density at the input of the receiver A/D 80 is <i>S_η</i>(<i>f</i>). Prior to receiver A/D 80, the received analog signal may be filtered by an anti-aliasing filter (i.e., receive filter 75) having a transfer function <i>H_a</i>(<i>f</i>). The effective discrete-time impulse response (EDIR) of the transmission channel of the OFDM system (including the transmit filter 65 and receive filter 75) is <i>h</i>(<i>n</i>). The output of the A/D 80 is input to a SCISF 90 having an impulse response <i>g</i>(<i>n</i>). <i>G</i>(ω) is the spectral response corresponding to <i>g</i>(<i>n</i>).

0075The expected signal energy <i>µ</i>(<i>k</i>) observed in frequency bin k at the output of the DFT120, which has a length of <i>NM</i>, is: <maths id="math0057" num="(36)"><math display="block"><mi>μ</mi><mfenced><mi>k</mi></mfenced><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>D</mi><mi>k</mi></msub><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>H</mi><mfenced><mfrac><mi mathvariant="italic">πk</mi><mi>M</mi></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>G</mi><mfenced><mfrac><mi mathvariant="italic">πk</mi><mi>M</mi></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>;</mo><mi>H</mi><mfenced><mi>ω</mi></mfenced><mo>=</mo><msub><mi>H</mi><mi>c</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced><mo>⁢</mo><msub><mi>H</mi><mi>a</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced></math><img file="EP1068704B1_D0057.tif" /></maths> where <i>C</i>₁ is a constant, 1/T is the sampling frequency and <i>D_k</i> is the transmitted power in frequency bin k. The noise power η(<i>k</i>) in bin k is: <maths id="math0058" num="(37)"><math display="block"><mi>η</mi><mfenced><mi>k</mi></mfenced><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><mfenced open="[" close="]"><msub><mi>S</mi><mi>η</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>G</mi><mfenced><mi>ω</mi></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><msup><mfenced open="|" close="|"><msub><mi>H</mi><mi>a</mi></msub><mfenced><mfrac><mi>ω</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced></mfenced><mn>2</mn></msup></mfenced><mo>*</mo><msub><mrow><mfenced open="[" close="]"><mfrac><mrow><msup><mi>sin</mi><mn>2</mn></msup><mfenced><mi>M</mi><mo>⁢</mo><mi>ω</mi></mfenced></mrow><mrow><msup><mi>sin</mi><mn>2</mn></msup><mfenced><mfrac><mi>ω</mi><mn>2</mn></mfrac></mfenced></mrow></mfrac></mfenced><mo>|</mo></mrow><mrow><mi>ω</mi><mo>=</mo><mfrac><mi mathvariant="italic">πk</mi><mi>M</mi></mfrac></mrow></msub><mo>,</mo></math><img file="EP1068704B1_D0058.tif" /></maths> where C₂ is a constant and * denotes a convolution of the discrete Fourier transforms. If the noise in the bands occupied by unused tones is sufficiently attenuated by the anti-alias filter (receive filter 75), η(<i>k</i>) is approximately: <maths id="math0059" num="(38)"><math display="block"><mi>η</mi><mfenced><mi>k</mi></mfenced><mo>≈</mo><msub><mi>C</mi><mn>3</mn></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>t</mi><mo>=</mo><msub><mi>M</mi><mn>1</mn></msub></mrow><msub><mi>M</mi><mn>2</mn></msub></munderover></mstyle><msub><mi>S</mi><mi>η</mi></msub><mfenced><mfrac><mi>l</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">MT</mi></mrow></mfrac></mfenced><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>G</mi><mfenced><mfrac><mi mathvariant="italic">πl</mi><mi>M</mi></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><msup><mfenced open="|" close="|"><msub><mi>H</mi><mi>a</mi></msub><mfenced><mfrac><mi>l</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><mfenced><mi>τ</mi><mo>⁢</mo><mfenced><mi>k</mi><mo>-</mo><mi>l</mi></mfenced><mo>+</mo><mi>τ</mi><mo>⁢</mo><mfenced><mn>2</mn><mo>⁢</mo><mi>M</mi><mo>-</mo><mi>k</mi><mo>-</mo><mi>l</mi></mfenced></mfenced><mo>,</mo></math><img