Frequency domain equalization of communication signals

<u>BACKGROUND OF THE INVENTION</u>

1.

Field of the Invention

0001The present invention relates generally to the field digital telecommunication, and more particularly to an adaptive equalization method for eliminating echoes at the receiver of a communication channel.

2.

Description of the Related Art

0002Many systems utilize wireless transmission of signals, including wireless telephone systems, wireless television systems, and other systems which perform wireless communication. Such systems generally include at least one transmitter which transmits signals through a channel to at least one receiver. During transmission of the signals through the channel, multiple reflections and diffraction from natural and/or man-made obstacles can occur. Such obstacles can include buildings, homes, vehicles, or natural terrain such as mountains or trees. The reflections and diffraction from these objects can create multi-path distortion of the transmitted signal. In open wireless channels, multipath reflections can introduce inter-symbol interference ISI into the received signal. In certain wired transmission systems, multipath reflections can also occur, e.g., micro-reflections due to impedance mismatch from various passive or active elements in the channel, such as taps, amplifiers, and cables, etc.

0003An equalizer is an electrical circuit positioned after signal down-conversion and before error-correction. The equalizer processes a signal to remove distortions introduced by the channel, for example, echoes/ghosts introduced by multiple paths in a radio/television channel. The signal output of the equalizer is "better" the more closely it approximates the signal input to the transmitter. Signals can be analog or digital. Digital signals may have many possible levels of amplitude/phases. Unique combinations of amplitude/phase are called symbols, and a digital signal may be represented as a symbol stream. For example, the North American High Definition Television (HDTV) Standard utilizes 8 amplitude levels with a DC offset. The transmitter produces one sideband with 8 amplitude levels and a vestigial carrier (8-VSB).

0004The equalizer in a radio/TV receiver is designed to compensate for distortions of the signal introduced by the channel. Typical distortions in a TV channel come from reflections from buildings and aircraft. Also, persons moving near an antenna can alter the relative strengths of individual reflection paths striking an antenna. Thus there is a need to compensate for echo ensembles which may be static or a mix of static and dynamic paths. For HDTV the symbol period is about 0.093 µsec and the echoes to be compensated can be up to 60 µsec away. Thus there can be a high degree of inter-symbol interference (ISI), up to 645 symbol periods away. Moving reflectors introduce Doppler shifts. The channel may also introduce noise into the signal, where the noise may be white or impulsive (bursting). For digital signals, noise can cause decision circuits to make symbol errors. The processing after the equalizer is designed to remove symbol errors through the use of trellis decoding, de-interleaving, and error correction (Reed-Solomon decoder in <figref idref="f0002">Figure 2</figref>). Randomizing a signal breaks up long strings of a single symbol, which neutralizes the effect of long strings of DC offsets in AC-coupled circuits. For HDTV, the symbol error rate at the input to the trellis decoder can be up to about 20% before the error correction processing breaks down.

0005Typically, an equalizer is a filter with a characteristic response of its outputs to its inputs. The filter characteristic may be altered to compensate for specific distortions introduced by a specific channel. Ideally, the equalizer response compensates for all the distortions introduced by the channel. However, the more complex the distortions, the more complex the compensation required, ultimately requiring a larger equalizer circuit For static distortions, the measure of compensation quality is the mean square error (MSE) across a long string of random symbols, plus any noise introduced by the equalizer as it fluctuates about the converged characteristic. For dynamic distortions the measure of quality includes a tracking component, such as the rate of convergence, or the allowable Doppler shift.

0006Various filter architectures have been used or proposed for digital signal equalizers. Filter architectures may be described by the features included in the signal data path and by the features in the control algorithm. The data path input and output are symbol streams in the time domain. Conventional digital filters keep the data in sequence in the time domain without explicit reference to frequency. However, the input symbol stream may be transformed to the frequency domain by means of a discrete Fourier transform (DFT), altered in the frequency domain, and then returned to the time domain by means of an inverse discrete Fourier transform (IDFT).

0007Frequency domain filters have been proposed for use in modems to receive high-bit rate data across digital subscriber lines (DSL) made of twisted pair cable, see <patcit id="pcit0001" dnum="US5809069A"><text>Polley et al. US patent No. 5,809,069</text></patcit>. In DSL modem applications, the symbol rate is allowed to vary depending on line quality, however the maximum symbol rate is limited by the receiver's analog sampling rate, currently at about 2.2 M samples per second. Two-way cross talk and multiple echoes are the main distortions for DSL, with Doppler shifts non-existent Since the DSL cable does not change very often, a complex training sequence may be used to determine equalizer characteristics, and then no further update is needed, perhaps for many days. These are very different requirements than for radio/television channels. Thus equalizers designed for DSL generally would not work for radio/television, especially for HDTV where the symbol rate is higher, about 10.8 M symbols per sec, and the channel fluctuates sometimes up to tens of Hertz.

0008Frequency domain digital equalizers have not been proposed for HDTV or other high-symbol-rate broadcast links. One reason is that frequency domain digital equalizers have a higher computational complexity, which in the past would have produced excessive receiver cost Thus control algorithms for frequency-domain HDTV equalizers are in a state of infancy compared to the prior art for time-domain equalizers. Many of the features used or proposed for HDTV time-domain equalizers may be applicable to frequency domain equalizers, but may not be necessary or desired. A brief summary of time domain equalizers follows, with a focus on their reported limitations.

0009Most digital filters are of the type Finite Impulse Response (FIR) or infinite impulse response (IIR). A FIR filter can be constructed of a tapped delay line and a summation node where each input to the summation is a tap signal multiplied by an independent coefficient (or "weight") for that tap. The more taps in the filter, the more capable it is to compensate for longer echo delays. An IIR filter can be formed by the combination of a component FIR filter and a digital adder. The adder sums the inputs to the IIR with the outputs of its component FIR filter and supplies IIR outputs, which outputs are also fed back to the input of the component FIR filter. Because of the feedback, IIR filters can cancel longer delayed echoes better than FIR filters, however, their stability is not assured.

0010The characteristics of a transmitted digital signal are generally known a priori. Therefore, at least in theory, it is possible to utilize such characteristics in a system of multipath detection and adaptive channel equalization. However, this approach to channel equalization has various problems. Accordingly, some signal communication standards utilize a training signal for the detection and characterization of multipath distortion. For example, television signal transmission systems recurrently transmit a training signal situated in a portion of the TV signal that is currently unused for video purposes, and this training signal is used for the detection and characterization of multipath distortion. Here it is presumed that the transmitted training signal will suffer the same multipath distortions as the rest of the television signal. The receiver can then examine the distorted training signal that is received and, with a priori knowledge of the distortion-free training signal, can calculate the characteristics of the transmission channel. The receiver can then calculate the characteristics required of a filter that will respond to the received signal, but will suppress the effects of multipath signals. A variety of different types of training or "ghost cancellation reference" signals have been described in patents and other technical publications.

0011The following is a quote from <patcit id="pcit0002" dnum="US5648987A"><text>US Patent No. 5,648,987 to Yang , et al., issued July 15, 1997</text></patcit>, pp.18-19, columns 2-3: <ul id="ul0001" list-style="none" compact="compact"><li>"In the digital television signals for broadcasting high-definition television (HDTV), each data field contains 313 data lines, and the fields are consecutively numbered modulo-two in order of their occurrence. Each line of data starts with a line synchronization code group of four symbols having successive values of +S, -S, -S and +S. The value +S is one level below the maximum positive data excursion, and the value -S is one level above the maximum negative data excursion. The lines of data are each of 77.3 microsecond duration, and there are 832 symbols per data line for a symbol rate of about 10 [megasymbols/second]. The initial line of each data field is a field synchronization code group that codes a training signal for channel-equalization and multipath signal suppression procedures. The training signal is a 511-sample pseudo-random sequence (or "PR-sequence") followed by three 63-sample PR sequences. This training signal is transmitted in accordance with a first logic convention in the first line of each odd-numbered data field and in accordance with a second logic convention in the first line of each even-numbered data field, the first and second logic conventions being one's complementary respective to each other. The reference sequence(s) can be analyzed, channel characterization determined and appropriate equalizing filter can be implemented. However, this process can be rather slow and is definitely not suitable for any multipath signal, such as some airplane flutter, that varies quite quickly with elapsed time.</li></ul>

0012Owing to the nature of the digital signal used in HDTV, the adaptation of the channel-equalization filter could be performed with every received symbol on a decision-directed basis (in the absence of the reference sequence). However, currently the limiting factor on the speed of initially equalizing the reception channel or of tracking a time-varying multipath is established by the processing speeds of the computing devices being utilized. Increasing the processing speeds of the computing devices will improve system performance until the point is reached at which all the computations and the subsequent updating of the filter coefficients can be realized with each newly received symbol or with a reasonably small group of newly received symbols.

0013Several methods of performing "adaptive equalization/multipath cancellation" are described in the literature. In simplest terms, the input signal is processed through an equalizer filter. The filter output, is "compared" to the desired output and based on a certain algorithm a correction to the filter parameters is computed and adapted to the filter. The process is continuously repeated until the equalized filter output is "correct", so multipath effects are attenuated sufficiently that they do not exceed levels prescribed as being "acceptable". To aid in developing an understanding of the nature of the computations involved, the reader is referred to the following: <ul id="ul0002" list-style="none" compact="compact"><li><nplcit id="ncit0001" npl-type="s"><text>J. C. Lee and C. K. Un, "Performance Analysis of Frequency-Domain Block LMS Adaptive Digital Filters," IEEE Trans. on Circuits and Systems, pp. 173-189, Vol. 36, No. 2, Feb. 1989</text></nplcit>.</li></ul>

0014The basic adaptive equalization/multipath cancellation equations are known from the above-listed reference to be: <maths id="math0001" num="(1)"><math display="block"><msup><mi>y</mi><mi>n</mi></msup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><msubsup><mi>W</mi><mi>k</mi><mi>m</mi></msubsup><mo>⋅</mo><msup><mi>X</mi><mfenced><mi mathvariant="italic">n</mi><mo>-</mo><mi mathvariant="italic">k</mi></mfenced></msup><mo>,</mo></math><img file="EP1525726B9_D0001.tif" /></maths> k= 0,1, ...,(N-1), and <i>mN</i>≤ <i>n</i> &lt; <i>(m+1)N</i><maths id="math0002" num="(2)"><math display="block"><msup><mi>e</mi><mi>n</mi></msup><mo>=</mo><msup><mi>y</mi><mi>n</mi></msup><mo>-</mo><msup><mi>d</mi><mi>n</mi></msup></math><img file="EP1525726B9_D0002.tif" /></maths><maths id="math0003" num="(3)"><math display="block"><msubsup><mi mathvariant="normal">Δ</mi><mi>k</mi><mi>m</mi></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi mathvariant="italic">mN</mi></mrow><mrow><mfenced open="[" close="]"><mfenced><mi>m</mi><mo>+</mo><mn>1</mn></mfenced><mo>⁢</mo><mi>N</mi></mfenced><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><msup><mi>e</mi><mi>j</mi></msup><mo>⋅</mo><msup><mi>X</mi><mfenced><mi>j</mi><mo>-</mo><mi>k</mi></mfenced></msup></math><img file="EP1525726B9_D0003.tif" /></maths><maths id="math0004" num="(4)"><math display="block"><msubsup><mi mathvariant="italic">W</mi><mi>k</mi><mfenced><mi>m</mi><mo>+</mo><mn>1</mn></mfenced></msubsup><mo>=</mo><msubsup><mi>W</mi><mi>k</mi><mi>m</mi></msubsup><mo>+</mo><mi>μ</mi><mo>⋅</mo><msubsup><mi mathvariant="normal">Δ</mi><mi>k</mi><mi>m</mi></msubsup></math><img file="EP1525726B9_D0004.tif" /></maths>

0015This adaptation algorithm is based on a group of N symbols and not on each symbol. Such an algorithm is identified as "Block LMS". It is known to have the same performance as the well-known LMS (least mean squares) algorithm when the channel varying speed is slower than the realized convergence with the block of N symbols. (Superscripted terms in these equations are not terms raised to "powers" indicated by the superscript. Rather the superscripts following general terms are a set of further indices for sets of specific terms, the specific terms in each set being indexed by subscripts following general terms.) A channel-equalization filter with coefficients W.sub.k (the parameter m is not shown here since it only indicates the number of updates) and input data X.sup.n (ghosted and/or equalization needed) generates equalized data y.sup.n according to equation (1). Since the equalization indicated by equation (1) must be done in real-time, standard practice is to implement that equalization using an appropriate FIR filter. When equalization is done using a training signal, an IIR filter suppresses multipath responses that are delayed respective to strongest signal better than an FIR filter having the same number of taps. In decision-directed equalization, the computation of weighting coefficients for the channel-equalization filter is based strictly on some observation that does not depend on or indicate the time relationship of multipath signals. When the computation procedure begins without knowledge of suitable initial values of the weighting coefficients, the procedure is referred to as "blind" equalization. Because the response of an IIR filter is regenerative in nature, errors introduced by "blind" equalization tend to be perpetuated and will be rarely eliminated by continuing calculation. Presumably this is the reason that, until the invention [<patcit id="pcit0003" dnum="US5648987A"><text>US patent No. 5,648,987</text></patcit>] was made, decision-directed equalization had invariably been used only with FIR channel-equalization filters."

