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Application of Blind Equalization Techniques to Voiceband and RF Modems
Copyright © IFAC Adaptive Systems in Control and Signal Processing, Grenoble, France, 1992
APPLICATION OF BLIND EQUALIZATION TECHNIQUES TO VOICEBAND AND RF MODEMS J.R. Treichler Applied Signal Technology Inc .• l6IJ Sobranle Way . Sunnyvale, CA 94086. USA
Abstract. The past decade has seen the development of a new class of algorithms for adaptively choosing the pulse response of a digital filter. These are termed "blind" algorithms since their use does not require a explicit training sequence to be transmitted by the signal's originator. This makes them particularly useful for multidestinational and broadcast communications. This paper describes the status of both the practical and theoretical progress in the area of blind adaptive algorithms. Several examples are provided of the design considerations associated with the introduction of a blind equalizer into data signal demodulator. Issues of algorithm choice, filter length, adaptation coefficients, and expected convergence rates are described, using a bypotheticallIDTV broadcast modem signal as the running example. After a review of the general state of analytic work on blind algorithms, special attention is focused on a theoretical problem with significant practical impact - the observed misconvergence of blind algorithms in the face of input signals wbicb are not sufficiently "wbite". Keywords. Adaptive systems; broadcasting; digital filters; digital signal processing; blind equalization; digital communications.
ital filters to equalize the effects of multipath-induced dispersion on quadrature-amplitude-modulated (QAM) signals bas encouraged both immediate application to practical problems and analysis of the algorithm's expected bebavior.
INfRODUCTION The past decade has seen the development of a new class of algorithms for adaptively cboosing the pulse response of a digital filter. These are termed "blind" algorithm since their use does not require a explicit training sequence to be transmitted by the signal's originator. Because of the removal of this restraint, the application of such algorithms is growing rapidly, especially in situations where a signal with inadequately known parameters must be initially acquired in the presence of interference and dispersion with no direct interaction with the transmitter. This is particularly important in multidestinational and broadcast systems.
This paper describes the status of both the practical and theoretical progress in the area of blind adaptive algorithms. More attention is given to the dispersiondirected scbemes owing to their somewbat greater maturity. On the practical side several examples are provided of the design considerations associated with the introduction of a blind equalizer into data signal demodulator. Specifically, issues of algorithm choice, filter length, adaptation coefficients, and expected convergence rates are described, using a bypothetical HDTV RF broadcast modem signal as the running example.
A number of different blind additive algorithms bave been put forward and studied, ranging from Sato's original PAM equalization scbeme to those using performance criteria based on polyspectra. Most of the practical emphasis, however, bas been focused on dispersion-directed techniques, and particularly on a class known both as Godard's algorithm and the constant modulus algorithm (CMA). The demonstrated ability of this class of algorithms to blindly adapt dig-
After a review of the general state of analytic work on blind algorithms, special attention is focused on a theoretical problem with significant practical impact - the observed misconvergence of blind algorithms in the face of input signals wbicb are not sufficiently "white". This paper describes several situations in
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complete absence of additive noise the received signal is badly distorted, leading to many incorrect decisions and hence error-ridden data.
which misconvergence occurs. These are documented with constellations photographs showing the behavior of an actual demodulator when subjected to a variety of input signals. THE EQUALIZATION PROBLEM IN The design of most radio communications systems assumes that the receiving antenna or sensor receives only the energy transmitted directly to it by the transmitter. It is often the case, however, that the transmitted signal is unintentionally reflected. refracted, or scattered to the receiver, making the resulting received signal a linear combination of delayed and scaled versions of the transmitted signal. An example is shown in Fig. 1 which depicts the multiple paths by which a digital HDTV signal might take from the transmitter to one receiving antenna. The signal is received directly but other versions are received also via both static and moving reflectors. Bello (1963) shows that such a ''multipath'' propagation channel can usually be described by a potentially time-varying finite duration impulse response, and characterizations have been developed for the multipath propagation experienced in troposcatter systems, lineof-sight microwave (Rummler, 1980) and aircraft-tosatellite links (Takhar, 1976), among others. This tapped delay line (IDL) model is shown in Fig. 2. In the HDTV example illustrated in Fig. 1 there are static, slowly changing, and rapidly changing multipath components. Depending on the physical distribution of the various contributing reflectors, the delay spread, that is, the time span over which the various signal components arrive, may be many tens of microseconds. So called ''nonminimum phase multipath" is also common, occurring when one of the delayed reflected components is stronger than the signal first received. In the case of digital HDTV transmission the corresponding IDL model may typically have hundreds of time-varying coefficients.
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This paper assumes that the HDTV data to be broadcast is sent via a 64-point quadrature-amplitude modulated (QAM) signal operating a 4.8 Msymbolsls as shown schernatically in Fig. 4. Using trellis-coding techniques this signal can reliably carry 24 Mbls through a 6 MHz-wide channel, the limit mandated by the Fee in the US for television broadcasting. Such QAM signals are usually characterized by their constellations - an overlay of all possible amplitude and phase combinations expected, 64 in this case. The left portion of Fig. 5 shows an ideal received 64QAM constellation while the right portion shows an example of actual received data plots on the same scale. None of the original signal structure is visible. Some of the distortion stems from additive noise seen at the receiver but most is due to the dispersion introduced by the propagation channel.
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The convolutional propagation model allows easy visualization of the effects of such a channel on a data signal. Fig. 3 shows the simple example of a binaryvalued pulsed input signal propagating through a channel mode led by three nonzero tenns. Even in the
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The usual and, to date, most fruitful way of dealing with the effects of frequency-selective multipath is to construct a filter which operates on the received signal to remove the filtering induced by the propaga-
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tion channel. This operation, termed correcting or equalizing the multipath channel, presents the demodulator with a signal which resembles the transmitted signal as closely as possible.
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The overall approach is shown in Fig. 6. The transmitted signal is unintentionally filtered by the propagation channel and exposed to additive noise. The received signal is filtered to compensate for the channel filtering and, as possible, to remove the additive noise. The output of the correction filter is demodulated in the conventional way. Our goal here is to determine how the characteristics (e.g., impulse or frequency response) of the correction filter will be chosen. The propagation channel is usually unknown a priori and, even if known, is usually time-varying. Thus we desire to find some method to choose the pulse response of the correction filter which presumes no initial knowledge of the channel (other than gross characteristics) and we desire that the method also serve to track any changes in the channel with time. To the extent possible it is also desirable that the filter determination method not require detailed knowledge of the modulating signal at the transmitter, especially its carrier frequency and phase. An FIR filter is most commonly used for the equalizing filter itself, as shown in Fig. 7. Once the length and tap spacing is determined, the issue remaining is how to choose the coefficients of the equalizing filter.