file="EP1068704B1_D0059.tif" /></maths> where τ(<i>n</i>) is defined as: <maths id="math0060" num="(39)"><math display="block"><mi>τ</mi><mfenced><mi>n</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∫</mo><mrow><mo>-</mo><mfrac><mi>π</mi><mi mathvariant="italic">NM</mi></mfrac></mrow><mfrac><mi>π</mi><mi mathvariant="italic">NM</mi></mfrac></msubsup></mstyle><mfenced open="[" close="]"><mfrac><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo>⁢</mo><mfenced><mi>M</mi><mo>⁢</mo><mfenced><mfrac><mi mathvariant="italic">πn</mi><mi mathvariant="italic">M</mi></mfrac><mo>-</mo><mi>λ</mi></mfenced></mfenced></mrow><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo>⁢</mo><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mfenced><mfrac><mi mathvariant="italic">πn</mi><mi mathvariant="italic">M</mi></mfrac><mo>-</mo><mi>λ</mi></mfenced></mfenced></mrow></mfrac></mfenced><mo>ⅆ</mo><mi>λ</mi><mo>,</mo></math><img file="EP1068704B1_D0060.tif" /></maths><i>m₁</i>...<i>m_N</i> are the used tones, and <i>C</i>₃ is a constant. A vector of frequency magnitudes <i>g</i> is defined as: <maths id="math0061" num="(40)"><math display="block"><mi>g</mi><mo>⁢</mo><munder><mi mathvariant="normal">Δ</mi><mo>=</mo></munder><mfenced open="[" close="]"><mtable><mtr><mtd><mrow><mo>|</mo><msup><mrow><mi>G</mi><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><msub><mi>m</mi><mn>1</mn></msub></mrow><mi>M</mi></mfrac></mfenced><mo>|</mo></mrow><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msup><mtable><mtr><mtd><mfenced open="|" close="|"><mtable><mtr><mtd><mi>G</mi></mtd></mtr></mtable><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><msub><mi>m</mi><mi>N</mi></msub></mrow><mi>M</mi></mfrac></mfenced></mfenced></mtd></mtr></mtable><mn>2</mn></msup></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>G</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mi>G</mi><mi>N</mi></msub></mtd></mtr></mtable></mfenced></math><img file="EP1068704B1_D0061.tif" /></maths> The SNR in frequency bin <i>k</i> is <maths id="math0062"><math display="inline"><mfrac><mrow><msub><mi>r</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi>G</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><mo>⁢</mo><mi>g</mi></mrow></mfrac></math><img file="EP1068704B1_D0062.tif" /></maths> for scalars <i>r_k</i> and vectors <i>s_k</i>. The scalars <i>r_k</i> are defined by: <maths id="math0063" num="(41)"><math display="block"><msub><mi>r</mi><mi>k</mi></msub><mo>⁢</mo><munder><mi mathvariant="normal">Δ</mi><mo>=</mo></munder><mo>⁢</mo><msub><mi>C</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>D</mi><mi>k</mi></msub><mrow><mo>|</mo><msup><mrow><mi>H</mi><mfenced><mfrac><mi mathvariant="italic">πk</mi><mi>M</mi></mfrac></mfenced><mo>|</mo></mrow><mn>2</mn></msup></mrow></math><img file="EP1068704B1_D0063.tif" /></maths> and <i>s_k(l),</i> the <i>l</i>th component <i>of s_k</i> is defined by: <maths id="math0064" num="(42)"><math display="block"><msub><mi>s</mi><mi>k</mi></msub><mfenced><mi>l</mi></mfenced><mo>=</mo><msub><mi>C</mi><mn>3</mn></msub><mo>⁢</mo><msub><mi>S</mi><mi>η</mi></msub><mfenced><mfrac><mi>l</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">MT</mi></mrow></mfrac></mfenced><mo>⁢</mo><msup><mfenced open="|" close="|"><msub><mi>H</mi><mi>a</mi></msub><mfenced><mfrac><mi>l</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πT</mi></mrow></mfrac></mfenced></mfenced><mn>2</mn></msup><mo>⁢</mo><mfenced><mi>τ</mi><mo>⁢</mo><mfenced><mi>k</mi><mo>-</mo><mi>l</mi></mfenced><mo>+</mo><mi>τ</mi><mo>⁢</mo><mfenced><mn>2</mn><mo>⁢</mo><mi>M</mi><mo>-</mo><mi>k</mi><mo>-</mo><mi>l</mi></mfenced></mfenced></math><img file="EP1068704B1_D0064.tif" /></maths>