0016As described in <patcit id="pcit0004" dnum="US5648987A"><text>US Patent No. 5,648,987 to Yang et al.</text></patcit>, equalization filters are known which cascade a finite-impulse-response (FIR) filter with an infinite-impulse-response (IIR) filter. The IIR filter can be formed from a digital adder with a component FIR filter. The coefficients of the component FIR filter in the IIR filter can be initially adjusted in response to information obtained from the training signals contained in portions of the transmitted data. This initial adjustment of the coefficients of the component FIR filter is performed to avoid the instability problems normally associated with IIR filters. Thereafter, Yang teaches that the coefficients of the component FIR filter can be computed as described in Yang using a further FIR filter to implement decision directed techniques in which best estimates of correct filter response are formed by quantizing actual filter response. Yang further teaches that, in equalization filters which cascade a finite-impulse-response (FIR) filter with the infinite-impulse-response (IIR) filter, the coefficients of the filters are independently adjusted.

0017The history of time-domain equalizers for North American HDTV is surveyed by M. Ghosh [1], who shows that the 8-VSB modulation was selected in part because of more effective equalization than competitive schemes. "The equalizer in the prototype built by Zenith was a DFE [Decision Feedback Equalizer] with 64 forward and 192 feedback taps [for the two digital filters] and was adapted using the standard LMS algorithm on the pseudo random noise (PN) sequence in the field sync segment. Since the field-sync segment arrives only once every field (i.e, about once every 24 ms), the overall rate of convergence of the equalizer can be quite slow." However, after convergence on a static echo pattern, the mean square error (MSE) was fairly good, and this helped 8-VSB to win the Grand Alliance recommendation. Testing on dynamic echoes was very limited at the time. HDTV manufacturers are free however to use any equalizer they like as long as it meets performance goals.

0018The summary from M. Ghosh in 1998 [1] : <ul id="ul0003" list-style="none" compact="compact"><li>"... the need for tracking time-varying channels indicates the use of blind algorithms. In this paper the advantages of a blind DFE structure were presented via the ATTC [Advanced Television Test Center] test results, as well as simulations with the Godard blind equalization algorithm. The advantages of using a blind DFE have since been validated by numerous field tests that have been conducted by HDTV receiver manufacturers. In addition to the advantages of the blind DFE in long multipath and dynamic multipath, a blind algorithm enables faster acquisition as well. With trained-only equalization it may take the equalizer 10-15 data fields to converge since the field sync occurs only once in each field. However, with blind equalization the equalizer converges in less than one field in most cases.</li></ul>

0019One of the main concerns in implementing an equalizer for digital television receivers is the number of taps required, which is on the order of 256. Hence, simple equalization algorithms can greatly reduce the hardware required to implement such a long equalizer. Future work in this area needs to concentrate on developing hardware efficient blind algorithms. The Godard algorithm and the RCA [Reduced Constellation Algorithm] both require multipliers in each of the tap update steps, which can add up to a large area requirement in silicon. Recent work in this area [1 [20]-[22]] has concentrated on sign-error versions of the Godard cost function, which does away with the need for multiplications in the tap update step and reduces implementation complexity. Faster algorithms with low complexity are also an area in need of further research, since the Godard algorithm can be quite slow in tracking rapidly varying channels.

0020Patent application <patcit id="pcit0005" dnum="US20020051087A1"><text>US 2002/0051087 A1 (Patel et al.), published May 2, 2002</text></patcit>, discloses a receiver for DTV signals with channel equalisation and echo suppression.

0021Finally, while the literature on the analyses of blind algorithms for linear equalizers is very rich, the same is not true for blind algorithms for DFE's with significant error propagation. HDTV receivers happen to be a very important commercial application for such structures, and there is a need for more analytical results in this area in order to prove conclusively some of the simulation results described in Section V. For now, most receiver manufacturers must rely heavily on simulation results of blind DFE's in order to design high-performance receivers for the real-world scenario of low SNR and long, possibly dynamic, multipath channels."

0022In summary the prior art for HDTV equalizers suffers from: <ol id="ol0001" compact="compact"><li>1. Slow convergence when using the PN sequence for training, and therefore an inability to track Doppler shifts or other dynamic multipath distortion.</li><li>2. Poor MSE (noisy) in converged state if a bigger step is used for faster convergence, although step size may be varied (Godard Algorithm).</li><li>3. Poor ability to center on the strongest path signal and cancel pre-echoes.</li><li>4. Limited range of echo delays due to steep cost rise to get beyond 200 taps.</li><li>5. Since a multiply is required for each tap, a large number of multipliers are needed for long delays, adding power consumption.</li><li>6. Stability issues when using IIR filters.</li></ol>

0023Accordingly, some objectives of embodiments of the present invention are to mitigate the above problems of time domain equalizers.

<u>SUMMARY</u>

0024Various embodiments of the present invention are directed to an equalization method. In one set of embodiments, the invention comprises a frequency domain equalization algorithm (FDEA). The FDEA is based on transforming the inbound symbol stream to the frequency domain, multiplying the real-time spectrum by a distortion-canceling spectrum, and then transforming the resulting spectrum back to the time domain for output of a symbol stream in the time domain. The frequency domain equalization algorithm may achieve a number of advantages over conventional digital filters. Such advantages may include the ability to compensate for echoes with greater time separation from the main path, the ability to compensate for pre-echoes as well as post-echoes, the ability to compensate for much more severe Doppler shifts, rapid convergence, and continuous updating.

0025Embodiments of the present invention employ a novel approach to the generation of the cancellation spectrum.

<u>BRIEF DESCRIPTION OF THE DRAWINGS</u>

0026A better understanding of the present invention can be obtained when the following detailed description is considered in conjunction with the following drawings, in which: <ul id="ul0004" list-style="none" compact="compact"><li><figref idref="f0001">Figures 1A &amp; 1B</figref> illustrate a communication system including a transmitter which transmits a wireless signal to a receiver through a channel, and further illustrates various multipath distortions that can occur;</li><li><figref idref="f0002">Figure 2</figref> illustrates a block diagram of a communication system, and includes an exemplary block diagram of the receiver;</li><li><figref idref="f0003">Figure 3</figref> is a high-level block diagram of the frequency domain equalization algorithm (FDEA) performed in the equalizer of <figref idref="f0002">Figure 2</figref>;</li><li><figref idref="f0004 f0005">Figures 4A-B</figref> is a flowchart of one set of embodiments of the frequency domain equalization algorithm (FDEA) performed by the equalizer of <figref idref="f0002">Figure 2</figref>;</li><li><figref idref="f0006">Figure 5</figref> is a graph illustrating the real part, imaginary part and amplitude envelope of a complex basis pulse;</li><li><figref idref="f0007">Figures 6A</figref> &amp; <figref idref="f0008">6B</figref> illustrate the real and imaginary parts of the PN511 correlation and shows two echoes;</li><li><figref idref="f0009">Figure 7</figref> is a close-up of the PN511 correlation showing well separated zero-phase pulses; and</li><li><figref idref="f0010">Figure 8</figref> is a close-up of the PN511 correlation showing three equal-amplitude zero-phase pulses approximately 110 samples behind the dominant echo.</li></ul>

0027While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the present invention as defined by the appended claims. As used herein the term "connected" means "directly or indirectly connected", and the term "coupled" means "directly or indirectly connected". The notation Re(z) is used to represent the real part of the complex number z. The notation Im(z) is used to represent the imaginary part of the complex number z. The notations exp(x) and <i>e^x</i> are used to represent the exponential function acting on the variable x. The notation z* is used to denote the complex conjugate of complex quantity z. The notation |z| is used to represent the modulus of the complex quantity z. The notation M^T is used to represent the transpose of matrix M. The notation (X) denotes the expected value of the random quantity X.

<u>DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS</u>

Figures 1A-B: Communication System

0028<ul id="ul0005" list-style="none" compact="compact"><li><figref idref="f0001">Figures 1A and 1B</figref> illustrate an exemplary communication system including a transmitter 102, a channel 104, and a receiver 106. As shown, the transmitter 102 may generate a signal which is intended to be received by the receiver 106. The signal generated by the transmitter 102 propagates through the channel 104 and is received by the receiver 106. As shown in <figref idref="f0001">Figure 1A</figref>, as the signal propagates through the channel 104, the signal may experience various types of multipath distortion, including reflections from the ground or from various objects such as objects 108 and 110. Examples of various types of objects that can introduce multipath distortion include buildings, homes, vehicles, airplanes, and natural phenomena such as non-level terrain, trees, etc. This multipath distortion may result in various echoes being received by the receiver 106.</li><li><figref idref="f0001">Figure 1B</figref> suggests that a direct path between the transmitter and receiver may be occluded. Thus, the "direct" echo component of the received signal may be weaker than later arriving multipath echo components. For example, the signal components reflecting off objects such as OBJ₁, OBJ₃ and OBJ₄ may be stronger than the direct path component due to the attenuating influence of object OBJ₂. As used herein, the term "echo" includes the signal component corresponding to the direct path between the transmitter and receiver as well as signal components corresponding to indirect paths.</li></ul>

0029In one set of embodiments of the invention, the receiver 106 includes an equalizer which operates to at least partially compensate for the multipath distortion introduced by the channel 104 into the signal. In one embodiment, the receiver equalizer includes one or more processors and one or more memory media, and the one or more memory media store a software program that is executable by the one or more processors to perform an equalization algorithm as described herein. In another embodiment, the receiver equalizer includes a programmable hardware element, such as a field programmable gate array (FPGA), which performs the equalization algorithm as described herein. In yet another embodiment, the receiver equalizer includes an integrated circuit designed to perform the equalization algorithm as described herein. Thus the receiver equalizer may be implemented in any of various ways, including combinations of the above, among others.

0030In one embodiment, the receiver equalizer is implemented by a software program executing on a processing system. The receiver equalizer described herein may be implemented in any of various systems. Exemplary systems include a digital television system, a wireless or cellular telephone system, a wireless networking system, etc. Embodiments of the present invention may be used in any of various systems which involve wireless transmission of signals from a transmitter through a transmission medium or channel to a receiver. Embodiments of the present invention may also be used in various types of wired systems where the nature of the wired medium introduces echoes.