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By far the most common techniques for choosing the proper equalizer coefficients are those which use a known and/or prearranged message waveform to permit modeling, and then correction, of the propagation channel. For virtually all voiceband telephone modems, a prearranged preamble or "training sequence" is used to set up the channel equalizer. This is shown in Fig. 8(a). By comparing the equalizer output with the locally stored version of the sequence
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Two classical approaches to adaptive equalization
A variant of this scheme is called "decision-direction". Shown in Fig. 8(b), it uses the demodulated output as the training reference. This technique is commonly used in data transmission systems after the equalizing filter is properly initialized and the output bit-error rate is low. The symbol decisions are then fed back to the equalizer and used as the reference signal. If the symbol error rate is low, then the demodulator's decisions closely mimic the transmitted sequence and hence the transmitted wavefonn. The use of the demodulated symbols as a locallygenerated reference signal is possible in digital systems because the finite number of levels in the transmitted source (e.g., four in QPSK) allows regeneration of the signal, largely removing the effects of noise and other minor perturbations, and because of the high likelihood that receiver decisions are correct In cases where the system input cannot be cleanly regenerated, as in most analog transmission systems, local derivation of a reference signal is not possible. If an actual or derived reference is available, then a plethora of sophisticated techniques
can be applied to compute the proper correction filter. For situations wbere no reference can be supplied or derived, as in conventional analog FM, these methods cannot be employed. Neither of these equalization approacbes is fully satisfactory for the JIDTV broadcast problem. The use of equalizer training sequences is very robust but is best suited to a point-to-point communications link, not a broadcast network wbere there are many receivers and eacb will need training upon its activation. Periodic transmission of training would permit spontaneous receiver activation but also reduces the transmission rate available for TV signals, already at a premium. Decision-direction, bowever, cannot be used alone since its operation requires that the symbol error rate be low. Some alternative is therefore needed to the use of prearranged training for initialization of the equalizer. BLIND EQUALIZATION The concept of adaptive equalization was the breakthrough needed to permit effective transmission of data signals. It allows, for example, signals at rates of up to 19.2 kbls to be sent over voiceband telepbone circuits wbicb would support no more than 2400 or at most 4800 bls without adaptive equalization. Even so those tecbniques suffered from the constraint that the equalizer must be initialized by "training" it at the beginning of eacb transmission. While not a practical concern for most "dialup" and leased line modems, it became an issue as engineers explored the application of digital versus analog modulation in broadcast and "multidrop" networks. In such systems it is higbly desirable that the transmission of known training sequences be avoided. Tbis led to the search for adaptation algorithms capable of "blindly" determining the equalizer's coefficients. The class of algorithms discussed in this paper was developed by at least two independent groups in the late 1970's. Godard's work (1980) was specifically motivated by the desire to use 16-QAM signals over voiceband multipoint networks. The wode by Treicbler and bis associates (1983) bad been initially motivated by the need for adaptive equalization of analog modulation, particularly FM. It was then sbown to extend to PSK and QAM signals as well. Even given these different motivations, both suggested the same algorithm.
fixed amplitude or modulus, while the presence of any additive degradation, wbether it be noise, interference, or multipath-induced distortion, caused instantaneous variations in this modulus. This is shown in Fig. 9. The constant modulus algorithm measures these variations and uses them to adapt the coefficients of adaptive filter to best remove these additive terms from the filter output. While others have been postulated, analyzed, and employed, the first constant modulus cost function lcm studied was given by J cm
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where A is the modulus of the undistorted constant envelope signal and y(k) is the complex-valued output of the adaptive filter. Note that 1cm;: 0 wben ly(k)1 =A and is positive otherwise. This particular form for lcm was cbosen since its gradient with respect to the coefficients of an FIR equalizer is easy to compute and requires no division or other normalization. Using this performance function as the starting point in the development of an approximate gradient descent yields the so-called ''2-2'' constant modulus algorithm for updating the filter coefficient vector W(l), W(l+l) = W(l)+J,1[ lyI 2_ A 2]y(k)X*(k) , (2)
where I is the update index,J..L is a small positive adaptation constant and X(k) is the regressor vector, the vector containing the data samples in the FIR delay line wben the output y(k) is computed. ImH
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As discussed in (Treichler, 1983) this algorithm proved very effective in blindly equalizing analog constant envelope signals as well as PSK and later QAM signals. Its insensitivity to carrier phase and frequency uncertainties increased its practical attractiveness even more. On the theoretical front the concept of constant modulus restoral was extended to the more general concept of adapting a digital filter to restore some property of the transmitted signal.
The constant modulus algorithm (CMA) was developed (Treicbler, 1983) as a method for adaptively cboosing the pulse response of a digital filter in sucb a way as to remove correlated and uncorrelated additive interference from a constant envelope signal, sucb as a frequency- or phase-modulated carrier. It was based on the observation that the complex-valued representation of a constant envelope signal had a
Godard (1980) developed exact! y the same algorithm from a different perspective. He observed that the 446
presence of group delay distortion in a voiceband transmission channel tended to disperse the received data symbols away from their intended QAM constellation points. He sought a gradient descent algorithm which would tend to reduce this dispersion. While not immediately intuitive, he postulated the algorithm shown above and proved that even though the constellation points do not fall on the circle of radius A the performance function is minimized at a tilter coefficient solution which minimizes the received signal dispersion, hence Godard's chosen description for the algorithm - "dispersion direction". The algorithm's effectiveness can be seen by examining the four plots shown in Fig. 10. The upper left plot is the same as that shown on the right of Fig. 5, a 64-QAM signal so completely dispersed that its structure is invisible. After 25K samples the CMAdirected equalizer has recovered the signal constellation to the degree than the points are clearly defined, thus permitting the use of decision direction. The constellation after successful carrier acquisition and decision-directed equalization is shown on the lower right.
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In addition to dealing with additive broadband noise and the dispersion effects induced by multipath propagation, the practical adaptive equalizer must also accommodate inband and out-of-band interference. When the interferers are relatively narrowband and limited in number, a high-order adaptive equalizer can suppress them without harming the signal of interest to any great degree. Strong narrowband interference does raise the potential of "capturing" the constant modulus algorithm, however, as described in (Treichler, 1983) and quantified to some degree in (Treichler, 1985). To cope with this possibility the adaptive equalizer can be initially operated as a "whitening filter" by minimizing the filter output power subject to the constraint that one filter coefficient is fixed. Such a filter will adapt to notch narrowband input signals. This has the effect of creating an initial filter for the CM A-direction mode which minimizes the probability of capture on any of the narrowband inputs .
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The demonstrated ability of the constant modulus algorithm to "blindly" adapt digital filters to equalize the effects of multipath-induced dispersion on QAM signals has encouraged analysis of the algorithm's expected behavior by a number of research groups. Several of these analyses contain proofs of the algorithm's guaranteed convergence to an optimum and satisfactory solution subject to a variety of conditions thought not to be practically taxing. As will be seen, these proofs do not tell the whole story. DEALING WITI-I NARROWBAND INTERFERENCE
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Consideration of all of these factors leads to the receiver architecture shown in Fig. 11. The incoming HDTV signal is tuned, filtered, gain controlled, and applied to the equalizer's filter. The filter output is carrier locked and the resulting soft decisions fed to a Viterbi decoder. The gradient update machinery is driven by one of four inputs, each used for a particular situation or phase of operation. In typical operation the whitening mode is used first, to notch strong narrow band interferers, the CMA mode is used next to "open the eye", and finally the decision-direction mode is used to optimize the equalizer's coefficients and track any environmental changes.