0076To determine an expression for <i>g</i> that maximizes system bit throughput, the capacity of each frequency bin <i>k</i> is approximated by log(1+ <i>SNR_k</i>). Accordingly, the optimal spectral profile is determined by minimizing the cost function <i>F</i>, where: <maths id="math0065" num="(43)"><math display="block"><mi>F</mi><mfenced><mi>g</mi></mfenced><mo>=</mo><mo>-</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><msub><mi>m</mi><mi mathvariant="normal">t</mi></msub></mrow><msub><mi>m</mi><mi>N</mi></msub></munderover></mstyle><mi>log</mi><mo>⁢</mo><mfenced><mn>1</mn><mo>+</mo><mfrac><mrow><msub><mi>r</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi mathvariant="normal">G</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><mo>⁢</mo><mi mathvariant="normal">g</mi></mrow></mfrac></mfenced></math><img file="EP1068704B1_D0065.tif" /></maths>

0077Since <i>G_k</i> = |<i>G</i>(π<i>m_k</i> / <i>M</i>)|², the minimization of the cost function is performed over all positive values of <i>G_k</i>, as: <maths id="math0066" num="(44)"><math display="block"><msub><mi>g</mi><mi mathvariant="italic">opt</mi></msub><mo>=</mo><mi>arg</mi><mspace width="1em" /><munder><mi>min</mi><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><mo>⁢</mo><mi>F</mi><mfenced><mi mathvariant="normal">g</mi></mfenced></math><img file="EP1068704B1_D0066.tif" /></maths> where: <maths id="math0067" num="(45)"><math display="block"><mi>G</mi><mo>=</mo><mfenced open="{" close="}"><mi mathvariant="normal">g</mi><mo>∈</mo><msup><mi>R</mi><mi>N</mi></msup><mo>:</mo><mfenced open="|" close="|"><mi mathvariant="normal">g</mi></mfenced><mo>=</mo><mn>1</mn><mo>,</mo><msub><mi mathvariant="normal">G</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>N</mi></mfenced><mn>.</mn></math><img file="EP1068704B1_D0067.tif" /></maths> A variety of constrained optimization strategies may be used to solve the above equations for <i>g_opt</i>.

0078Once the optimal impulse response <i>g_opt</i> and desired spectral response <i>G_d</i>(ω) (which may be expressed as <i>G_d</i>(π<i>m_k</i> / <i>M</i>) for a system having <i>M</i> tones) have been determined, a training process is used to adapt the SCISF 90 so that its impulse response <i>g</i> matches the desired spectral response. As shown in <figref idref="f0010">Fig. 13</figref>, the training process may be generalized as a feedback system. A reference sequence <i>x_k</i> is input to the system. This corresponds to inputting a predetermined reference bit sequence to a transmitter. The reference sequence passes through a transmission channel 410 having frequency response <i>H</i>(<i>f</i>) (including the physical transmission channel and the transmit and receive filters). Additive noise η<i>_k</i> from the transmission channel is represented in this general model as an external input 420 to the system. The resulting signal <i>y_k</i> is input to a filter 430 having a frequency response <i>G</i>(<i>f</i>), e.g., a SCISF. The output of the filter 430 is then passed to an adaptation processor 440, which computes an error signal based on the feedback loop 450 and adapts the filter accordingly. The adaptation processor may, for example, use the LMS algorithm described above.

0079The reference sequence <i>x_k</i> is also input to the feedback loop 450, which passes the reference sequence <i>x_k</i> through a scaling filter 460 with frequency characteristic <i>Q</i>(<i>f</i>). The frequency characteristic <i>Q</i>(<i>f</i>) of the scaling filter 460 (which may be expressed as a set of frequency domain scaling factors <i>Q_k</i>) is determined so that the SCISF adapts to the desired spectral response. The output of the scaling filter 460 is used a reference for the calculation of the error signal in the adaptation processor 440, as described above.