0031As described herein, in one set of embodiments, the present invention comprises a computational method for characterizing a transmission channel, and this channel characterization may then be utilized to perform frequency-domain equalization on a received signal stream. The computational method may operate on the received signal stream to estimate a channel spectrum for the transmission channel. The channel spectrum estimate may then be used to remove (i.e., deconvolve) the effects of the transmission channel from the received signal spectrum.

0032In one embodiment, the equalization algorithm may take advantage of known data sequences and the known statistical properties of unknown data sequences in signal transmissions (e.g., 8VSB transmissions) to keep track of the delays and phase shifts encountered in realistic reception conditions. The equalization algorithm may exploit statistical properties of the signal and depend on the use of embedded training sequences of symbols.

0033In the Advanced Television Systems Committee (ATSC) standard for HDTV transmission using 8VSB modulation, the training sequence is a pseudorandom noise (PN) sequence of length 511, denoted as PN511. Correlation of the incoming data with the known PN511 series is used to identify multipathing arrivals (i.e., echoes). However, this correlation processing may fail to reveal lower amplitude echoes that are strong enough to make the data unintelligible.

0034In the presence of dynamic multipathing, the relative phases of the various echoes change as a consequence of unequal Doppler shifts. The PN511 sequence recurs at intervals of approximately 24.2 milliseconds at the start of each data field. The ATSC standard calls for the usability of 8VSB by receivers traveling at highway speeds, which may induce Doppler shifts in excess of 100 Hz. Even at a 10 Hz shift, the phase change incurred in the recurrence interval of the PN511 is about 90°. Obviously, the updates of phase obtained from the PN511 correlation are far too infrequent to track the changes in phase adequately.

0035By itself, the use of the training sequence is not in general sufficient to equalize 8VSB signals in the presence of dynamic multipathing. Statistical properties may be exploited to attempt to solve this problem. However, low complexity recursive least squares (RLS) algorithms may not converge fast enough to be effective (especially in the case of 8VSB signals). Thus it appears that the only widely usable algorithms will be high-complexity, near-optimal methods that approximate as closely as possible the direct removal of multipath distortion by deconvolution.

0036Background on 8VSB modulation is deemed appropriate. The details of the 8VSB modulation to be used for HDTV transmission are given in the ATSC (Advanced Television Systems Committee) standard in Document A/53B, dated 7 August 2001. For the purposes of the present application, it is sufficient to describe the 8VSB analog time series as a bandlimited random signal with a known spectrum. The signal is transmitted with a root raised cosine spectrum, and the receiver applies a further filtering by an identical root raised cosine spectrum to yield a raised cosine spectrum for the equalized symbols. The correctly normalized and equalized signal yields eight possible output values, being the odd integers from -7 to +7 inclusively. In practice, the presence of additive noise and errors in equalization causes deviations from these values. The final output value is taken to be the closest valid odd integer to the observed output. This rounding operation is termed "data slicing". The variance of the sliced values, assuming equal probability, is 21. In cases where the signal is determinate, as in the PN511 sequence and the segment synchronization blocks, the symbols assume the values of +5 and -5 only, with variance 25.

0037The deconvolved signal may be normalized to achieve these variance conditions as a final step of equalization before data slicing.

0038The essential difficulties with the reception of 8VSB transmissions arise from the effects of multipathing and dynamic propagation paths. The symbol rate for HDTV transmission by 8VSB is approximately 10.76 MHz, yielding a symbol period of 0.093 µs. Observed delays in field tests are measured in the tens of microseconds, leading to severe intersymbol interference. In order to reverse the effects of interference, it is desirable to rapidly and continuously update the measurements of the amplitude, phase and delay associated with each echo. Here the term "echo" is used to denote the contribution of any single path to the received wavefield, including the dominant path (i.e., the path that contributes the strongest signal component). With a suitable parallel computing engine, it is in principle possible to use a high complexity algorithm that provides nearly optimal equalization.

0039The method outlined herein may use cross correlations with the PN511 training sequence to determine the properties of the largest echoes. When the input signal to noise ratio (SNR) is above the theoretical minimum (approximately 14.8 dB, based on a symbol error rate of 0.2), the largest echoes are always detectable in the PN511 correlation in the presence of a realistic number of echoes (e.g., less than 20 echoes). The detection of echoes is limited by the fixed variance of this correlation. If conditions are static, it may be possible to average these cross correlations to reduce the variance, but generally a more frequently sampled measurement of the channel properties is needed. The method described herein is not limited to cross correlation with the PN511 sequence. Rather, the method also employs autocorrelation of the signal sequences to determine echoes, independent of the presence of an embedded PN511 sequence. In doing so, the method may rapidly reduce the variance of the estimated autocorrelation by stacking (equivalently, averaging) individual autocorrelations to permit the detection of echoes that are too small to be revealed by correlation with the PN511. It is noted that dynamic conditions may limit the number of autocorrelations that may be averaged. These correlations are computed efficiently by the use of the fast Fourier transform (FFT), although the circular buffer effect caused by the FFT assumption of a periodically repeated time series should be taken into account.

0040While the use of averaged autocorrelations permits the detection of small echoes, it also may introduce ambiguity in the phase and delay of any given echo. In addition, the correlations of the separate echoes with each other leads to a profusion of autocorrelation peaks. These autocorrelation peaks should preferably be winnowed to identify the actual delay of a minor echo with respect to any major echo, most commonly the dominant path. These matters create the greatest difficulties in the development of a practical algorithm.

0041In one embodiment, the method described herein also addresses the difficulties caused by delays that are not integer multiples of the sample interval, due to propagation or to errors in phase lock in the conversion to baseband.

Figure 2 - Communication System Block Diagram

0042<figref idref="f0002">Figure 2</figref> is a block diagram of a communication system (e.g., the communication system exemplified in <figref idref="f0001">Figures 1A-B</figref>). <figref idref="f0002">Figure 2</figref> illustrates a transmitter block 102 which couples through a channel 104 to a receiver block 106. <figref idref="f0002">Figure 2</figref> also illustrates one embodiment of the receiver 106. As shown in this exemplary embodiment, the receiver 106 may comprise an RF down-converter 122, an analog to digital (A/D) conversion unit 124, an equalizer 126 (e.g., an 8-VSB equalizer), a Trellis decoder 128, a deinterleaver 130, a Reed-Solomon decoder 132, and a derandomizer 134.

0043The RF down-converter 122 may operate to receive an RF (radio frequency) signal, and convert the RF signal to baseband. The A/D conversion unit 124 may operate to convert the baseband analog signal to a digital data stream. The equalizer 126 may operate to adjust the digital data stream to compensate for multipath distortions. In one set of embodiments, the equalizer 126 may be an 8-VSB equalizer (as suggested in <figref idref="f0002">Figure 2</figref>). The equalizer 126 outputs symbols to a Trellis decoder 128. The Trellis decoder 128 performs forward error correction to remove errors in the symbol stream. The Trellis decoder 128 provides its output to the deinterleaver 130. The deinterleaver 130 operates to deinterleave the received symbols. The deinterleaver 130 provides its output to Reed-Solomon decoder 132. The Reed-Solomon decoder 132 performs further error correction and provides its output to derandomizer 134. The derandomizer 134 then outputs the symbol stream.

Figure 3 - FDEA Block Diagram

0044<figref idref="f0003">Figure 3</figref> presents an overview of the frequency domain equalization algorithm (FDEA) according to one set of embodiments. The block diagram of <figref idref="f0003">Figure 3</figref> may be used to realize equalizer 126 of <figref idref="f0002">Figure 2</figref>.

0045As shown, the equalizer 126 may include an input for receiving a symbol stream. The received symbol stream is the received version of the transmitted symbol stream, where the received symbol stream may include the effects of multipath distortion.

0046The equalizer may include an FFT block 202 which performs a Fourier transform on the received symbol stream to generate a signal spectrum. The FFT block 202 provides the signal spectrum to a multiplier block 217. The FFT block 202 also provides the signal spectrum to each of an auto correlation block 204 and a training sequence correlation block 206.

0047The auto correlation block 204 performs a frequency-domain autocorrelation operation on the signal spectrum to obtain a power spectrum.

0048Where the received signal corresponds to a transmitted training sequence, the training sequence correlation block 206 performs a frequency-domain cross correlation operation relative to the training sequence to provide synchronization and to estimate the properties of the strongest echoes. Each of the auto correlation block 204 and the training sequence correlation block 206 provides its outputs to an inverse FFT block (IFFT) 208.

0049The IFFT block 208 performs an inverse Fourier transform on its received frequency domain signal to convert the received signal back into the time domain. The IFFT block 208 provides its output to an Amplitude Phase Delays block 210.

0050The Amplitude Phase Delays block 210 receives a signal from the IFFT block 208 and also receives a signal from Test Equalizer block 212. The Amplitude Phase Delays block 210 operates to obtain the amplitude, phase, and delay of each echo. The Amplitude Phase Delays block 210 provides its output to a Generate Inverse Echo Spectrum block 214.

0051The Generate Inverse Echo Spectrum block 214 provides an output to a Post-Echo Adaptive Equalizer (Recursion) block 216. The Generate Inverse Echo Spectrum block 214 also provides an inverse echo spectrum output to the convolution block 217.

0052The multiplication block 217 operates to multiply the signal received from the FFT block 202 with the inverse echo spectrum received from the Generate Inverse Echo Spectrum block 214. Multiplication of the signal received from the FFT block 202 with the inverse echo spectrum operates to remove the echoes (the effects of multipath distortion) from the input signal. Thus the multiplication block 217 provides as an output the received symbol stream adjusted to remove at least a portion of the echoes associated with multipath distortion. The multiplication block 217 provides its output to an IFFT block 218. The IFFT block 218 performs the inverse Fourier transform on the received signal to convert the adjusted symbol stream back into the time domain. The time domain symbol stream is then provided to the multiplexer 220.

0053The Post-Echo Adaptive Equalizer (Recursion) block 216 receives an input from the Generate Inverse Echo Spectrum block 214. The Post-Echo Adaptive Equalizer (Recursion) block 216 also receives inphase (I) and quadrature (Q) inputs from the synchronous detector or by forming the analytic signal of the input stream. The Post-Echo Adaptive Equalizer (Recursion) block 216 provides an output to the multiplexer 220. The multiplexer 220 selects between the output from the IFFT block 218 and the output from the Echo Adaptive Equalizer (Recursion) block 216 and provides an output symbol stream. The Post-Echo Adaptive Equalizer (or recursion equalization) block 216 is optional in some embodiments.

0054In one set of embodiments, the FDEA control may be broken down into the following functions: <ol id="ol0002" compact="compact"><li>1. Synchronize Signal <ol id="ol0003"><li>(a) Cross-correlate the stored training sequence with incoming data to locate the training sequence in the data stream.</li><li>(b) Locate peaks in the cross-correlation that are above a threshold T₁ for signal equalization and symbol synchronization.</li></ol></li><li>2. Locate Echoes (Done periodically) <ol id="ol0004" compact="compact"><li>(a) Form the analytic signal corresponding to the average autocorrelation by inverse Fourier transforming an average of power spectra (over nonnegative frequencies);</li><li>(b) Form the amplitude envelope of the average autocorrelation;</li><li>(c) Determine threshold T₂ and peaks of the amplitude envelope using the threshold T₂;</li><li>(d) Determine the correct placement of echoes (including dominant echo, pre-echoes and post-echoes).</li></ol></li><li>3. Generate Echo Sequence <ol id="ol0005" compact="compact"><li>(a) Determine amplitudes and phases of echoes in echo sequence;</li><li>(b) Determine a subwindow for equalization;</li><li>(c) Combine amplitudes, phases, and delays to generate the echo sequence.</li></ol></li><li>4. Form the Inverse Echo Sequence <ol id="ol0006" compact="compact"><li>(a) Use stabilized deconvolution to form the inverse echo spectrum;</li><li>(b) Invert the stabilized spectrum to obtain the inverse echo sequence;</li><li>(c) Determine the optimum equalization segments in the data window from the properties of the inverse echo sequence.</li></ol></li><li>5. Deconvolve the echo sequence from the data sequences</li><li>6. Do Final Inphase (I) Signal Equalization <ol id="ol0007" compact="compact"><li>(a) Do final spectral shaping on equalized data;</li><li>(b) Perform amplitude normalization on the I symbol sequence;</li><li>(c) "Dataslice" the normalized I symbol sequence.</li></ol></li></ol>

0055The above functions are described in detail below.