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After 25 kbauds and carrier phase lock After 50 kbauds Fig. 10 Scatter plots during equalizer acquisition While this class of algorithms is probably the best known, others preceded it and more have been developed since. Proakis and Stein (1990) report such development efforts in the early 1970's. Sato (1975) published an algorithm for blind equalization of pulse-amplitude modulation (PAM). The CMNdispersion-direction algorithms can in fact be viewed as two-dimensional extensions of Sato's approach. Godard published a scheme termed the "reduced constellation" algorithm, or RCA, which grouped constellation points together to decrease the required accuracy of the feedback in a decision-direction approach. Since the emergence of CMNdispersiondirection, improvements and extensions have been suggested, including those by Benveniste and Goursat (1984) and Picchi and Prati (1987). Blind equaliza-
A CURRENT PROBLEM: MISCONVERGENCE DUE TO INCOMPLETE INPUT EXCITATION As noted in the section Blind Equalization the demonstrated ability of the constant modulus algorithm (CMA) to "blindly" adapt digital filters to equalize the effects of multipath-induced dispersion on quadrature amplitude modulated (QAM) signals has 447
encouraged analysis of the algorithm's expected behavior by a number of research groups. Several of these analyses (Godard, 1980; Treichler, 1983; Benveniste, 1984; Foschini, 1985) contain proofs of the algorithm's guaranteed convergence to an optimum and satisfactory solution subject to a variety of conditions thought not to be practically taxing. In practical application, however, a number of circumstances have occurred in which CMA fails to converge, or, equally bad from a practical standpoint. converges to a solution which fails to equalize the input signal.
be needed to equalize the signal to the degree obtained with no multipath in Fig. 12(a).
(a) Spectrum and Constellation in the Absence of Multipath
Fig. 11 Block diagram of a broadcast receiver based on a multimode equalizer
(b) Spectrum and Constellation in the Presence of
Heavy Specular Multipath
This section describes several situations in which misconvergence occurs, suggesting that a firmer analytical understanding is needed of the behavior of blind algorithms in the presence of cyclostationary and/or quasi-periodic, non-white inputs. While this analytical understanding is not yet established, the practical experience reported here can be used immediately by those designing new digital communications systems. We will again use the broadcast lIDTV problem as a case in point
Fig. 12 Successful operation of a CM A-based adaptive equalizer with an NEC 16-QAM signal, with and without multipath Observed Misconvergence and Apparent Cause Sil:nall:eneration model. Sixteen-level quadratureamplitude modulation is commonly used in terrestrial line-of-sight microwave transmission of telephone signals. A common design for the transmitter section of a 16-QAM modem is shown in Fig. 13.
Baseline Observation As discussed in previous sections and illustrated in Fig. 11, CM A-directed equalization is used in the initial acquisition or reacquisition of a QAM signal. The success of CMA in this acquisition mode can be seen by examining Fig. 12. Fig. 12(a) shows the spectrum of a 33 megabaud, 16-QAM signal from an NEC modem, plus the equalized, carrier-locked constellation. In this case there is no dispersion impressed on the signal other than the pulse shaping imposed at the transmitter. Fig. 12(b) shows the spectrum and resulting constellation in the presence of a strong specular multipath. The presence of multipath is indicated in the scalloping seen in the power spectrum of the received signal. Without equalization the signal's "eye" is closed and demodulation is impossible. The CMA-directed equalizer, however, opens the eye and permits successful acquisition of the decisiondirected modes. The residual effects of the multipath propagation are still seen after full acquisition, as evidenced by the large constellation "points". The multipath in this case is so strong that an equalizer with hundreds of taps (rather than the 64 used here) would
Fig.13 Block diagram of a generic 16-QAM modulator for digital radios Multiplexing equipment is used to combine up to several thousand PCM telephone channels into a single serial stream, which is then "randomized" in some fashion to "whiten" the bit stream, making the spectrum of the transmitted signal uniform in amplitude across its assigned band and thereby reducing any interference to spectraIly adjacent signals. A common method of performing this randomization is to cyclically add (modulo-2) a selected sequence of length L, which is typically a section of a pseudorandom (PN) sequence of equal or somewhat greater length. The combined sequence then broken into 4-
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bit nibbles and each transmitted as one of the 16 possible symbols present in a 16-QAM signal. An implication of this technique is that when the mu 1tiplexed telephone signal is essentially idle, for example, in the middle of the night, the added sequence dominates. As a result the transmitted signal is repetitive with period at least L, and perhaps U4, if Lis divisible by 4 as it commonly is.
with two missing tributaries can appear as a nonsquare 8-QAM pattern. The resulting signal is essentially a quadrature partial response (QPR) waveform, a class known to be problematical to equalize.
Experimental observations. The implications to blind equalization algorithms can be seen by examining the constellation photographs seen in Fig. 14. Fig. 14(a) shows the constellation of a properly processed 16-QAM signal operating at 25 megasymbols/s. Achieving this condition requires gain adjustment, symbol timing synchronization, adaptive equalization, and carrier frequency acquisition. Fig. 14(b) shows the received signal when only gain adjustment and real-to-quadrature conversion have been done. Clearly the "eye" is closed. After symbol timing synchronization is attained and when L, the additive sequence length, is very long (e.g., greater than 1O~, the constant modulus algorithm (CM A) can be reliably employed to open the eye, even without carrier frequency acquisition. Fig. 14(c) shows this constellation. The appearance of the three rings indicates the opening of the eye. These rings correspond to the rotation-induced blurring of the 16-QAM pattern caused by lack of carrier frequency lock.
(a) Successful recovery of a 16-QAM, 25 megabaud Signal
(b) Constellation of Received Signal, with symbol timing achieved but without equalization or carrier recovery
If the additive sequence length is much lower, then false convergence of CMA can occur. Such a case is shown in Fig. 14(d). In this case the length Lis 256, four times the length of the adaptive equalizer's regressor vector. (Commercial QAM modems are built which use randomizers as short as 128.) Note that in this situation the algorithm converges to a single "fuzzy" ring, and that the constellation's eye is not open. Thus blind equalization fails and with it the complete acquisition procedure for the demodulator.