0080Using the general feedback system shown in <figref idref="f0010">Fig. 13</figref>, a SCISF having an impulse response <i>g</i> may be trained to minimize the error ∥<i>q</i> * <i>x</i> - <i>x</i> * <i>g</i> * <i>h</i> - η * <i>g</i>∥². The resulting filter matches <i>P₂</i>(ω) in the frequency domain in a least-squares sense, where: <maths id="math0068" num="(46)"><math display="block"><msub><mi>P</mi><mn>2</mn></msub><mfenced><mi>ω</mi></mfenced><mo>=</mo><mfrac><mrow><msub><mi>S</mi><mi>x</mi></msub><mfenced><mi>ω</mi></mfenced><mo>⁢</mo><mi>H</mi><mo>*</mo><mfenced><mi>ω</mi></mfenced><mo>⁢</mo><mi>Q</mi><mfenced><mi>ω</mi></mfenced></mrow><mrow><msub><mi>S</mi><mi>x</mi></msub><mfenced><mi>ω</mi></mfenced><mrow><mo>|</mo><msup><mrow><mi>H</mi><mfenced><mi>ω</mi></mfenced><mo>|</mo></mrow><mn>2</mn></msup></mrow><mo>+</mo><msub><mi>S</mi><mi>η</mi></msub><mfenced><mi>ω</mi></mfenced></mrow></mfrac><mo>,</mo></math><img file="EP1068704B1_D0068.tif" /></maths><i>S_x</i>(ω) is the power-spectral density at the input of the system, H(ω) is the frequency response of the effective discrete-time impulse response (EDIR) of the transmission channel, <i>S_η</i> (<i>ω</i>) is the power spectral density of the additive noise, and <i>Q</i>(ω) is the spectral response of the scaling filter 460 having impulse response <i>q</i>.

0081The solution for <i>g_opt</i> in the equations above specifies only the magnitude of the spectral response of the SCISF. If the SCISF is a FIR filter, a linear phase characteristic may be used. If the length of the SCISF is <i>n_g</i>, the desired values of <i>G</i>(ω) for the frequency bins of interest are: <maths id="math0069" num="(47)"><math display="block"><msub><mi>G</mi><mi>d</mi></msub><mfenced><mi>π</mi><mo>⁢</mo><msub><mi>M</mi><mi>k</mi></msub><mo>/</mo><mi>M</mi></mfenced><mo>⁢</mo><munder><mi mathvariant="normal">Δ</mi><mo>=</mo></munder><mo>⁢</mo><msqrt><msub><mi>g</mi><mi mathvariant="italic">opt</mi></msub><mfenced><mi>k</mi></mfenced></msqrt><mo>⁢</mo><mi>exp</mi><mfenced><mfrac><mrow><mo>-</mo><msub><mi mathvariant="italic">jπM</mi><mi>k</mi></msub><mo>⁢</mo><mfenced><msub><mi>n</mi><mi>g</mi></msub><mo>-</mo><mn>1</mn></mfenced></mrow><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></mfrac></mfenced><mn>.</mn></math><img file="EP1068704B1_D0069.tif" /></maths>

0082The values Q_k are defined by: <maths id="math0070" num="(48)"><math display="block"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>G</mi><mi>d</mi></msub><mfenced><msub><mi mathvariant="italic">πm</mi><mi>k</mi></msub><mo>/</mo><mi>M</mi></mfenced><mo>⁢</mo><mfenced><msub><mi>S</mi><mi>x</mi></msub><mfenced><mi mathvariant="italic">jπk</mi><mo>/</mo><mi>M</mi></mfenced><mo>⁢</mo><msup><mfenced open="|" close="|"><mi>H</mi><mfenced><mi mathvariant="italic">jπk</mi><mo>/</mo><mi>M</mi></mfenced></mfenced><mn>2</mn></msup><mo>+</mo><msub><mi>S</mi><mi>η</mi></msub><mfenced><mi mathvariant="italic">jπk</mi><mo>/</mo><mi>M</mi></mfenced></mfenced></mrow><mrow><msub><mi>S</mi><mi>η</mi></msub><mfenced><mi mathvariant="italic">jπk</mi><mo>/</mo><mi>M</mi></mfenced><mo>⁢</mo><mi>H</mi><mo>*</mo><mfenced><mi mathvariant="italic">jπk</mi><mo>/</mo><mi>M</mi></mfenced></mrow></mfrac></math><img file="EP1068704B1_D0070.tif" /></maths> The values of <i>Q_k</i> may be computed during an initial training period and may be periodically updated during operation of the communication system.