Figures 4A-B: Flowchart Description of FDEA

0056<figref idref="f0004 f0005">Figures 4A-B</figref> present a flowchart description of one set of embodiments of the computational method.

0057FFT module 202 operates on windows of samples of the input signal, and computes a fast Fourier transform (FFT) of each window to generate a corresponding signal spectrum. Thus, FFT module 202 generates a stream S₁ of signal spectra. Successive windows may overlap as indicated below.

0058Deconvolution module 242 performs a stabilized deconvolution operation on each signal spectrum of the stream S₁, using a current estimate of the inverse echo spectrum (i.e., a stabilized reciprocal of the current echo spectrum estimate), to obtain a stream S₂ of deconvolved spectra. Module 268 may provide the current estimate of the inverse echo spectrum.

0059Deconvolution module 242 may perform the stabilized deconvolution on a signal spectrum of the stream S₁ by multiplying the signal spectrum by the current inverse echo spectrum estimate to obtain a corresponding deconvolved spectrum of the stream S₂.

0060The computational method may generate updates of the echo spectrum estimate to track changes in phases, amplitudes and delay times of echoes.

0061Inverse transform module 218 performs an inverse FFT on each deconvolved spectrum of the stream S₂ to generate a stream S₃ of deconvolved blocks of samples in the time-domain. The deconvolved blocks of the stream S₃ may be referred to herein as equalization windows as they contain equalized samples. (The stabilized deconvolution operation is interpreted as a frequency-domain equalization process.)

0062Module 258 may perform normalization and data slicing on each deconvolved block of stream S₃ to generate a stream S₄ of output blocks. The stream S₄ of output blocks may be forwarded to output module 296.

0063Output module 296 operates on the output blocks of the stream S₄ to recover a sequence of information symbols. The information symbols may be used to generate an analog signal which drives an output device. The output device may be a television (e.g., an HDTV television), a monitor (e.g., an LCD display or CRT display), a set of one or more speakers, a transmit antenna, a network interface device (e.g., an Ethernet card).

0064Autocorrelation module 204 performs a frequency-domain autocorrelation operation on each signal spectrum of the stream S₁ to generate a stream S₅ of power spectra. Averaging module 244 may perform an averaging operation on the stream S₅ of power spectra to generate a stream S₆ of averaged power spectra. The averaging operation may be implemented with a recursive filter (e.g., an IIR filter of the form given by equation (17)) which has a decaying memory of past inputs. Thus, the averaging operation may be referred to herein as a "stack and forget" operation. The recursive filter has an associated time constant which is an indicator of the time required to effectively forget a given input stimulus to the recursive filter. In some alternative embodiments, the averaging operation may be implemented with an FIR filter.

0065The inverse transform module 208A may perform an inverse transform on an averaged power spectrum from the stream S₆ approximately once per time constant of the recursive filter. The inverse transform generates an averaged autocorrelation signal.

0066Module 246 may perform a pulse decomposition on the analytic signal of the autocorrelation function to obtain estimates for the locations (in time) and complex coefficients for peaks in the envelope of the analytic signal of the autocorrelation function. The section below entitled "Pulse Decomposition" describes the pulse decomposition process mostly in terms of the cross-correlation function. However, this process applies also to the autocorrelation function.

0067Cross-correlation module 206 performs a frequency-domain cross-correlation between each signal spectrum of the stream S₁ and the spectrum of a known training sequence to generate a stream S₇ of cross-correlation spectra.

0068In an initial echo acquisition mode, inverse transform module 208B may perform an inverse transform on each cross-correlation spectrum of the stream S₇ to determine a stream S₈ of cross-correlation signals, and detection module 248 may scan an amplitude envelope of each cross-correlation signal of the stream S₈ to determine a first cross-correlation signal containing a sufficiently large pulse indicating an occurrence of the training sequence in the received signal stream. Once the location L_TS of the training sequence in the received signal stream has been established, inverse transform module 208B may operate less frequently.

0069For some signal transmissions, it is valid to assume that the training signal occurs periodically (e.g., once per field of an 8VSB transmission). Thus, in response to establishing the location L_TS, the windowing of the received sample stream may be adjusted so that the training sequence occurs in the middle of every N^th window. Therefore, the inverse transform module 208B may perform an inverse transform on every N^th cross-correlation spectrum from the stream S₇.

0070Pulse decomposition module 254 may operate on a cross-correlation signal generated by the inverse transform module 208B and its corresponding amplitude envelope generated by the detection module 248 to determine complex coefficients (containing amplitude and phase information) and delay times for a set of major echoes. See the section below entitled "Pulse Decomposition" for a description of the pulse decomposition operation.

0071Echo sequencing module 252 may invoke a full update of the echo sequence using the cross correlation information and autocorrelation information, or, invoke a partial update of the echo phases using autocorrelation information alone. Full updates may be performed when necessary (e.g., when the channel is changed or an echo delay changes measurably). Partial updates may be performed at a higher rate than full updates in order to maintain the quality of equalization.

0072In step 269, a full update test may be performed to determine if a full update of the echo sequence is needed. The full update test may include determining if the delay of any echo of the current set of echoes has changed by an amount greater than a delay difference threshold. If one or more of the delays have changed by more than the threshold amount, a full update of the set of echoes may be invoked by passing control to step 266. If none of the delays have changed by more than the threshold amount, control may pass to coefficient update module 270.

0073Coefficient update module 270 may update the complex coefficients of the current set of echoes based on current estimates D_E of the delay times of the current set of echoes. For example, coefficient update module 270 may update the complex coefficients as indicated by expression (30). For example, the current estimates D_E of the delay times may be the delay time values computed during the most recent full update, or perhaps, the output of a predictive filtering operation on the sets of delay times computed during the full updates.

0074Doppler shift module 278 may compute differences of the echo phases between a current update and a previous update of the complex coefficients. The current update of the complex coefficients may be supplied by phase update module 270 or refinement module 280.

0075Rate control module 299 may update a recursion coefficient (e.g., the coefficient α of IIR filter (17)) based on the computed phase differences. The magnitude of the recursion coefficient may be a decreasing function of the maximum of the absolute values of the phase differences. For example, the recursion coefficient may be set equal to 0.3 for very fast-changing phases or 0.95 for slowly changing phases. The recursion coefficient is used in the averaging operation performed by averaging module 244.

0076In step 266, a single pulse test may be performed to determine if the number N_P of pulses, of sufficient amplitude, detected in the amplitude envelope of the most recent cross-correlation signal (or alternatively, the next available cross-correlation signal), is equal to one. If the number N_P equals one, control may pass to module 274. Otherwise (i.e., if the number N_P is greater than one) control may pass to module 272.

0077Sieving module 274 identifies minor echoes up to a complex-conjugate symmetry by operating on a filtered autocorrelation signal according to a pulse-sieving algorithm described below. The complex-conjugate symmetry implies that the delay times of the identified minor echoes have ambiguous signs.

0078Resolution module 282 operates on the minor echoes identified by sieving module 274 to resolve signs of delay times and complex coefficients of the minor echoes. Resolution module 282 may invoke a processing loop comprising steps 294, 286, 290 and 292.

0079Removal module 272 generates a power spectrum P_M for the major echoes (determined by the pulse decomposition module 254).

0080Identification module 276 estimates delay times and complex coefficients of the minor echoes using the power spectrum P_M of the major echoes and the received power spectrum. Identification module 276 may operate according to the algorithm described below in connection with equations (35) and (36). Identification module 276 calls inverse FFT module 288.

0081Refinement module 280 generates an update of the complete echo sequence (including delay times and complex coefficients) using (a) the major echo data computed by pulse decomposition module 254 and (b) minor echo data supplied by the resolution module 282 or the identification module 276.

0082Module 268 may receive data K_E defining a current estimate of the echo sequence from the coefficient update module 270 or refinement module 280. The echo sequence is defined by the complex coefficients and the time delays of the major echoes and minor echoes. In the case of a full update, the refinement module 280 supplies the updated time delays and updated complex coefficients it has just computed. In the case of a partial update, the coefficient update module 270 may supply the updated complex coefficients and the current estimates D_E of the delay times.

0083Module 268 may compute an estimate of the inverse echo spectrum from the data K_E. In particular, module 268 may first compute an estimate of the echo spectrum (i.e., the channel spectrum) according to the expression (28) from the data K_E, and then compute a stabilized reciprocal R_S of the echo spectrum estimate. The inverse echo spectrum estimate R_S may be forwarded to deconvolution module 242.

0084Module 268 may additionally compute an inverse echo sequence estimate by performing an inverse FFT on the inverse echo spectrum estimate R_S. The inverse echo sequence estimate may be forwarded to subwindow determination module 284.

0085Not all parts of the current equalization window are of equal quality because of the finite damping time of the inverse echo sequence estimate. Thus, subwindow determination module 284 may determine a subwindow of the current equalization window based on the inverse echo sequence estimate, i.e., a subwindow over which the quality of equalization may be deemed to be sufficient. Subwindow determination may be performed according to the methodology described in the section entitled "Data Normalization and Slicing".

0086A specification of the subwindow may be forwarded to module 258. Module 258 may discard equalized data outside the subwindow from the current equalization window.

0087Stabilization update module 250 may compute an update for one or more stabilization parameters (e.g., the parameter epsilon of expression (37)) using results generated by the subwindow determination module 284 (e.g., infinity norms of the inverse echo sequence computed on subintervals of the current equalization window). The one or more updated stabilization parameters may be forwarded to module 268 through subwindow determination module 284.

0088Module 268 may recompute the inverse echo spectrum estimate and the inverse echo sequence estimate from the echo spectrum estimate using the one or more updated stabilization parameters. The recomputed inverse echo spectrum estimate may be forwarded to deconvolution module 242. The recomputed inverse echo sequence estimate may be forwarded to subwindow determination module 284.

<u>Signal processing and measurements</u>

0089We presume wide sense stationarity of the 8VSB signal to simplify the calculation of expectations, so that expectations are given by averages over time. We also omit additive noise in most of the discussions. We let the sample period be 1, so that the frequency range is from -0.5 to 0.5 (-n to π in radian frequency). In the present application, we use 2 samples per symbol, so that the symbol frequency scales to 0.25.

0090The discrete Fourier transform is defined by <maths id="math0005" num="(1)"><math display="block"><msub><mi>s</mi><mi>k</mi></msub><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>n</mi></munder></mstyle><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πikn</mi><mo>/</mo><mi>N</mi></mrow></msup><mo>⁢</mo><msub><mi>u</mi><mi>n</mi></msub></mstyle></math><img file="EP1525726B9_D0005.tif" /></maths> The sum is from n = 0 to n = N-1. In the present work, N is a power of 2. The series s is termed the spectrum of the sampled time series u. The (signed) frequency is given by k/N for k ≤ N/2, and (k-N)/N for N/2 &lt; k ≤ N-1. The transform back to the time domain is <maths id="math0006" num="(2)"><math display="block"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>⁢</mo><msup><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><mi>e</mi></mstyle><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πikn</mi><mo>/</mo><mi>N</mi></mrow></msup><mo>⁢</mo><msub><mi>s</mi><mi>k</mi></msub></math><img file="EP1525726B9_D0006.tif" /></maths>

0091The time series we use are complex analytical signals, where the imaginary part is given by the Hilbert transform of the real part. We obtain the Hilbert transform by transforming to the frequency domain, multiplying positive frequency terms by -i and negative frequency terms by i. The zeroth element is multiplied by zero. An inverse transform to the time domain then yields the Hilbert transform. This convention is analogous to convolution with 1/πt.