(c) Constellation of the Same Signal After Successful Blind Equalization but Before Carrier Acquisition; Note the open "eye"
Considerable laboratory experimentation has been done using various constellation sizes, equalizer length, PN sequence lengths, and various equalizer parameters. Invariably these experiments indicate that when the sequence length L is long compared to the filter and its principal adaptation modes, the convergence is reliable. When the sequence length is short, however, that is, commensurate in length with the equalizer, then false convergence is common, but even then not guaranteed. Some specific cases of interest include the following : • Idle multiplexer tributaries, with tributary-level randomizing: With no randomizing of the composite signal, this often leads to the limiting case of unequal constellation point distribution - points which are completely missing. A 16-QAM signal
(d) Constellation of the Same Signal After Unsuccessful Blind Equalization; Eye is not open and carrier acquisition is impossible Fig. 14 Particular instance of CMA misconvergence traced to input insufficiency 449
• Lightly loaded tributaries. PN reset randomizer. sequence length L commensurate with the equalizer's length: In this case the input is essentially periodic. producing a "liney" spectrum. Examples of this phenomenon are shown in Fig. 15. Fig. 15(a) shows the power spectrum of a 16-QAM modulator driven by a very long (224 _1) PN sequence. This results in symbols which are independent and identically distributed over the limited period of the spectrum calculation. Fig. 15(b) shows the spectrum when driven by a very short sequence. L=63 in this case. The spectrum is no longer relatively smooth but is completely composed of spectrallines. The same effect appears in Fig. 15(c) where L=255 and the line spacing is four times closer. When confronted with a line-dominated spectrum a CMA-directed adaptive filter can often converge to a solution which isolates a single spectral component. While inappropriate for equalization. the converged solution minimizes the CMA performance criterion even bener than the "proper" solution would. This "capture" phenomenon is discussed in (Treichler. 1985). • Lightly loaded tributaries. PN reset randomizer. sequence length L much longer than the equalizer but much shorter than the nominal convergence time: In this case the signal autocorrelation matrix appears to have full rank at all times. but the adaptive algorithm's control loop is driven periodically by the cyclic nature of the input since its frequency is 100 high for the loop to track.
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Suspected cause. "Input sufficiency" is immediately assumed in essentially all analyses of blind equalizer convergence. In particular it is always presumed that the transmitted constellation points are identically distributed and appear independently from one another (Le .• the symbols are lID). It is further presumed that the adaptive equalizer has such high order that its length in no way affects the convergence of the adaptive algorithm. Ding (1990) has demonstrated the shortcomings in the second assumption but the first was thought to be reasonably representative of practical applications. In fact, it is not and usually will not unless the signal is specifically designed to achieve it The high-capacity telephony signals described previously (Signal generation model) tend to fail these input sufficiency tests for one or more of the following reasons:
Fig. 15 The impact of periodic PN inputs on the spectra of 16-QAM signals • "Randoming" is not complete or adequate. The degree of randomization needed to avoid adjacent channel interference is much less than that needed to provide statistical stationarity over the convergence interval of an adaptive equalizer. In general a modem designer would choose a short sequence to minimize the time needed for demultiplexer synchronization. • Differential encoding correlates otherwise uncorrelated symbols. Implications to the Design of an HDTV Broadcast System
• Input tributaries are sometimes lightly loaded (e.g .• in the middle of the night)
While all of the theoretical underpinnings are not yet fully understood. the experimental results attained to date and reported in the paper carry lessons to the designer of an HDTV broadcast system. These include. at the least, the following:
• Sometimes tributaries are completely missing (e.g .• when a new radio system is installed with excess capacity)
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• Avoid source coding and FEC designs which result in data sequences which are idle, almost idle, or periodic for long intervals of time
Ding, Z., (1990). Application Aspects of Blind Adaptive Equalizers in QAM Data Communications, Ph.D. Dissertation, Comell University.
• Avoid the use of "short" randomizers, particularly of the additive (e.g., PN reset) type
Foschini, GJ., (1985). Equalizing Without Altering or Detecting Data, AT&TTechnical Journal, Vol. 64, pp. 1885-1911.
•
If only one signal constellation is to be employed, consider using a performance function which makes more use of that lmowledge, such as the COMA scheme discussed in (de Victoria, 1991).
Godard, D.N., (1980). Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems,lEEE Transactions on Communications, Vol. COM-28, pp. 1867-1875, November, 1980.
CONCLUSIONS
Jablon, N.K., (1992). Joint Blind Equalization, Carrier Recovery, and TlDling Recovery for Highorder QAM Signal Constellations, to appear in IEEE Transactions on Signal Processing, June 1992.
About a decade ago the concept of blind equalization was brought forward by a number of research groups. Since then a number of algorithms have been suggested, analytical work has been done, and operational hardware using them has been successfully designed, built, and shipped. The latter fact is a testament to functional need for blind equalization. Even so much remains to be done in the analytical theater. The algorithms currently in use, CMNdispersiondirection, for example, are lmown to be very slow to convergence and, as illustrated in the section A Cur-
Picchi, G . and Prati, G., (1987). Blind Equalization and Carrier Recovery Using "stop-and-go" Decision-directed Algorithm, IEEE Transactions on Communications, Vol. COM-35, No. 9, pp. 877-887. Proakis, J .G. and Stein, S., (1990). Personal communication, Ruidoso, NM.
rent Problem: Misconvergence Due to Incomplete Input Excitation, to lack robustness in the face of sig-
Rummler. W.D., (1980). Tlme- and FrequencyDomain Representation of Multipath Fading on Line-of-Sight Microwave Paths, Bell System Technical Journal, Vol. 59, No. 5, pp. 763796.
nals not adhering the stated regularity assumptions. More work needs to be done to understand the algorithms we're currently using and to develop better ones.
Sato, Y., (1975). A Method of Self-recovering Equalization for Multilevel Amplitude-modulation Systems, IEEE Transactions on Communications, Vol. COM-23, pp. 697-682, June 1975.Takhar, G.S. and Gupta, S.c., (1976). Discrete Estimation of Continuous AngleModulated Channels for Aeronautical Communication, IEEE Transactions on Communications, Vol. COM-24, No. 3.
ACKNOWLEOOMENTS The long term contributions of Mike Larimore and Rick Johnson to the analytical and practical understanding of adaptive digital filters is highly appreciated. Also gratefully aclmowledged is the incredible effort put in by the staff at Applied Signal Technology, Inc. in order to create real products out of the theoretical ideas. Vin Wolff and his crew contributed the photographs of signal spectra and constellations.
Treichler, J.R. and Agee, B.G., (1983). ANew Approach to Multipath Correction of Constant Modulus Signals, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-31, pp. 459-472.
REFERENCES Bello, P. A., (1963). Characterization of Randomly TlDle-Invariant Linear Channels, IEEE Transactions on Communication Systems, Vol. CS11, pp. 360-393.
Treichler, J.R. and Larimore, M.G., (1985). The Tone Capture Properties of CM A-based Interference Suppressors, IEEE Transactions on Acoustics. Speech, and Signal Processing, Vol. ASSP-33, pp. 946-958.
Benveniste, A. and Goursat, M., (1984). Blind Equalizers, IEEE Transactions on Communications, Vol. COM-32, pp. 871-883. de Victoria, EL., (1991). An Adaptive Blind Equalization Algorithm for QAM and QPR Modulations: The Concentrically Ordered Modulus Algorithm, Proceedings of the 25th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, November 4-6.