0083As shown in <figref idref="f0011 f0012 f0013">Figs. 14-16</figref>, the general feedback training process may be used to perform an initial training of a SCISF followed by periodic training analogous to the process described above with respect to <figref idref="f0007 f0008 f0009">Figs. 10-12</figref>. One difference between the techniques is that a scaled reference signal (<i>x</i> * <i>q</i>)<i>_k</i> is used rather than an unscaled reference <i>x_k</i>.

0084Referring to <figref idref="f0011">Fig. 14</figref>, to perform the initial training, a predetermined sequence of bits <i>x_k</i> is input to the transmitter. The transmitted signal is filtered and noise-corrupted by the transmission channel, resulting in a received sequence {<i>y_k</i>} at the output of the A/D 80 in the receiver. The SCISF 90 filters and transforms the received sequence {<i>y_k</i>} into an output sequence {<i>x̂_k</i>}. The output sequence {<i>x̂_k</i>} is compared (in a signal comparator 205) to a scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i>.

0085The scaled reference sequence is computed from a copy of the predetermined sequence <i>x_k</i> that is stored in memory 210 in the receiver. As a first step, the predetermined sequence is input to a serial to parallel converter 510 and a DFT 515. The resulting frequency domain signal is input to a scaling filter 520 which applies the set of frequency domain scaling factors <i>Q_k</i> that causes the SCISF to adapt to the desired spectral response, as discussed above. The scaled signal is input to an inverse discrete Fourier transform 330, a parallel to serial converter 340 and a cyclic prefix adder 350, resulting in a scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i>. The comparison of the output sequence {<i>x̂_k</i>} to the scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i> results in an error signal <i>e_k</i> that is input to the LMS algorithm processor 215 along with sequence {<i>x̂_k</i>}. Alternatively, a frequency domain reference (e.g., a predetermined bit sequence that has been processed by a serial to parallel converter and DFT) may be stored in memory in the receiver, which would eliminate the need for the serial to parallel converter and discrete Fourier transform in the feedback loop.

0086Following the initial training, the SCISF is periodically trained during operation of the communication system. A sequence of communication data bits <i>x_k</i> is input to the transmitter. Referring to <figref idref="f0012">Fig. 15</figref>, the transmitted signal is filtered and noise-corrupted by the transmission channel, resulting in a received sequence {<i>y_k</i>} at the output of the A/D 80 in the receiver. The SCISF 90 filters and transforms the received sequence {<i>y_k</i>} into an output sequence {<i>x̂'_k</i>}.

0087The received sequence {<i>y_k</i>} is also input to a delay 305 and then to a secondary SCISF 300 that has the same coefficients as the primary SCISF 90 following the initial training. The secondary SCISF 300 provides output sequence {<i>x̂_k</i>}, which is compared to a reference sequence during the periodic training process. Such a configuration allows periodic training to be performed without disruption to the operation of the communication system. The secondary SCISF 300 is periodically or continuously trained using an algorithm similar to that used for the initial training. The new coefficients of the secondary SCISF 300 are periodically copied to the primary SCISF 90.

0088To compute the reference sequence, the data output of the decoder 140 is input to an encoder 320. The resulting frequency domain signal is input to a scaling filter 520 which applies the set of frequency domain scaling factors <i>Q_k</i> that causes the SCISF to adapt to the desired spectral response, as discussed above. The scaled signal is input to an inverse discrete Fourier transform 330, a parallel to serial converter 340 and a cyclic prefix adder 350, resulting in a scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i>. The comparison of the output sequence {<i>x̂_k</i>} to the scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i> results in an error signal <i>e_k</i> that is input to the LMS algorithm processor 215. The training process determines coefficients for the secondary SCISF 300 so that the output {<i>x̂_k</i>} matches the scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i> as closely as possible in a least squares sense, i.e., the mean square error between the output and the reference sequence is minimized. Periodically, the coefficients of the secondary SCISF 300 are copied to the primary SCISF 90.