0092We define <maths id="math0007" num="(3)"><math display="block"><mo>〈</mo><mi>f</mi><mo>〉</mo><mo>=</mo><munder><mi>lim</mi><mrow><mi>N</mi><mo>→</mo><mi>∞</mi></mrow></munder><mo>⁢</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>∑</mo><msub><mi>f</mi><mi>n</mi></msub></math><img file="EP1525726B9_D0007.tif" /></maths> and <maths id="math0008" num="(4)"><math display="block"><mo>〈</mo><msup><mi>f</mi><mo>*</mo></msup><mo>⁢</mo><mi>g</mi><mo>〉</mo><mo>=</mo><munder><mi>lim</mi><mrow><mi>N</mi><mo>→</mo><mi>∞</mi></mrow></munder><mo>⁢</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>∑</mo><msubsup><mi>f</mi><mi>n</mi><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>g</mi><mi>n</mi></msub><mo>⁢</mo><msup><mrow><mo>〈</mo><msup><mi>g</mi><mo>*</mo></msup><mo>⁢</mo><mi>f</mi><mo>〉</mo></mrow><mo>*</mo></msup></math><img file="EP1525726B9_D0008.tif" /></maths> where * denotes complex conjugation.

Variance of summed echoes

0093Let f(t) be the analytical signal associated with an 8VSB transmission. The sampled data from the k^th echo may be written as <maths id="math0009" num="(5)"><math display="block"><msub><mi>f</mi><mrow><mi>n</mi><mo>-</mo><msub><mi>τ</mi><mi>k</mi></msub></mrow></msub><mo>=</mo><mi>f</mi><mo>⁢</mo><mfenced><mi>n</mi><mo>-</mo><msub><mi>τ</mi><mi>k</mi></msub></mfenced></math><img file="EP1525726B9_D0009.tif" /></maths> An arbitrary echo-contaminated sequence may be written as <maths id="math0010" num="(6)"><math display="block"><msub><mi>g</mi><mi>n</mi></msub><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msub><mi>a</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>-</mo><msub><mi>τ</mi><mi>k</mi></msub></mrow></msub></mstyle></math><img file="EP1525726B9_D0010.tif" /></maths>

0094We take f to be a zero-mean function (〈f〉 = 0). Then the variance of g is <maths id="math0011" num="(7)"><math display="block"><mo>〈</mo><mi>g</mi><mo>*</mo><mi>g</mi><mo>〉</mo><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder></mstyle><msubsup><mi>a</mi><mi>j</mi><mo>*</mo></msubsup><mo>〈</mo><msubsup><mi>f</mi><mrow><mi>n</mi><mo>-</mo><msub><mi>τ</mi><mi>j</mi></msub></mrow><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>-</mo><msub><mi>τ</mi><mi>k</mi></msub></mrow></msub><mo>〉</mo></mstyle><msub><mi>a</mi><mi>k</mi></msub><mo>=</mo><msup><mi mathvariant="bold">a</mi><mo mathvariant="bold">*</mo></msup><mo>⁢</mo><mi mathvariant="bold">Γa</mi></math><img file="EP1525726B9_D0011.tif" /></maths> where the jk element of the hermitian matrix Γ is the complex correlation shown under the summation in (7). For delay differences |τ_j-τ_k| larger than the symbol time, the off-diagonal elements of r are small. For most purposes,

0095we may take Γ to be the identity matrix times the variance of f, σ²= 〈f*f〉. We then approximate the variance of a sum of echoes to be <maths id="math0012" num="(8)"><math display="block"><mo>〈</mo><msup><mi>g</mi><mo>*</mo></msup><mo>⁢</mo><mi>g</mi><mo>〉</mo><mo>≈</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>∑</mo><msup><mfenced open="|" close="|"><msub><mi>a</mi><mi>k</mi></msub></mfenced><mn>2</mn></msup></math><img file="EP1525726B9_D0012.tif" /></maths>

Correlation with PN511

0096We take the PN sequence to be nonzero only for one point per symbol, without convolution with any bandlimited pulse. This representation is therefore not an analytic signal: it is taken to be real. At the on-sample points it has the value specified in the ATSC standard. Define <maths id="math0013" num="(9)"><math display="block"><msub><mi>g</mi><mi>n</mi></msub><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msub><mi>p</mi><mi>k</mi></msub><mo>⁢</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub></mstyle></math><img file="EP1525726B9_D0013.tif" /></maths> where p_k is the PN511 sequence. This finite series will have some random hash (that is, g is a random sequence) in the absence of a PN sequence in f. We may show that the variance of (9) is <maths id="math0014" num="(10)"><math display="block"><mo>〈</mo><msup><mi>g</mi><mo>*</mo></msup><mo>⁢</mo><mi>g</mi><mo>〉</mo><mo>≈</mo><mn>511</mn><mo>×</mo><msubsup><mi>σ</mi><mi>p</mi><mn>2</mn></msubsup><mo>⁢</mo><msup><mi>σ</mi><mn>2</mn></msup></math><img file="EP1525726B9_D0014.tif" /></maths> Here, σ_p² is the variance of the (nonzero) elements of the PN sequence. If there is a PN sequence imbedded in f, there will be a peak of amplitude 511×σ_p². The ratio of the peak amplitude to the RMS level of g is (511)^1/2σ_p/σ. In the presence of multiple echoes, this ratio becomes, using the approximation in (8), <maths id="math0015" num="(11)"><math display="block"><mfrac><mi>correlation peak</mi><mi>RMS variation</mi></mfrac><mo>=</mo><mfrac><mrow><msup><mfenced><mn>511</mn></mfenced><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mspace width="1em" /></mrow></msup><mo>⁢</mo><msub><mi>σ</mi><mi>p</mi></msub><mfenced open="|" close="|"><msub><mi>a</mi><mi>j</mi></msub></mfenced></mrow><mrow><mi>σ</mi><mo>⁢</mo><msup><mfenced><mo>∑</mo><msup><mfenced open="|" close="|"><msub><mi>a</mi><mi>k</mi></msub></mfenced><mn>2</mn></msup></mfenced><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>≈</mo><mfrac><msup><mfenced><mn>511</mn></mfenced><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>m</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></math><img file="EP1525726B9_D0015.tif" /></maths> for the j^th echo. The rightmost expression gives a worst-case estimate for m equal strength echoes. In order to avoid errors in identification, it is important that we set the amplitude threshold sufficiently high for acceptance of a peak. A ratio of approximately 5 in (11) gives an acceptable error rate (approximately 0.001 errors in 4096 points). In the best case of a single dominant path and a single weaker path, the weaker path may be detectable to approximately 13 dB, but -10 dB is probably the lowest reliable level below the maximum peak. The background random variation is lower within 511 symbol-periods of large correlation peaks, due to the self-orthogonality of the PN sequence. However, detectability degrades in the presence of two or more large echoes.

Variance of autocorrelation

0097If we define <maths id="math0016" num="(12)"><math display="block"><msub><mi>g</mi><mi>m</mi></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><msubsup><mi>f</mi><mi>k</mi><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>f</mi><mrow><mi>k</mi><mo>+</mo><mi>m</mi></mrow></msub></math><img file="EP1525726B9_D0016.tif" /></maths> and evaluate the variance of g, we obtain <maths id="math0017" num="(13)"><math display="block"><mo>〈</mo><msup><mi>g</mi><mo>*</mo></msup><mo>⁢</mo><mi>g</mi><mo>〉</mo><mo>=</mo><msup><mi mathvariant="italic">Nσ</mi><mn>4</mn></msup><mo>=</mo><msubsup><mi mathvariant="italic">σ</mi><mi mathvariant="italic">a</mi><mn mathvariant="normal">2</mn></msubsup></math><img file="EP1525726B9_D0017.tif" /></maths>

0098This is the expected value for disjoint windows of N points (m > N). In the presence of multiple arrivals, this becomes <maths id="math0018" num="(14)"><math display="block"><mo>〈</mo><msup><mi>g</mi><mo>*</mo></msup><mo>⁢</mo><mi>g</mi><mo>〉</mo><mo>=</mo><msup><mi mathvariant="italic">Nσ</mi><mn>4</mn></msup><mo>⁢</mo><msup><mfenced><mo>∑</mo><msup><mfenced open="|" close="|"><msub><mi>a</mi><mi>k</mi></msub></mfenced><mn>2</mn></msup></mfenced><mn>2</mn></msup></math><img file="EP1525726B9_D0018.tif" /></maths>

0099This result neglects the effects caused by independent sets of identical delay-time differences. We expect such occurrences to be rare and to have modest effects. For autocorrelations of data in a window, we expect to see peaks of (complex) amplitude <maths id="math0019" num="(15)"><math display="block"><msub><mi>q</mi><mi mathvariant="italic">peak</mi></msub><mo>=</mo><msubsup><mi>a</mi><mi>j</mi><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>a</mi><mi>k</mi></msub><mo>⁢</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>⁢</mo><mfenced><mi>N</mi><mo>-</mo><mfenced open="|" close="|"><msub><mi>n</mi><mi>j</mi></msub><mo>-</mo><msub><mi>n</mi><mi>k</mi></msub></mfenced></mfenced></math><img file="EP1525726B9_D0019.tif" /></maths> where n_k is the delay of the kth echo. The time over which the data are correlated is reduced by the relative delay between the echoes, which affects the peak amplitudes.

0100The peaks are deterministic if the echo properties are constant over time, while the realizations of g may be treated as random. Thus, with a stack of M autocorrelations, the ratio of peak amplitudes to the underlying hash behaves as <maths id="math0020" num="(16)"><math display="block"><msup><mi>M</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⁢</mo><msup><mi>N</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>⁢</mo><mfrac><mfenced open="|" close="|"><msubsup><mi>a</mi><mi>j</mi><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>a</mi><mi>k</mi></msub></mfenced><mfenced><mo>∑</mo><msup><mfenced open="|" close="|"><msub><mi>a</mi><mi>l</mi></msub></mfenced><mn>2</mn></msup></mfenced></mfrac></math><img file="EP1525726B9_D0020.tif" /></maths> neglecting the amplitude effect in (15) due to the (generally) small delay differences. In principle, we may stack as many autocorrelations as we wish to reveal arbitrarily small echoes, but in practice the number of realizations may be limited by time variations in the properties of the echoes.

Autocorrelation stacking for detection of minor echoes

0101In practical computations, we work with finite windows (N = 4096 in the present work) and ultimately finite data. In the following, we use the terms correlation and autocorrelation to denote finite sums of the form <maths id="math0021" num="(4')"><math display="block"><msub><mi>r</mi><mi>m</mi></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover></mstyle><msubsup><mi>f</mi><mi>k</mi><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>g</mi><mrow><mfenced><mi>k</mi><mo>+</mo><mi>m</mi></mfenced><mo>⁢</mo><msub><mi>mod</mi><mspace width="1em" /></msub><mo>⁢</mo><mi>N</mi></mrow></msub></math><img file="EP1525726B9_D0021.tif" /></maths> Note that we omit normalization by N and employ a circular buffer. We may regard the realizations of the autocorrelation as being composed of a random and a deterministic component, as shown above. Instead of directly summing a set of autocorrelations, we use a stacking that has a fading memory of older autocorrelations but continuously updates the estimate of the stacked values using more recent data. Let q be the nominally constant deterministic component, and let n_k be the k^th realization of the random component. We define the continuously updated stack as <maths id="math0022" num="(17)"><math display="block"><msub><mi mathvariant="bold">y</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">=</mo><mi mathvariant="bold">q</mi><mo mathvariant="normal">+</mo><msub><mi mathvariant="bold">n</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">+</mo><mi>α</mi><mo>⁢</mo><msub><mi mathvariant="bold">y</mi><mrow><mi mathvariant="normal">k</mi><mo mathvariant="normal">-</mo><mn mathvariant="normal">1</mn></mrow></msub><mn mathvariant="normal">.</mn></math><img file="EP1525726B9_D0022.tif" /></maths>

0102The quantities q, n_k, y_k and y_k-l are interpreted as vectors (or sequences) of length N. Thus, the additions indicated in expression (17) are vector additions. The deterministic component y of y_k tends to the value <maths id="math0023" num="(18)"><math display="block"><mi mathvariant="bold">y</mi><mo mathvariant="bold">→</mo><mi mathvariant="bold">q</mi><mo mathvariant="normal">/</mo><mfenced><mn mathvariant="normal">1</mn><mo mathvariant="normal">-</mo><mi mathvariant="normal">α</mi></mfenced></math><img file="EP1525726B9_D0023.tif" /></maths>

0103The estimate of q is then asymptotically (1-α)y.