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APPLICATION OF BLIND EQUALIZATION TECHNIQUES TO VOICEBAND AND RF MODEMS J.R. Treichler Applied Signal Technology Inc .• l6IJ Sobranle Way . Sunnyvale, CA 94086. USA
Abstract. The past decade has seen the development of a new class of algorithms for adaptively choosing the pulse response of a digital filter. These are termed "blind" algorithms since their use does not require a explicit training sequence to be transmitted by the signal's originator. This makes them particularly useful for multidestinational and broadcast communications. This paper describes the status of both the practical and theoretical progress in the area of blind adaptive algorithms. Several examples are provided of the design considerations associated with the introduction of a blind equalizer into data signal demodulator. Issues of algorithm choice, filter length, adaptation coefficients, and expected convergence rates are described, using a bypotheticallIDTV broadcast modem signal as the running example. After a review of the general state of analytic work on blind algorithms, special attention is focused on a theoretical problem with significant practical impact - the observed misconvergence of blind algorithms in the face of input signals wbicb are not sufficiently "wbite". Keywords. Adaptive systems; broadcasting; digital filters; digital signal processing; blind equalization; digital communications.
ital filters to equalize the effects of multipath-induced dispersion on quadrature-amplitude-modulated (QAM) signals bas encouraged both immediate application to practical problems and analysis of the algorithm's expected bebavior.
INfRODUCTION The past decade has seen the development of a new class of algorithms for adaptively cboosing the pulse response of a digital filter. These are termed "blind" algorithm since their use does not require a explicit training sequence to be transmitted by the signal's originator. Because of the removal of this restraint, the application of such algorithms is growing rapidly, especially in situations where a signal with inadequately known parameters must be initially acquired in the presence of interference and dispersion with no direct interaction with the transmitter. This is particularly important in multidestinational and broadcast systems.
This paper describes the status of both the practical and theoretical progress in the area of blind adaptive algorithms. More attention is given to the dispersiondirected scbemes owing to their somewbat greater maturity. On the practical side several examples are provided of the design considerations associated with the introduction of a blind equalizer into data signal demodulator. Specifically, issues of algorithm choice, filter length, adaptation coefficients, and expected convergence rates are described, using a bypothetical HDTV RF broadcast modem signal as the running example.
A number of different blind additive algorithms bave been put forward and studied, ranging from Sato's original PAM equalization scbeme to those using performance criteria based on polyspectra. Most of the practical emphasis, however, bas been focused on dispersion-directed techniques, and particularly on a class known both as Godard's algorithm and the constant modulus algorithm (CMA). The demonstrated ability of this class of algorithms to blindly adapt dig-
After a review of the general state of analytic work on blind algorithms, special attention is focused on a theoretical problem with significant practical impact - the observed misconvergence of blind algorithms in the face of input signals wbicb are not sufficiently "white". This paper describes several situations in
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complete absence of additive noise the received signal is badly distorted, leading to many incorrect decisions and hence error-ridden data.
which misconvergence occurs. These are documented with constellations photographs showing the behavior of an actual demodulator when subjected to a variety of input signals. THE EQUALIZATION PROBLEM IN The design of most radio communications systems assumes that the receiving antenna or sensor receives only the energy transmitted directly to it by the transmitter. It is often the case, however, that the transmitted signal is unintentionally reflected. refracted, or scattered to the receiver, making the resulting received signal a linear combination of delayed and scaled versions of the transmitted signal. An example is shown in Fig. 1 which depicts the multiple paths by which a digital HDTV signal might take from the transmitter to one receiving antenna. The signal is received directly but other versions are received also via both static and moving reflectors. Bello (1963) shows that such a ''multipath'' propagation channel can usually be described by a potentially time-varying finite duration impulse response, and characterizations have been developed for the multipath propagation experienced in troposcatter systems, lineof-sight microwave (Rummler, 1980) and aircraft-tosatellite links (Takhar, 1976), among others. This tapped delay line (IDL) model is shown in Fig. 2. In the HDTV example illustrated in Fig. 1 there are static, slowly changing, and rapidly changing multipath components. Depending on the physical distribution of the various contributing reflectors, the delay spread, that is, the time span over which the various signal components arrive, may be many tens of microseconds. So called ''nonminimum phase multipath" is also common, occurring when one of the delayed reflected components is stronger than the signal first received. In the case of digital HDTV transmission the corresponding IDL model may typically have hundreds of time-varying coefficients.
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This paper assumes that the HDTV data to be broadcast is sent via a 64-point quadrature-amplitude modulated (QAM) signal operating a 4.8 Msymbolsls as shown schernatically in Fig. 4. Using trellis-coding techniques this signal can reliably carry 24 Mbls through a 6 MHz-wide channel, the limit mandated by the Fee in the US for television broadcasting. Such QAM signals are usually characterized by their constellations - an overlay of all possible amplitude and phase combinations expected, 64 in this case. The left portion of Fig. 5 shows an ideal received 64QAM constellation while the right portion shows an example of actual received data plots on the same scale. None of the original signal structure is visible. Some of the distortion stems from additive noise seen at the receiver but most is due to the dispersion introduced by the propagation channel.
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The convolutional propagation model allows easy visualization of the effects of such a channel on a data signal. Fig. 3 shows the simple example of a binaryvalued pulsed input signal propagating through a channel mode led by three nonzero tenns. Even in the
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The usual and, to date, most fruitful way of dealing with the effects of frequency-selective multipath is to construct a filter which operates on the received signal to remove the filtering induced by the propaga-
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tion channel. This operation, termed correcting or equalizing the multipath channel, presents the demodulator with a signal which resembles the transmitted signal as closely as possible.
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The overall approach is shown in Fig. 6. The transmitted signal is unintentionally filtered by the propagation channel and exposed to additive noise. The received signal is filtered to compensate for the channel filtering and, as possible, to remove the additive noise. The output of the correction filter is demodulated in the conventional way. Our goal here is to determine how the characteristics (e.g., impulse or frequency response) of the correction filter will be chosen. The propagation channel is usually unknown a priori and, even if known, is usually time-varying. Thus we desire to find some method to choose the pulse response of the correction filter which presumes no initial knowledge of the channel (other than gross characteristics) and we desire that the method also serve to track any changes in the channel with time. To the extent possible it is also desirable that the filter determination method not require detailed knowledge of the modulating signal at the transmitter, especially its carrier frequency and phase. An FIR filter is most commonly used for the equalizing filter itself, as shown in Fig. 7. Once the length and tap spacing is determined, the issue remaining is how to choose the coefficients of the equalizing filter.