0089Alternatively, as shown in <figref idref="f0013">Fig. 16</figref>, the periodic training may be performed with a single SCISF 90. In this configuration, a received sequence {<i>y_k</i>} is output by the A/D 80 in the receiver. The SCISF 90 filters and transforms the received sequence {<i>y_k</i>} into an output sequence {<i>x̂'_k</i>}. The received sequence {<i>y_k</i>} is also input to a delay. After the received sequence {<i>y_k</i>} passes through the SCISF 90, a data switch 360 is changed from position A to position B, allowing the delayed received sequence to make a second pass through the SCISF 90. An output switch 370 also may be opened, so that data is not output during the training process. In addition, the SCISF coefficients are controlled by the LMS algorithm during the training process.

0090A reference sequence is computed as in the configuration of <figref idref="f0012">Fig. 15</figref>. The data output of the decoder 140 is input to an encoder 320. The resulting frequency domain signal is input to a scaling filter 520 which applies the set of frequency domain scaling factors <i>Q_k</i> that causes the SCISF to adapt to the desired spectral response, as discussed above. The scaled signal is input to an inverse discrete Fourier transform 330, a parallel to serial converter 340 and a cyclic prefix adder 350, resulting in a scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i>. The scaled reference sequence is input to the LMS algorithm processor.

0091The output sequence {<i>x̂_k</i>} of the second pass through the SCISF 90 is compared (in signal comparator 205) to the reference sequence (<i>x</i>*<i>q</i>)<i>_k</i>. As noted above, the received sequence {<i>y_k</i>} passes through a delay 305 before being input to the SCISF 90 for the second pass. The delay 305 compensates for the processing delay in the demodulation/decoding chain and the encoding/modulation chain. The comparison results in an error signal <i>e_k</i> that is input to the LMS algorithm processor 215. The training process determines coefficients for the SCISF 90 so that the output {<i>x̂_k</i>} matches the scaled reference sequence (<i>x</i>*<i>q</i>)<i>_k</i> as closely as possible in a least squares sense, i.e., the mean square error between the output and the reference sequence is minimized. The coefficients of the SCISF 90 then are updated to the coefficients determined in the training process.

0092In a third examplary embodiment, the system dynamically selects the length of the cyclic prefix (CP) to maximize data throughput for a communication channel having a particular noise profile. As discussed above, a CP is added to each symbol prior to transmission through the communication channel to reduce the effects of ISI. However, because the CP constitutes redundant data, increasing the length of the CP reduces the efficiency of the communication system. Hence, to maximize efficiency, the length of the CP must be as short as the noise characteristics of the communication channel permit.

0093For a DMT communication system with <i>M</i> tones, the maximum sample rate <i>W</i> (samples/second) for a particular channel depends, in part, on the available bandwidth and hardware limitations. The sample rate includes communication data and CP bits. For a CP length of <i>n_c</i>, the maximum symbol rate (which includes communication data, but not the CP) is <i>W</i> /(2<i>M</i> + <i>n_c</i>).

0094Before determining the optimal CP length, the SCISF should be initially trained to the channel. However, a communication system need not have a SCISF to employ the CP optimization algorithm. It is noted that the SCISF coefficients determined during the training process do not depend on the CP length. The capacity of the sub-channel may be approximated as log(1+<i>SNR_i</i>) bits per second, so the number of bits per symbol is ∑<i>_i</i> log(1+<i>SNR_i</i>). For a CP length of <i>n_c</i>, the maximum bit rate is expressed as a function of the cyclic prefix as: <maths id="math0071" num="(49)"><math display="block"><msub><mi>B</mi><mi>a</mi></msub><mfenced><msub><mi>n</mi><mi>c</mi></msub></mfenced><mo>=</mo><mfrac><mrow><mi>W</mi><mstyle displaystyle="false"><munder><mo>∑</mo><mi>i</mi></munder><mo>⁢</mo><mi>log</mi><mo>⁢</mo><mfenced><mn>1</mn><mo>+</mo><msub><mi mathvariant="italic">SNR</mi><mi>i</mi></msub></mfenced></mstyle></mrow><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi><mo>+</mo><msub><mi>n</mi><mi>c</mi></msub></mrow></mfrac></math><img file="EP1068704B1_D0071.tif" /></maths> The optimal CP length is determined by computing the maximum bit rate for a set of candidate values of CP length and finding the length that maximizes <i>B_a</i> (<i>n_c</i>).