0104The variance of an arbitrary element of the noise term after stacking is <maths id="math0024" num="(19)"><math display="block"><mrow><mo>(</mo><msup><mrow><mo>∑</mo><msup><mfenced open="|" close="|"><msub><mi>a</mi><mi>i</mi></msub></mfenced><mn>2</mn></msup><mo>)</mo></mrow><mn>2</mn></msup></mrow><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>/</mo><mfenced><mn>1</mn><mo>-</mo><msup><mi>α</mi><mn>2</mn></msup></mfenced></math><img file="EP1525726B9_D0024.tif" /></maths> in the presence of multiple echoes, and using (14). After normalizing to obtain q directly, the variance is reduced by a factor of (1-α)² to yield a noise power of <maths id="math0025" num="(20)"><math display="block"><mfrac><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>α</mi><mo>⁢</mo><msup><mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mfenced><mn>1</mn><mo>-</mo><msup><mi>α</mi><mn>2</mn></msup></mfenced></mfrac><mo>⁢</mo><msup><mfenced><mo>∑</mo><mrow><mo>|</mo><msub><mi>a</mi><mi>i</mi></msub><mo>⁢</mo><msup><mrow><mo>|</mo></mrow><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>⁢</mo><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>α</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow></mfrac><mo>⁢</mo><msup><mfenced><mo>∑</mo><msup><mfenced open="|" close="|"><msub><mi>a</mi><mi>i</mi></msub></mfenced><mn>2</mn></msup></mfenced><mn>2</mn></msup><mo>⁢</mo><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup></math><img file="EP1525726B9_D0025.tif" /></maths>

0105The choice of α = 0.8 yields a leading factor (1-α)/ (1+α) = 1/9, and thus the equivalent noise reduction of stacking 9 autocorrelations. The contribution of the 10^th previous autocorrelation is reduced to 11%, so that the most recent realizations are strongly weighted, as was intended. Smaller values of a lead to faster decay of the contribution of older autocorrelations, but result in less reduction of noise. In highly dynamic situations, the decay time may have to be reduced in this manner.

Practical calculations and effects of finite bandwidth

0106The relatively fast roll-off of the 8VSB spectrum creates substantial ringing that makes peak selection difficult. To mitigate this problem, we use a weighting function in the frequency domain to multiply the products of spectra encountered in the calculation of correlations: <maths id="math0026" num="(21)"><math display="block"><mi>w</mi><mfenced><mi>f</mi></mfenced><mo>=</mo><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>+</mo><mi>cos</mi><mn>4</mn><mo>⁢</mo><mi mathvariant="italic">πf</mi></mtd><mtd><mfenced open="|" close="|"><mi>f</mi></mfenced><mo>&lt;</mo><mn>0.25</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfenced open="|" close="|"><mi>f</mi></mfenced><mo>≥</mo><mn>0.25</mn></mtd></mtr></mtable></math><img file="EP1525726B9_D0026.tif" /></maths>

0107Because of the finite bandwidth of the data, the time resolution of the measured delays is limited. The filtering of spectral products by (21) slightly worsens the time resolution, but it has the beneficial result of greatly simplifying the correlation peaks.

0108The choice of the limiting frequency of f = ±0.25 reflects the use of 2 samples per symbol and the fact that the roll-off in the raised cosine spectrum occurs at half the Nyquist frequency. The Fourier transform of w filters the band-passed form of a spike (constant spectrum) and yields the real part of the basis for decomposition of pulses in the time domain: <maths id="math0027" num="(22)"><math display="block"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><munderover><mo>∫</mo><mrow><mo>-</mo><mn>0.25</mn></mrow><mn>0.25</mn></munderover><mi mathvariant="italic">df</mi><mspace width="1em" /><msup><mi mathvariant="italic">e</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πift</mi></mrow></msup><mo>⁢</mo><mi>w</mi><mfenced><mi>f</mi></mfenced><mo>=</mo><mfrac><mrow><mi>sin</mi><mfrac><mi mathvariant="italic">π</mi><mn>2</mn></mfrac><mo>⁢</mo><mi>t</mi></mrow><mi mathvariant="italic">πt</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mfenced open="[" close="]"><mfrac><mrow><mi>sin</mi><mfrac><mi mathvariant="italic">π</mi><mn>2</mn></mfrac><mo>⁢</mo><mfenced><mi>t</mi><mo>+</mo><mn>2</mn></mfenced></mrow><mrow><mi mathvariant="italic">π</mi><mo>⁢</mo><mfenced><mi>t</mi><mo>+</mo><mn>2</mn></mfenced></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>sin</mi><mfrac><mi mathvariant="italic">π</mi><mn>2</mn></mfrac><mo>⁢</mo><mfenced><mi>t</mi><mo>-</mo><mn>2</mn></mfenced></mrow><mrow><mi mathvariant="bold-italic">π</mi><mo>⁢</mo><mfenced><mi>t</mi><mo>-</mo><mn>2</mn></mfenced></mrow></mfrac></mfenced></math><img file="EP1525726B9_D0027.tif" /></maths>

0109The small amount of roll-off in the signal spectrum before frequency 0.25 deviates from the assumption of a white spectrum in (22), but the errors we incur are very slight (-75 dB) because of the small magnitude of w in this region.

0110We rewrite (22) in terms of the sinc function as <maths id="math0028" num="(23)"><math display="block"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="{" close="}"><mi>sinc</mi><mfenced><mfrac><mi mathvariant="italic">πt</mi><mn>2</mn></mfrac></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mfenced open="[" close="]"><mi>sinc</mi><mo>⁢</mo><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><mfenced><mi>t</mi><mo>-</mo><mn>2</mn></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mi>sinc</mi><mo>⁢</mo><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><mfenced><mi>t</mi><mo>+</mo><mn>2</mn></mfenced></mrow><mn>2</mn></mfrac></mfenced></mfenced></mfenced></math><img file="EP1525726B9_D0028.tif" /></maths>

0111The Hilbert transform of (23) may be written <maths id="math0029" num="(24)"><math display="block"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="{" close="}"><mi>hsinc</mi><mfenced><mfrac><mi mathvariant="italic">πt</mi><mn>2</mn></mfrac></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mfenced open="[" close="]"><mi>hsinc</mi><mo>⁢</mo><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><mfenced><mi>t</mi><mo>-</mo><mn>2</mn></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mi>hsinc</mi><mo>⁢</mo><mfenced><mfrac><mrow><mi>π</mi><mo>⁢</mo><mfenced><mi>t</mi><mo>+</mo><mn>2</mn></mfenced></mrow><mn>2</mn></mfrac></mfenced></mfenced></mfenced></math><img file="EP1525726B9_D0029.tif" /></maths>

0112The sinc and hsinc functions are given by <maths id="math0030" num="(25)"><math display="block"><mi>sinc</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>sin</mi><mspace width="1em" /><mi>x</mi></mrow><mi>x</mi></mfrac></math><img file="EP1525726B9_D0030.tif" /></maths><maths id="math0031" num="(26)"><math display="block"><mi>hsinc</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mspace width="1em" /><mi>x</mi><mspace width="1em" /></mrow><mi>x</mi></mfrac></math><img file="EP1525726B9_D0031.tif" /></maths>

0113The complex pulse <maths id="math0032" num="(27)"><math display="block"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>+</mo><mi mathvariant="italic">ih</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>2</mn><munderover><mo>∫</mo><mn>0</mn><mn>0.25</mn></munderover><mi mathvariant="italic">df</mi><mo>⁢</mo><msup><mi mathvariant="italic">e</mi><mrow><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πift</mi></mrow></msup><mo>⁢</mo><mi>w</mi><mfenced><mi>f</mi></mfenced></math><img file="EP1525726B9_D0032.tif" /></maths>

0114is the basis for representation of the weighted correlation or autocorrelation peaks. The one-sided spectrum is the converse of the Kramers-Kronig relation: instead of causality in time, the Hilbert transform relation between the real and imaginary parts of the complex time series leads to "causality" in frequency.

0115<figref idref="f0006">Figure 5</figref> shows the complex pulse used to analyze the pulses in the time domain derived from the weighted correlations. In particular, note that the envelope has only a single maximum. The complex pulse was derived from equations (22) through (27). The peak value is 1. The abscissa is samples, 2 samples per symbol.

Pulse decomposition

0116The most troublesome aspect of the equalization problem has been the accurate definition of the echo sequence that defines the propagation channel. The presumed model of the channel is a set of discrete echoes. In the frequency domain, this model has the form <maths id="math0033" num="(28)"><math display="block"><mi>E</mi><mfenced><mi>f</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder><mo>⁢</mo><msub><mi>a</mi><mi>k</mi></msub></mstyle><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mo>⁢</mo><msub><mi mathvariant="italic">πifτ</mi><mi>k</mi></msub></mrow></msup></math><img file="EP1525726B9_D0033.tif" /></maths>

0117The weighted correlation formed by tapering this function with w(f) from (21) and transforming to the time domain is a sum of pulses of the form (27) with delays τ_k and with the amplitude and phase of the coefficients a_k. With ever longer training sequences, we could approximate this simple result with greater and greater accuracy. However, the correlation with the PN511 sequence shows more complexity, as shown in the following figures. Because of the finite bandwidth, the pulses may mutually interfere, making the recognition of separate echoes more difficult. Features of the PN511 correlation are visible, particularly the pulse amplitudes and the variability of the RMS variation near the correlation peaks. Because the data and the basis pulses are band-limited, the recovery of the locations of the pulses is ill-posed. If we were to allow a pulse to arise at every possible sample, the resulting linear system would actually be singular.

0118<figref idref="f0007 f0008">Figure 6</figref> illustrates the PN511 correlation showing two echoes. Peak values and background variances are consistent with theory. Numerical modeling uses reference PN series with values ±0.5, but the 8VSB signal is properly normalized. There is an additional factor of 0.5 present because of the average of the weighting function w(f). Predicted peak = 638.75, observed = 632. Predicted RMS variation = 52, observed = 51. Note the reduction of variation near pulses.

0119<figref idref="f0009">Figure 7</figref> is a detailed graph of <figref idref="f0007 f0008">Figure 6</figref> showing well separated zero-phase pulses. <figref idref="f0010">Figure 8</figref> is a detailed figure of PN511 correlation with three equal-amplitude zero-phase pulses approximately 110 samples behind the dominant echo. Simple peak-finding would not be adequate for this case.

0120We overcome the "ill-posedness" of the problem by limiting the number of possible echoes in any interval and testing the reduction of variance for each set of delay times. The finest increment of time used is on the order of one-half of the sample interval. The greatest error that can be made in timing is then 0.25 samples, which leads to a maximum phase error of ±22.5° at half the Nyquist frequency. The number of echoes in a given time segment varies from a minimum equal to the number of distinguishable peaks in the envelope of the PN511 correlation up to an adaptable limit. The endpoint in the fitting process comes when the residual drops to the one-sigma level of the background variation. The choice of delay times to be used is not exhaustive, but is guided by the properties of the least squares problem.