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By far the most common techniques for choosing the proper equalizer coefficients are those which use a known and/or prearranged message waveform to permit modeling, and then correction, of the propagation channel. For virtually all voiceband telephone modems, a prearranged preamble or "training sequence" is used to set up the channel equalizer. This is shown in Fig. 8(a). By comparing the equalizer output with the locally stored version of the sequence
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Two classical approaches to adaptive equalization
A variant of this scheme is called "decision-direction". Shown in Fig. 8(b), it uses the demodulated output as the training reference. This technique is commonly used in data transmission systems after the equalizing filter is properly initialized and the output bit-error rate is low. The symbol decisions are then fed back to the equalizer and used as the reference signal. If the symbol error rate is low, then the demodulator's decisions closely mimic the transmitted sequence and hence the transmitted wavefonn. The use of the demodulated symbols as a locallygenerated reference signal is possible in digital systems because the finite number of levels in the transmitted source (e.g., four in QPSK) allows regeneration of the signal, largely removing the effects of noise and other minor perturbations, and because of the high likelihood that receiver decisions are correct In cases where the system input cannot be cleanly regenerated, as in most analog transmission systems, local derivation of a reference signal is not possible. If an actual or derived reference is available, then a plethora of sophisticated techniques
can be applied to compute the proper correction filter. For situations wbere no reference can be supplied or derived, as in conventional analog FM, these methods cannot be employed. Neither of these equalization approacbes is fully satisfactory for the JIDTV broadcast problem. The use of equalizer training sequences is very robust but is best suited to a point-to-point communications link, not a broadcast network wbere there are many receivers and eacb will need training upon its activation. Periodic transmission of training would permit spontaneous receiver activation but also reduces the transmission rate available for TV signals, already at a premium. Decision-direction, bowever, cannot be used alone since its operation requires that the symbol error rate be low. Some alternative is therefore needed to the use of prearranged training for initialization of the equalizer. BLIND EQUALIZATION The concept of adaptive equalization was the breakthrough needed to permit effective transmission of data signals. It allows, for example, signals at rates of up to 19.2 kbls to be sent over voiceband telepbone circuits wbicb would support no more than 2400 or at most 4800 bls without adaptive equalization. Even so those tecbniques suffered from the constraint that the equalizer must be initialized by "training" it at the beginning of eacb transmission. While not a practical concern for most "dialup" and leased line modems, it became an issue as engineers explored the application of digital versus analog modulation in broadcast and "multidrop" networks. In such systems it is higbly desirable that the transmission of known training sequences be avoided. Tbis led to the search for adaptation algorithms capable of "blindly" determining the equalizer's coefficients. The class of algorithms discussed in this paper was developed by at least two independent groups in the late 1970's. Godard's work (1980) was specifically motivated by the desire to use 16-QAM signals over voiceband multipoint networks. The wode by Treicbler and bis associates (1983) bad been initially motivated by the need for adaptive equalization of analog modulation, particularly FM. It was then sbown to extend to PSK and QAM signals as well. Even given these different motivations, both suggested the same algorithm.
fixed amplitude or modulus, while the presence of any additive degradation, wbether it be noise, interference, or multipath-induced distortion, caused instantaneous variations in this modulus. This is shown in Fig. 9. The constant modulus algorithm measures these variations and uses them to adapt the coefficients of adaptive filter to best remove these additive terms from the filter output. While others have been postulated, analyzed, and employed, the first constant modulus cost function lcm studied was given by J cm
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where A is the modulus of the undistorted constant envelope signal and y(k) is the complex-valued output of the adaptive filter. Note that 1cm;: 0 wben ly(k)1 =A and is positive otherwise. This particular form for lcm was cbosen since its gradient with respect to the coefficients of an FIR equalizer is easy to compute and requires no division or other normalization. Using this performance function as the starting point in the development of an approximate gradient descent yields the so-called ''2-2'' constant modulus algorithm for updating the filter coefficient vector W(l), W(l+l) = W(l)+J,1[ lyI 2_ A 2]y(k)X*(k) , (2)
where I is the update index,J..L is a small positive adaptation constant and X(k) is the regressor vector, the vector containing the data samples in the FIR delay line wben the output y(k) is computed. ImH
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As discussed in (Treichler, 1983) this algorithm proved very effective in blindly equalizing analog constant envelope signals as well as PSK and later QAM signals. Its insensitivity to carrier phase and frequency uncertainties increased its practical attractiveness even more. On the theoretical front the concept of constant modulus restoral was extended to the more general concept of adapting a digital filter to restore some property of the transmitted signal.
The constant modulus algorithm (CMA) was developed (Treicbler, 1983) as a method for adaptively cboosing the pulse response of a digital filter in sucb a way as to remove correlated and uncorrelated additive interference from a constant envelope signal, sucb as a frequency- or phase-modulated carrier. It was based on the observation that the complex-valued representation of a constant envelope signal had a
Godard (1980) developed exact! y the same algorithm from a different perspective. He observed that the 446
presence of group delay distortion in a voiceband transmission channel tended to disperse the received data symbols away from their intended QAM constellation points. He sought a gradient descent algorithm which would tend to reduce this dispersion. While not immediately intuitive, he postulated the algorithm shown above and proved that even though the constellation points do not fall on the circle of radius A the performance function is minimized at a tilter coefficient solution which minimizes the received signal dispersion, hence Godard's chosen description for the algorithm - "dispersion direction". The algorithm's effectiveness can be seen by examining the four plots shown in Fig. 10. The upper left plot is the same as that shown on the right of Fig. 5, a 64-QAM signal so completely dispersed that its structure is invisible. After 25K samples the CMAdirected equalizer has recovered the signal constellation to the degree than the points are clearly defined, thus permitting the use of decision direction. The constellation after successful carrier acquisition and decision-directed equalization is shown on the lower right.
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In addition to dealing with additive broadband noise and the dispersion effects induced by multipath propagation, the practical adaptive equalizer must also accommodate inband and out-of-band interference. When the interferers are relatively narrowband and limited in number, a high-order adaptive equalizer can suppress them without harming the signal of interest to any great degree. Strong narrowband interference does raise the potential of "capturing" the constant modulus algorithm, however, as described in (Treichler, 1983) and quantified to some degree in (Treichler, 1985). To cope with this possibility the adaptive equalizer can be initially operated as a "whitening filter" by minimizing the filter output power subject to the constraint that one filter coefficient is fixed. Such a filter will adapt to notch narrowband input signals. This has the effect of creating an initial filter for the CM A-direction mode which minimizes the probability of capture on any of the narrowband inputs .
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The demonstrated ability of the constant modulus algorithm to "blindly" adapt digital filters to equalize the effects of multipath-induced dispersion on QAM signals has encouraged analysis of the algorithm's expected behavior by a number of research groups. Several of these analyses contain proofs of the algorithm's guaranteed convergence to an optimum and satisfactory solution subject to a variety of conditions thought not to be practically taxing. As will be seen, these proofs do not tell the whole story. DEALING WITI-I NARROWBAND INTERFERENCE
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Consideration of all of these factors leads to the receiver architecture shown in Fig. 11. The incoming HDTV signal is tuned, filtered, gain controlled, and applied to the equalizer's filter. The filter output is carrier locked and the resulting soft decisions fed to a Viterbi decoder. The gradient update machinery is driven by one of four inputs, each used for a particular situation or phase of operation. In typical operation the whitening mode is used first, to notch strong narrow band interferers, the CMA mode is used next to "open the eye", and finally the decision-direction mode is used to optimize the equalizer's coefficients and track any environmental changes.