0095The signal to noise ratio <i>SNR_i</i> of each subchannel is determined by measuring the received signal and noise power and computing the ratio of the two. The noise power γ<i>_i</i> for the i^<i>t</i>h bin may be measured by transmitting a data communication sequence and computing the average of the squares of the errors measured at the output of the receiver DFT. The total received power (signal and noise) δ<i>_i</i> for the <i>i</i>^th bin may be measured by computing the average of the squares of the outputs of the receiver DFT. The signal to noise ratio is determined from the expression: δ<i>_i</i>/γ<i>_i</i> =1+<i>SNR_i</i>. Since the signal to noise ratio is determined in the receiver, the computed bit rate <i>B_a</i>(<i>n_c</i>) must be transmitted back to the transmitter. The transmitter compares the bit rate to the values computed for other candidate CP lengths and selects the CP length <i>n_c</i> with the highest maximum bit rate <i>B_a</i>(<i>n_c</i>).

0096<figref idref="f0014 f0015 f0016 f0017">Figs. 17-24</figref> show performance simulations for test systems based on system parameters and test loops described in <nplcit id="ncit0009" npl-type="s"><text>HDSL Alliance SDMT VDSL Draft Standard Proposal, Technical report, ANSI, 1998</text></nplcit>; and <nplcit id="ncit0010" npl-type="s"><text>Very-high-speed digital subscriber lines: System requirements, T1E1.4/97-131R1, Technical report, ANSI, 1997</text></nplcit>. The results are for a VDSL system working over test loops 2 and 6 of length 4500 feet in the upstream direction. The system has a sampling frequency of 11.04 MHz. Noise is generated by near-end cross talk from an interfering ADSL and an interfering HDSL and white noise at a level of -140 dBm. The SCISF used in the simulations is length-15 FIR. A version of the Normalized LMS algorithm is used to train the SCISF during an initial training period using a predetermined transmitted sequence.

0097<figref idref="f0014 f0015">Figs. 17-20</figref> show the simulated system performance for a communication system having the parameters defined for Test Loop 2, which is 4500 feet in length. <figref idref="f0014">Fig. 17</figref> shows channel frequency response with and without a SCISF. The SCISF provides a much more uniform frequency response across the frequency band of interest and significantly improves the signal to noise ratio (SNR) in the higher frequency bins. <figref idref="f0014">Fig. 18</figref> is a plot of the error signal (10 log |<i>x̂_k</i> - <i>x_k</i>) during the training process. The error decreases rapidly during the first few iterations and is nearly converged after only 20-30 iterations. <figref idref="f0015">Fig. 19</figref> is a plot of transmitted power spectral density, received power spectral density and the additive noise power spectral density over the used subchannels at the output of the receiver A/D. <figref idref="f0015">Fig. 20</figref> is a plot of SNR at the input to the receiver A/D, which is the maximum attainable SNR. The plot also shows the SNR at the output of the receiver DFT without a SCISF and the SNR at the outputs of the receiver DFT using an adapted SCISF.

0098<figref idref="f0016 f0017">Figs. 21-24</figref> show the simulated system performance for a communication system having the parameters defined for Test Loop 6, which is 4500 feet in length. <figref idref="f0016">Fig. 21</figref> shows channel frequency response with and without a SCISF. <figref idref="f0016">Fig. 22</figref> is a plot of the error signal (10 log |<i>x̂_k</i> - <i>x_k</i>|) during the training process. <figref idref="f0017">Fig. 23</figref> is a plot of transmitted power spectral density, received power spectral density and the additive noise power spectral density over the used subchannels at the output of the receiver A/D. <figref idref="f0017">Fig. 24</figref> is a plot of SNR at the input to the receiver A/D. The plot also shows the SNR at the output of the receiver DFT without a SCISF and the SNR at the outputs of the receiver DFT using an adapted SCISF.

0099The embodiments of the invention are defined by the scope of the following claims.