0121The fitting problem is nonlinear because of the arbitrary and variable imposition of a limited number of echoes, and because of the nonlinearity of varying the delay times. Only the coefficients of the expansion occur linearly, and the problem is analyzed as a succession of linear least squares problems for the coefficients.

0122The linear subproblems have the form <maths id="math0034" num="(29)"><math display="block"><msup><mfenced open="|" close="|"><mi mathvariant="bold">y</mi><mo mathvariant="bold">-</mo><mi mathvariant="bold">Ca</mi></mfenced><mn>2</mn></msup><mo>=</mo><mi>minimum</mi></math><img file="EP1525726B9_D0034.tif" /></maths> where y is a segment or a set of segments of the observed correlation. The columns of C are segmented in time exactly the same as in y, and each column has a single zero-phase pulse (27) with a specified delay and evaluated at the same time points as are in y.

0123The segmentation into intervals of the time samples in y, and consequently also in the columns of C, depends on the temporal distribution of correlation peaks. Pulses that are close enough to interfere with each other (up to approximately 30 samples apart) are treated together in a single fitting problem (29). The portions to be fitted need only extend approximately ±6 samples from the envelope peak, or without a gap over regions where peaks are less than ∼10 samples apart. This limits the fitting intervals to regions where the signal dominates. An interval may contain several peaks, but any single column of C has only one pulse in it, with its time samples segmented as in y.

0124The coefficients in vector a form a linear combination of the columns of C that minimizes the misfit in (29). Since (29) is always an overdetermined problem, the solution for a is <maths id="math0035" num="(30)"><math display="block"><mi mathvariant="bold">a</mi><mo mathvariant="normal">=</mo><msup><mfenced><msup><mi mathvariant="bold">C</mi><mrow><mo mathvariant="normal">*</mo><mi mathvariant="bold-italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="bold">C</mi></mfenced><mrow><mo mathvariant="normal">-</mo><mn mathvariant="bold">1</mn></mrow></msup><mo>⁢</mo><msup><mi mathvariant="bold">C</mi><mrow><mo mathvariant="normal">*</mo><mi mathvariant="bold-italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="bold">y</mi></math><img file="EP1525726B9_D0035.tif" /></maths> where *^T denote the adjoint (complex conjugate transpose). For pulses in C that are adequately separated in time, the inverse in (30) exists. If we substitute (30) in (29), we obtain <maths id="math0036" num="(31)"><math display="block"><msup><mfenced open="|" close="|"><mfenced><mi mathvariant="bold">I</mi><mo mathvariant="bold">-</mo><mi mathvariant="bold">C</mi><mo>⁢</mo><msup><mfenced><msup><mi mathvariant="bold">C</mi><mrow><mo mathvariant="bold">*</mo><mi mathvariant="italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="bold">C</mi></mfenced><mrow><mo mathvariant="bold">-</mo><mn mathvariant="bold">1</mn></mrow></msup><mo>⁢</mo><msup><mi mathvariant="bold">C</mi><mrow><mo mathvariant="bold">*</mo><mi mathvariant="italic">T</mi></mrow></msup></mfenced><mo>⁢</mo><mi mathvariant="bold">y</mi></mfenced><mn mathvariant="bold">2</mn></msup><mo>=</mo><mi>minimum</mi></math><img file="EP1525726B9_D0036.tif" /></maths>

0125If we define <maths id="math0037" num="(32)"><math display="block"><mi mathvariant="normal">M</mi><mo mathvariant="normal">⁢</mo><mo mathvariant="normal">=</mo><mtext mathvariant="normal"> </mtext><mi mathvariant="normal">C</mi><mo>⁢</mo><msup><mfenced><msup><mi mathvariant="normal">C</mi><mrow><mo mathvariant="normal">*</mo><mi mathvariant="italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="normal">C</mi></mfenced><mrow><mo mathvariant="normal">-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⁢</mo><msup><mi mathvariant="normal">C</mi><mrow><mo mathvariant="normal">*</mo><mi mathvariant="italic">T</mi></mrow></msup></math><img file="EP1525726B9_D0037.tif" /></maths> it is apparent that MM = M. That is, M is a projection operator, so that its nonzero eigenvalues are equal to 1. Using these properties of M, we can show that <maths id="math0038" num="(33)"><math display="block"><msup><mi mathvariant="bold">y</mi><mrow><mo mathvariant="bold">*</mo><mi mathvariant="bold-italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="bold">My</mi><mo mathvariant="bold">=</mo><msup><mi mathvariant="bold">y</mi><mrow><mo mathvariant="bold">*</mo><mi mathvariant="bold-italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="bold">C</mi><mo>⁢</mo><msup><mfenced><msup><mi mathvariant="bold">C</mi><mrow><mo mathvariant="bold">*</mo><mi mathvariant="bold-italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="bold">C</mi></mfenced><mrow><mo mathvariant="bold">-</mo><mn mathvariant="bold">1</mn></mrow></msup><mo>⁢</mo><msup><mi mathvariant="bold">C</mi><mrow><mo mathvariant="bold">*</mo><mi mathvariant="bold-italic">T</mi></mrow></msup><mo>⁢</mo><mi mathvariant="bold">y</mi><mo>=</mo><mi>maximum</mi></math><img file="EP1525726B9_D0038.tif" /></maths> is equivalent to (31).

0126The variations performed on (33) comprise moving the pulses in C around and adding new pulses (columns) to C as well. Incorporating these changes can be done fairly methodically, since the eigenvector decomposition of the span of C is equivalently done by diagonalizing the Gram matrix C*^TC or by simply conducting a Gram-Schmidt orthogonalization. The Gram-Schmidt method is not in general the most desirable means of accomplishing this decomposition, but it has the great advantage of making the addition of new columns or the variation of existing columns very straightforward to analyze.

0127In practice, the starting number of columns in C in a multiple-peak interval would be the number of distinguishable envelope peaks. The quadratic form in (33) is maximized with respect to the delays of the pulses. The changes in the delays are determined by a step-length damped Newton-Raphson method, and the C matrices are recalculated when any delay changes by more than one-half sample. After convergence, the envelope of the residual y - Ca is then examined for detectable peaks to determine if additional peaks are to be incorporated. The linear system becomes unstable as the peaks get closer than 2 samples apart, but in practical work, the residual after fitting the contribution of two such closely spaced peaks with a single peak would not be detectable. Experience shows that the resulting equalization is still satisfactory.

0128Once the reduction of the variance in (29) is sufficient, or the Gram matrix is too nearly singular, the process is terminated. All independent groups of echoes are evaluated in this manner. The coefficients in the a vectors are taken to be the complex amplitudes of the echo sequence, and the delay times are retained as well. Note that none of the delay times is taken to be the zero reference time. The zero reference time always coincides with a sample time, so that the symbols will line up on-sample upon equalization.

Retrieval of minor echoes from the autocorrelation

0129If there is only one echo detected by the PN511 correlation, then the determination of the remainder of the echo sequence is straightforward but tedious. Let the largest echo be denoted by subscript 0. Then <maths id="math0039" num="(34)"><math display="block"><mtable columnalign="left"><mtr><mtd><mrow><mo>|</mo><mi>E</mi><mfenced><mi>f</mi></mfenced><mo>⁢</mo><msup><mrow><mo>|</mo></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>j</mi></munder></mstyle><msup><mrow><msubsup><mi>a</mi><mi>j</mi><mo>*</mo></msubsup><mo>⁢</mo><mi>e</mi></mrow><mrow><mn>2</mn><mo>⁢</mo><msub><mi mathvariant="italic">πifτ</mi><mi>j</mi></msub></mrow></msup></mstyle><msup><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msub><mi>a</mi><mi>k</mi></msub><mi>e</mi></mstyle><msub><mrow><mo>-</mo><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πifτ</mi></mrow><mi>k</mi></msub></msup></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mstyle displaystyle="false"><mstyle displaystyle="true"><mo>=</mo><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msup><mfenced open="|" close="|"><msub><mi>a</mi><mi>k</mi></msub></mfenced><mn>2</mn></msup><mo>+</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder><mo>⁢</mo><msubsup><mi>a</mi><mn>0</mn><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>a</mi><mi>k</mi></msub><mo>⁢</mo><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πif</mi><mo>⁢</mo><mfenced><msub><mi>τ</mi><mi>k</mi></msub><mo>-</mo><msub><mi>τ</mi><mn>0</mn></msub></mfenced></mrow></msup><mo>+</mo><munder><mo>∑</mo><mi>k</mi></munder><mo>⁢</mo><msub><mi>a</mi><mn>0</mn></msub><mo>⁢</mo><msubsup><mi>a</mi><mi>k</mi><mo>*</mo></msubsup><mo>⁢</mo><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πif</mi><mo>⁢</mo><mfenced><msub><mi>τ</mi><mn>0</mn></msub><mo>-</mo><msub><mi>τ</mi><mi>k</mi></msub></mfenced></mrow></msup><mo>+</mo><munder><mo>∑</mo><mrow><mi>k</mi><mo>≠</mo><mi>j</mi><mo>;</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo>></mo><mn>0</mn></mrow></munder><mo>⁢</mo><msubsup><mi>a</mi><mi>j</mi><mo>*</mo></msubsup><mo>⁢</mo><msub><mi>a</mi><mi>k</mi></msub><mo>⁢</mo><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mo>⁢</mo><mi mathvariant="italic">πif</mi><mo>⁢</mo><mfenced><msub><mi>τ</mi><mi>j</mi></msub><mo>-</mo><msub><mi>τ</mi><mi>k</mi></msub></mfenced></mrow></msup></mstyle></mstyle></mtd></mtr></mtable></math><img file="EP1525726B9_D0039.tif" /></maths>

0130The constant term transforms to the central peak of the autocorrelation at zero time. The next two sums are called the primary terms of the autocorrelation, which include a factor of the dominant echo or its conjugate. The transform of these terms will be pulses at times ±(τ_k -τ₀) with conjugate phases at the opposite delays. The last summation contains the secondary terms, which are typically at least 10 dB down from the primary terms. Their lower amplitudes and their occurrence at the difference of the delay times of the primary terms characterize the secondary terms. Since we do not know a priori whether the delay times are positive or negative, we discriminate against peaks that occur at either the differences or the sums of the primary delay times. There is a possibility for error at this point, but experience indicates this to be a fairly effective method.

0131We first consider only positive delay times, since we have no basis for any other choice. The algorithm proceeds by ordering and eliminating the peaks in the autocorrelation, and thus, may be referred to herein as a "peak sieving algorithm". <ol id="ol0008" compact="compact"><li>1. Set a threshold for acceptance of peaks based on autocorrelation variance</li><li>2. Tabulate all peaks above threshold</li><li>3. Sort peaks by amplitude.</li><li>4. Assign complex amplitude and a positive delay to the two largest echoes</li><li>5. Difference and sum known delays, save in table</li><li>6. Reject peaks at difference or sum times as secondary</li><li>7. Accept next largest peak, assign amplitude and positive delay</li><li>8. Terminate if no more peaks</li><li>9. Loop on update of difference and sum table</li></ol>

0132At this point, all primaries are determined, but may not be correctly placed in time. If there are M peaks, we test the quality of equalization of the known PN511 sequence for the 2^M choices of signs. Once the signs are established, it is unnecessary to re-examine them unless the channel is changed or reacquired.