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After 25 kbauds and carrier phase lock After 50 kbauds Fig. 10 Scatter plots during equalizer acquisition While this class of algorithms is probably the best known, others preceded it and more have been developed since. Proakis and Stein (1990) report such development efforts in the early 1970's. Sato (1975) published an algorithm for blind equalization of pulse-amplitude modulation (PAM). The CMNdispersion-direction algorithms can in fact be viewed as two-dimensional extensions of Sato's approach. Godard published a scheme termed the "reduced constellation" algorithm, or RCA, which grouped constellation points together to decrease the required accuracy of the feedback in a decision-direction approach. Since the emergence of CMNdispersiondirection, improvements and extensions have been suggested, including those by Benveniste and Goursat (1984) and Picchi and Prati (1987). Blind equaliza-
A CURRENT PROBLEM: MISCONVERGENCE DUE TO INCOMPLETE INPUT EXCITATION As noted in the section Blind Equalization the demonstrated ability of the constant modulus algorithm (CMA) to "blindly" adapt digital filters to equalize the effects of multipath-induced dispersion on quadrature amplitude modulated (QAM) signals has 447
encouraged analysis of the algorithm's expected behavior by a number of research groups. Several of these analyses (Godard, 1980; Treichler, 1983; Benveniste, 1984; Foschini, 1985) contain proofs of the algorithm's guaranteed convergence to an optimum and satisfactory solution subject to a variety of conditions thought not to be practically taxing. In practical application, however, a number of circumstances have occurred in which CMA fails to converge, or, equally bad from a practical standpoint. converges to a solution which fails to equalize the input signal.
be needed to equalize the signal to the degree obtained with no multipath in Fig. 12(a).
(a) Spectrum and Constellation in the Absence of Multipath
Fig. 11 Block diagram of a broadcast receiver based on a multimode equalizer
(b) Spectrum and Constellation in the Presence of
Heavy Specular Multipath
This section describes several situations in which misconvergence occurs, suggesting that a firmer analytical understanding is needed of the behavior of blind algorithms in the presence of cyclostationary and/or quasi-periodic, non-white inputs. While this analytical understanding is not yet established, the practical experience reported here can be used immediately by those designing new digital communications systems. We will again use the broadcast lIDTV problem as a case in point
Fig. 12 Successful operation of a CM A-based adaptive equalizer with an NEC 16-QAM signal, with and without multipath Observed Misconvergence and Apparent Cause Sil:nall:eneration model. Sixteen-level quadratureamplitude modulation is commonly used in terrestrial line-of-sight microwave transmission of telephone signals. A common design for the transmitter section of a 16-QAM modem is shown in Fig. 13.
Baseline Observation As discussed in previous sections and illustrated in Fig. 11, CM A-directed equalization is used in the initial acquisition or reacquisition of a QAM signal. The success of CMA in this acquisition mode can be seen by examining Fig. 12. Fig. 12(a) shows the spectrum of a 33 megabaud, 16-QAM signal from an NEC modem, plus the equalized, carrier-locked constellation. In this case there is no dispersion impressed on the signal other than the pulse shaping imposed at the transmitter. Fig. 12(b) shows the spectrum and resulting constellation in the presence of a strong specular multipath. The presence of multipath is indicated in the scalloping seen in the power spectrum of the received signal. Without equalization the signal's "eye" is closed and demodulation is impossible. The CMA-directed equalizer, however, opens the eye and permits successful acquisition of the decisiondirected modes. The residual effects of the multipath propagation are still seen after full acquisition, as evidenced by the large constellation "points". The multipath in this case is so strong that an equalizer with hundreds of taps (rather than the 64 used here) would
Fig.13 Block diagram of a generic 16-QAM modulator for digital radios Multiplexing equipment is used to combine up to several thousand PCM telephone channels into a single serial stream, which is then "randomized" in some fashion to "whiten" the bit stream, making the spectrum of the transmitted signal uniform in amplitude across its assigned band and thereby reducing any interference to spectraIly adjacent signals. A common method of performing this randomization is to cyclically add (modulo-2) a selected sequence of length L, which is typically a section of a pseudorandom (PN) sequence of equal or somewhat greater length. The combined sequence then broken into 4-
448
bit nibbles and each transmitted as one of the 16 possible symbols present in a 16-QAM signal. An implication of this technique is that when the mu 1tiplexed telephone signal is essentially idle, for example, in the middle of the night, the added sequence dominates. As a result the transmitted signal is repetitive with period at least L, and perhaps U4, if Lis divisible by 4 as it commonly is.
with two missing tributaries can appear as a nonsquare 8-QAM pattern. The resulting signal is essentially a quadrature partial response (QPR) waveform, a class known to be problematical to equalize.
Experimental observations. The implications to blind equalization algorithms can be seen by examining the constellation photographs seen in Fig. 14. Fig. 14(a) shows the constellation of a properly processed 16-QAM signal operating at 25 megasymbols/s. Achieving this condition requires gain adjustment, symbol timing synchronization, adaptive equalization, and carrier frequency acquisition. Fig. 14(b) shows the received signal when only gain adjustment and real-to-quadrature conversion have been done. Clearly the "eye" is closed. After symbol timing synchronization is attained and when L, the additive sequence length, is very long (e.g., greater than 1O~, the constant modulus algorithm (CM A) can be reliably employed to open the eye, even without carrier frequency acquisition. Fig. 14(c) shows this constellation. The appearance of the three rings indicates the opening of the eye. These rings correspond to the rotation-induced blurring of the 16-QAM pattern caused by lack of carrier frequency lock.
(a) Successful recovery of a 16-QAM, 25 megabaud Signal
(b) Constellation of Received Signal, with symbol timing achieved but without equalization or carrier recovery
If the additive sequence length is much lower, then false convergence of CMA can occur. Such a case is shown in Fig. 14(d). In this case the length Lis 256, four times the length of the adaptive equalizer's regressor vector. (Commercial QAM modems are built which use randomizers as short as 128.) Note that in this situation the algorithm converges to a single "fuzzy" ring, and that the constellation's eye is not open. Thus blind equalization fails and with it the complete acquisition procedure for the demodulator.
(c) Constellation of the Same Signal After Successful Blind Equalization but Before Carrier Acquisition; Note the open "eye"
Considerable laboratory experimentation has been done using various constellation sizes, equalizer length, PN sequence lengths, and various equalizer parameters. Invariably these experiments indicate that when the sequence length L is long compared to the filter and its principal adaptation modes, the convergence is reliable. When the sequence length is short, however, that is, commensurate in length with the equalizer, then false convergence is common, but even then not guaranteed. Some specific cases of interest include the following : • Idle multiplexer tributaries, with tributary-level randomizing: With no randomizing of the composite signal, this often leads to the limiting case of unequal constellation point distribution - points which are completely missing. A 16-QAM signal
(d) Constellation of the Same Signal After Unsuccessful Blind Equalization; Eye is not open and carrier acquisition is impossible Fig. 14 Particular instance of CMA misconvergence traced to input insufficiency 449
• Lightly loaded tributaries. PN reset randomizer. sequence length L commensurate with the equalizer's length: In this case the input is essentially periodic. producing a "liney" spectrum. Examples of this phenomenon are shown in Fig. 15. Fig. 15(a) shows the power spectrum of a 16-QAM modulator driven by a very long (224 _1) PN sequence. This results in symbols which are independent and identically distributed over the limited period of the spectrum calculation. Fig. 15(b) shows the spectrum when driven by a very short sequence. L=63 in this case. The spectrum is no longer relatively smooth but is completely composed of spectrallines. The same effect appears in Fig. 15(c) where L=255 and the line spacing is four times closer. When confronted with a line-dominated spectrum a CMA-directed adaptive filter can often converge to a solution which isolates a single spectral component. While inappropriate for equalization. the converged solution minimizes the CMA performance criterion even bener than the "proper" solution would. This "capture" phenomenon is discussed in (Treichler. 1985). • Lightly loaded tributaries. PN reset randomizer. sequence length L much longer than the equalizer but much shorter than the nominal convergence time: In this case the signal autocorrelation matrix appears to have full rank at all times. but the adaptive algorithm's control loop is driven periodically by the cyclic nature of the input since its frequency is 100 high for the loop to track.
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Suspected cause. "Input sufficiency" is immediately assumed in essentially all analyses of blind equalizer convergence. In particular it is always presumed that the transmitted constellation points are identically distributed and appear independently from one another (Le .• the symbols are lID). It is further presumed that the adaptive equalizer has such high order that its length in no way affects the convergence of the adaptive algorithm. Ding (1990) has demonstrated the shortcomings in the second assumption but the first was thought to be reasonably representative of practical applications. In fact, it is not and usually will not unless the signal is specifically designed to achieve it The high-capacity telephony signals described previously (Signal generation model) tend to fail these input sufficiency tests for one or more of the following reasons:
Fig. 15 The impact of periodic PN inputs on the spectra of 16-QAM signals • "Randoming" is not complete or adequate. The degree of randomization needed to avoid adjacent channel interference is much less than that needed to provide statistical stationarity over the convergence interval of an adaptive equalizer. In general a modem designer would choose a short sequence to minimize the time needed for demultiplexer synchronization. • Differential encoding correlates otherwise uncorrelated symbols. Implications to the Design of an HDTV Broadcast System
• Input tributaries are sometimes lightly loaded (e.g .• in the middle of the night)
While all of the theoretical underpinnings are not yet fully understood. the experimental results attained to date and reported in the paper carry lessons to the designer of an HDTV broadcast system. These include. at the least, the following:
• Sometimes tributaries are completely missing (e.g .• when a new radio system is installed with excess capacity)
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• Avoid source coding and FEC designs which result in data sequences which are idle, almost idle, or periodic for long intervals of time
Ding, Z., (1990). Application Aspects of Blind Adaptive Equalizers in QAM Data Communications, Ph.D. Dissertation, Comell University.
• Avoid the use of "short" randomizers, particularly of the additive (e.g., PN reset) type
Foschini, GJ., (1985). Equalizing Without Altering or Detecting Data, AT&TTechnical Journal, Vol. 64, pp. 1885-1911.
•
If only one signal constellation is to be employed, consider using a performance function which makes more use of that lmowledge, such as the COMA scheme discussed in (de Victoria, 1991).
Godard, D.N., (1980). Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems,lEEE Transactions on Communications, Vol. COM-28, pp. 1867-1875, November, 1980.
CONCLUSIONS
Jablon, N.K., (1992). Joint Blind Equalization, Carrier Recovery, and TlDling Recovery for Highorder QAM Signal Constellations, to appear in IEEE Transactions on Signal Processing, June 1992.
About a decade ago the concept of blind equalization was brought forward by a number of research groups. Since then a number of algorithms have been suggested, analytical work has been done, and operational hardware using them has been successfully designed, built, and shipped. The latter fact is a testament to functional need for blind equalization. Even so much remains to be done in the analytical theater. The algorithms currently in use, CMNdispersiondirection, for example, are lmown to be very slow to convergence and, as illustrated in the section A Cur-
Picchi, G . and Prati, G., (1987). Blind Equalization and Carrier Recovery Using "stop-and-go" Decision-directed Algorithm, IEEE Transactions on Communications, Vol. COM-35, No. 9, pp. 877-887. Proakis, J .G. and Stein, S., (1990). Personal communication, Ruidoso, NM.
rent Problem: Misconvergence Due to Incomplete Input Excitation, to lack robustness in the face of sig-
Rummler. W.D., (1980). Tlme- and FrequencyDomain Representation of Multipath Fading on Line-of-Sight Microwave Paths, Bell System Technical Journal, Vol. 59, No. 5, pp. 763796.
nals not adhering the stated regularity assumptions. More work needs to be done to understand the algorithms we're currently using and to develop better ones.
Sato, Y., (1975). A Method of Self-recovering Equalization for Multilevel Amplitude-modulation Systems, IEEE Transactions on Communications, Vol. COM-23, pp. 697-682, June 1975.Takhar, G.S. and Gupta, S.c., (1976). Discrete Estimation of Continuous AngleModulated Channels for Aeronautical Communication, IEEE Transactions on Communications, Vol. COM-24, No. 3.
ACKNOWLEOOMENTS The long term contributions of Mike Larimore and Rick Johnson to the analytical and practical understanding of adaptive digital filters is highly appreciated. Also gratefully aclmowledged is the incredible effort put in by the staff at Applied Signal Technology, Inc. in order to create real products out of the theoretical ideas. Vin Wolff and his crew contributed the photographs of signal spectra and constellations.
Treichler, J.R. and Agee, B.G., (1983). ANew Approach to Multipath Correction of Constant Modulus Signals, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-31, pp. 459-472.
REFERENCES Bello, P. A., (1963). Characterization of Randomly TlDle-Invariant Linear Channels, IEEE Transactions on Communication Systems, Vol. CS11, pp. 360-393.
Treichler, J.R. and Larimore, M.G., (1985). The Tone Capture Properties of CM A-based Interference Suppressors, IEEE Transactions on Acoustics. Speech, and Signal Processing, Vol. ASSP-33, pp. 946-958.
Benveniste, A. and Goursat, M., (1984). Blind Equalizers, IEEE Transactions on Communications, Vol. COM-32, pp. 871-883. de Victoria, EL., (1991). An Adaptive Blind Equalization Algorithm for QAM and QPR Modulations: The Concentrically Ordered Modulus Algorithm, Proceedings of the 25th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, November 4-6.
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