0133The pathological case of echoes located at delays of +τ and -τ from the dominant path may cause errors. Obtaining the properties of the minor echoes in the presence of more than one major echo is surprisingly easier. In fact, the method is more effective if the number of major echoes is greater than 2. The idea is very simple. We write the echo sequence in terms of its large echoes, F, and small echoes g, both functions of frequency. Then <maths id="math0040" num="(35)"><math display="block"><msup><mi>E</mi><mo>*</mo></msup><mo>⁢</mo><mi>E</mi><mo>=</mo><mfenced><msup><mi>F</mi><mo>*</mo></msup><mo>+</mo><msup><mi>g</mi><mo>*</mo></msup></mfenced><mo>⁢</mo><mfenced><mi>F</mi><mo>+</mo><mi>g</mi></mfenced><mo>=</mo><msup><mfenced open="|" close="|"><mi>F</mi></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>F</mi><mo>*</mo></msup><mo>⁢</mo><mi>g</mi><mo>+</mo><msup><mi mathvariant="italic">Fg</mi><mo>*</mo></msup><mo>+</mo><msup><mfenced open="|" close="|"><mi>g</mi></mfenced><mn>2</mn></msup></math><img file="EP1525726B9_D0040.tif" /></maths>

0134We subtract the known term |F|², and neglect the quadratic term |g|². We now approximate division by F* in a manner that avoids instabilities due to spectral minima: <maths id="math0041" num="(36)"><math display="block"><mover><mi>g</mi><mo>^</mo></mover><mo>=</mo><mfrac><mi>F</mi><mrow><msup><mfenced open="|" close="|"><mi>F</mi></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></mfrac><mo>⁢</mo><mfenced><msup><mfenced open="|" close="|"><mi>E</mi></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced open="|" close="|"><mi>F</mi></mfenced><mn>2</mn></msup></mfenced><mo>≈</mo><mi>g</mi><mo>+</mo><mfrac><msup><mi>F</mi><mn>2</mn></msup><mrow><msup><mfenced open="|" close="|"><mi>F</mi></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></mfrac><mo>⁢</mo><msup><mi>g</mi><mo>*</mo></msup><mo>+</mo><mi>O</mi><mfenced><msup><mi>g</mi><mn>2</mn></msup></mfenced></math><img file="EP1525726B9_D0041.tif" /></maths> omitting a term of O(ε²). This operation is a form of stabilized deconvolution that limits the effect of zeroes near or on the real frequency axis. The estimate g in (36) is transformed to the time domain for identification of the low amplitude echoes. The sparse train of pulses in the time domain representation of g stands out prominently compared to the transform of the following term, which is generally highly oscillatory and lower amplitude. As the spikes are resolved and pulse-decomposed in g, they are moved to F to improve the definition of the remaining echoes and to reduce the error from the neglected terms. The echo sequence is refined until there is no detectable residue in g. Remarkably, the features of g are placed correctly in time, and no further processing is required.

0135The only caveat regarding this procedure is a possible instability when there is a high degree of symmetry in F, such as when there are only two echoes in F that have nearly equal strength and conjugate phases. The ambiguities induced by the near symmetry in F may make the identification of g more difficult.

Deconvolution

0136Once the echo sequence has been determined, a stabilized deconvolution is applied to remove the effects of the channel. In the frequency domain, the echo-contaminated symbols have the form E(f)s(f), where s(f) is the spectrum of the original symbols. To recover s, we approximate division by E(f) in a stabilized manner: <maths id="math0042" num="(37)"><math display="block"><mover><mi>s</mi><mo>^</mo></mover><mfenced><mi>f</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>E</mi><mo>*</mo></msup><mfenced><mi>f</mi></mfenced></mrow><mrow><mo>|</mo><mi>E</mi><mfenced><mi>f</mi></mfenced><mo>⁢</mo><msup><mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow></mfrac><mfenced open="[" close="]"><mi>E</mi><mfenced><mi>f</mi></mfenced><mo>⁢</mo><mi>s</mi><mfenced><mi>f</mi></mfenced></mfenced></math><img file="EP1525726B9_D0042.tif" /></maths>

0137The stabilization term ε² serves to move the poles of the deconvolution filter away from the real frequency axis. Poles cause resonant behavior with ringing persisting for a time given by the reciprocal of the width of the resonance. This width scales as the distance of the pole from the axis, hence the effect of the stabilization is to lower the Q of the resonance. By increasing s, we may cause the ringing of any resonance to decay sufficiently fast to prevent significant "wrap-around" in the finite time widow available for processing. Larger vales of ε also reduce the enhancement of additive noise that plagues deconvolution.

0138The increase of s cannot be continued indefinitely, since the estimate in (37) tends to the convolution of the symbols with the autocorrelation of the echo sequence instead of a deconvolution. There are always echo sequences that cannot be adequately deconvolved in any given finite window.

Recursion

0139Recursive filtering is possible in some circumstances, most notably when a nearly 0 dB echo comes after the dominant path. For a single subsidiary echo of relative amplitude -a, the inverse echo sequence is a sequence of pulses of amplitude a^k at time kτ, where τ is the delay of the echo. If the amplitude does not decay to a relatively low value at the end of the finite processing window, even with the stabilization described above, then a recursive filter may be necessary. The stability of the recursive filter is assured in the case where |a| &lt; 1, but recursion based on more complex echo sequences is a subject worthy of investigation.. Recursion should also work in the presence of continuous phase variations in a.

Data normalization and slicing

0140After deconvolution, the data are filtered with the root-raised-cosine filter specified in the ATSC standard. Normalization is accomplished by requiring the variance of the equalized and filtered on-sample real values (I) to be 21 (automatic variance control). Data slicing consists of assigning the nearest odd integer (from -7 to 7) to the measured value of the output.

0141Because of the difficulties associated with analog phase locking, it is recommended that feedback be supplied by the equalizer to change the phasing. The quantity <maths id="math0043" num="(38)"><math display="block"><mi>u</mi><mo>=</mo><mo>〈</mo><mi mathvariant="italic">IQ</mi><mo>〉</mo></math><img file="EP1525726B9_D0043.tif" /></maths> is positive if the sampling is late and negative if early. The equalizer may provide a voltage based on (38) to apply to the local oscillator (LO) or the digital synchronous detector to correct the delay. It is possible to remove any delay at the point of deconvolution as long as the error in LO or synchronous detector frequency is small (e.g., less than 10 Hz). The bandwidth of the feedback to the oscillator should be small (e.g., less than 10 Hz) to avoid phase jitter that might derail the equalizer.

0142Not all parts of the equalization window are of equal quality because of the finite damping time of the inverse echo sequence. Generally, the most reliably equalized symbols occur where the inverse echo sequence has the lowest amplitudes. We partition the data window into a given number, typically 8-12, of non-overlapping subintervals and determine the maximum absolute amplitude (infinity norm) of the inverse echo sequence in each interval. The best equalization interval is the set of successive subintervals with the lowest average maximum amplitude. The computational burden of equalization is inversely proportional to the length of the subwindow in which there is an acceptable error rate, since this length limits the increment by which we may advance the processing window through the data. In difficult cases, the equalization algorithm balances two sources of errors. An insufficient amount of stabilization in (37) may cause the inverse echo sequence to wrap around in the processing window, contaminating the output symbols. If the stabilization is increased to damp the inverse echo sequence more severely, errors arise from the departure of (37) from simple deconvolution. Between these effects, it is possible that the region in which the error rate is acceptable will shrink to an unusably short interval, and equalization will fail.

Primary data flow path (indicated by bold arrows in Figures 4A-B)

0143The inphase (I) data from the digital output of the synchronous detector are the input to the equalizer. These data are Fourier transformed and the positive frequencies are retained. The data spectrum is subjected to stabilized deconvolution by the current echo spectrum and shaped by the root-raised-cosine filter prescribed by the ATSC standard. The deconvolved and shaped spectrum (positive frequencies only) is inverse Fourier transformed to the time domain to yield the equalized I and Q samples. The I component is required (by the ATSC standard) to satisfy the condition that the variance of its on-symbol values be 21 (in the absence of a PN sequence) for data in the current equalization interval. The average of the product of I and Q is updated to provide a uniform time-shift to the echo spectrum deconvolution and to adjust local oscillator phase or synchronous detector oscillator phase. Corrections to the oscillators are applied with low bandwidth to avoid sudden phase jitter in the input data. The data are sliced by rounding the sampled I value to the nearest odd integer between -7 and +7 inclusively. These integer values are the output of the equalizer.

Initialization and maintenance of training sequence location

0144When the receiver is turned on or the channel is changed, the echo sequence is unknown or may not be currently valid. Thus, a procedure is invoked to determine the echo sequence from the incoming data. The first step in this procedure is the detection of the training sequence (e.g., PN511 sequence). The spectrum of the input data is multiplied by the complex conjugate of the spectrum of the known training sequence to obtain a product spectrum (i.e., the cross-correlation spectrum). The positive frequency amplitudes in the product spectrum are retained and the negative frequency amplitudes are set to zero. The inverse Fourier transform of the product spectrum is computed to obtain the analytical signal of the cross-correlation function in the time domain. If the cross-correlation contains peaks that are sufficiently higher than the RMS variation of the cross-correlation, then the training sequence has been detected. The current field (e.g., in 8VSB transmission, a field comprising 313x832 = 260416 symbols = 520832 samples at 2 samples per symbol) is then analyzed for the stability of the autocorrelation function, while the start of the data window is adjusted so that the correlation peaks in the cross-correlation occur near the middle of the window.

0145The schedule of jumps in the data window may be chosen to preserve the location of the correlation peaks in the cross-correlation. For example, in 8VSB transmissions, each field contains 254 blocks of 2048 samples plus 640 extra samples. It is desirable to preserve the recurrence of the training sequence and the phase of the on-symbol samples in the data window. An arrangement that satisfies both of these requirements is to use a shift of 2050 samples for 188 times, and a shift of 2052 samples for the remaining 66 times. The number of equalized symbols recovered is constrained to equal the number of symbols in each shift (1025 or 1026) to preserve the location of the symbols with respect to the optimum equalization interval in the data window.

0146The correlation peaks in the cross-correlation may be decomposed in terms of a basis pulse as outlined above to determine delays and complex amplitudes of the largest echoes.

Autocorrelation processing

0147The autocorrelation spectrum may be stacked (i.e., averaged) with an exponentially fading memory as detailed in (17). In some embodiments, stacking is performed only on the real component of the power spectrum over nonnegative frequencies.

0148Information on the stability of the autocorrelation peak phases may be used to set the time constant for the stacking. The stability information may be obtained during the PN511 acquisition phase and continuously throughout data acquisition. The stability information may be computed by observing the time rate of change of the peak phases.

0149The analytical signal form of the autocorrelation in the time domain may be evaluated less frequently than the FFT of the incoming data. In some embodiments, the time constant for the stacking is used to determine the period between updates of the autocorrelation function. For example, the update period may be set equal to the stacking time constant.

0150The stacking time constant is a measure of the amount of time required for filter (17) to forget an input stimulus. For example, the stacking time constant may be interpreted as the value 1/(1-α), or more generally, B_C/(1-α), where B_C is a positive constant, where α is defined by equation (17).

0151The minor echo component of the echo sequence is determined from the analysis of the autocorrelation function as outlined above. Once the delays are established, the amplitude and phase of all the echoes are adjusted continuously (e.g., once per N_U blocks where N_U equals the stacking time constant) to account for the effects of dynamic propagation paths.

Circuit Observations

0152The FDEA algorithm may be implemented as a set of software programs which are executable on a set of one or more computers.

0153In one set of embodiments, the FDEA algorithm may be computed on a single-chip integrated circuit (IC), fabricated in deep-submicron semiconductor technologies suitable for very large scale integration (VLSI).

0154In another set of embodiments, the FDEA may be computed on a set of two or more integrated circuits.

0155Although the system and method of the present invention has been described in connection with various sets of embodiments, it is not intended to be limited to the specific form set forth herein, but on the contrary, it is intended to cover such alternatives, modifications, and equivalents, as can be reasonably included within the spirit and scope of the invention as defined by the appended claims.