Berg Literature Survey

My objective in this article is to add evidence for the belief espoused above regarding the two elds of (i) signal processing and (ii) control and estimation. I focus here on the interaction of specialties from (i), i.e., adaptive ltering, and (ii), i.e., adaptive identi cation and control. For evidence, I use three basic communication systems applications of adaptive ltering to evoke system identi cation (and adaptive control) style problems amenable to analytical tools popular over (at least) the last decade. The applications raise a number of un(der)solved issues that challenge existing theory. The connections drawn to adaptive identi cation and control problems (and relevant analytical tools) suggest approaches to these challenges. [Johnson ASIL 95] C.R. Johnson, Jr. et. al.,\On fractionally-spaced equalizer design for digital microwave radio channels," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 290-294, October 1995. Abstract: Recent advances in blind identi cation of fractionally-spaced models for digital communication channels and blind fractionally-spaced equalizer adaptation rely on the assumption that the time span chosen for the fractionally-spaced equalizer exceeds that of the channel. This paper considers time-domain formulas minimizing the mean-squared symbol recovery error achieved by a nite-length FIR fractionally-spaced equalizer with a time span shorter that the channel impulse response time span for white zero-mean QAM sources in the presence of white zero-mean channel noise. For minimum mean-squared error designs the symbol error rates achievable are plotted versus the ratio of the source variance to the channel noise variance (with the channel model power normalized to achieve a received signal of unit variance) for di erent fractionally-spaced equalizer lengths on 64-QAM for several T/2-spaced channel models derived from experimental data. Our intent is to fuel the ongoing debate about fractionally-spaced equalizer length selection. [Johnson IJACSP 95] C.R. Johnson, Jr. and B.D.O. Anderson,\Godard blind equalizer error surface charateristics: White, zero-Mean, binary case,"International Journal Of Adaptive Control And Signal Processing, vol. 9, pp. 301-324, 1995. (Filed in BERG library.) Comments : Derives compact descriptions of gradient and Hessian expressions of the BPSK CMA cost function for baud-spaced sampling. The paper then catalogues the stationary points of this cost function by the di erent manners in which the gradient of the cost can be zeroed. A key result is that the Ding class equalizer (corresponding to the nullspace of the channel covolution matrix) is such that the parameterization has most of its energy at the ends of the tapped delay line, rather towards its center. The paper does not show arbitrarily low CMA error for a long, but nite, equalizer. {tje Abstract: We study the Godard/CMA error surface in adapting the impulse response coe cients of a nite-length, tapped-delay-line lter to equalize a white, zero-mean, binary ( 1) source that su ers ISI due to a linear channel. Our results arise primarily from examination of the expressions for the gradient and the Hessian in the baud-spaced equalizer parameter space of the cost-function in this particular setting. This paper presents results useful in topological assessment of procedures for initializing a gradient descent algorithm on this multimodal surface in the euqalizer parameter space. [Jones ASIL 95] D.L. Jones,\A normalized constant-modulus algorithm," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 694-697, October 1995. Abstract: The constant-modulus algorithm (CMA), while the most commonly used blind equalization technique, converges very slowly. We propose a \normalized" constant-modulus algo39 rithm (analogous to the widely used normalized LMS algorithm) with an adjustable step size that greatly increases the convergence rate for noise colorings with large eigenvalue spreads. The normalized step size is proportional to that required to achieve the desired modulus with the current data vector. Only a few extra operations per update are required. Many applications now using the constant modulus algorithm should achieve greatly improved convergence rates at almost negligible computational increase by adapting the new normalized CMA algorithm. [Kamel ICC 94] R.E. Kamel and Y. Bar-Ness,\Anchored constant modulus algorithm (ACMA) for blind equalization," in Proc. International Conference on Communications (New Orleans, LA, USA), pp. 1571-5, 1-5 May 1994. Abstract: Blind equalization is a technique of adapting an equalizer without the need of a training sequence. The constant modulus algorithm (CMA) is one of the rst known blind equalization algorithms. The cost function of the CMA exhibits local minima, which are the primary cause of the ill-convergence of the CMA. Anchoring the CMA improves the performance of the CMA in terms of ill-convergence. This technique is used with the linear and the decision feedback equalizers. It is shown that the adaptive equalizer will always remove intersymbol interference (ISI) as long as the channel gain exceeds a certain critical value (7 Refs.) [Kamel ELET 94] R.E. Kamel, Y. Barness,\Blind maximum-likelihood sequence estimation of digital sequences in presence of intersymbol interference,"Electronics Letters, vol. 30, no. 7, pp. 537-539, March 1994. Abstract: A blind maximum likelihood (ML) sequence estimator for unknown linear dispersive channels is described. The estimator assumes a channel model with quantised parameters. A channel trellis and a data trellis are de ned to search for the ML channel and data estimates using the Viterbi algorithm (VA). This approach provides a good performance/complexity tradeo (3 Refs.) [Karaoguz TBC 91] J. Karaoguz, S.H. Ardalan,\Use of blind equalization for teletext broadcast systems,"IEEE Transcations on Broadcasting, vol. 37, no. 2, pp. 44-54, 1991. Abstract: A new application of blind equalization that appears to ameliorate the delays and improve the reliability of teletext systems is presented. The compatibility with the existing TV system and the cost-e ectiveness of this application were considered for practical purposes. Equalizer adaptation does not require a knowledge of the transmitted data sequence or carrier phase recovery. Therefore, a reduction in the existing teletext circuitry has been achieved. In addition, the proposed quadrature phase shift keying (QPSK) modulation scheme doubles the teletext data transmission rate and uses the allocated TV bandwidth e ciently. The performance of the proposed blind equalizer has been evaluated in the presence of a multipath broadcast channel. Simulations have shown that substantial improvements are observed in the bit-error-rates for both two-phase PSK and QPSK modulation schemes (21 Refs.) [Karaoguz ISCS 93] J. Karaoguz,\An unsupervisedGaussian cluster formation technique as a Bussgang blind deconvolution algorithm," in Proc. 1993 IEEE International Symposium on Circuits and Systems (Chicago, IL), pp. 719-22 vol.1, 3-6 May 1993. Abstract: A decision-directed-type blind equalization algorithm is reformulated. It is based on the unsupervised Gaussian cluster formation technique. It is shown that the end result is closely related to the Bussgang algorithm and its implementation simplicity compares to the Bussgang blind deconvolution algorithm. It is also shown that along with the new algorithm, 40 popular algorithms such as the Sato, Godard, maximum-level-error (MLE), and Bussgang blind equalization algorithms can all be categorized as decisiondirected-type blind equalization algorithms which use a nonlinear estimator at the output of the equalizer to generate a decision-directed estimated error (11 Refs.) [Kechriotis MIL 92] G. Kechriotis, E. Zervas, E.S. Manolakos,\Using recurrent neural networks for blind equalization of linear and nonlinear communications channels," in Proc. MILCOM (San Diego, CA), pp. 784-788, October 1992. Abstract: A recurrent neural network (RNN) equalizer for blind equalization of linear and nonlinear communication channels is proposed. RNNs have the ability to learn dynamical mappings of arbitrary complexity and therefore present a natural choice for implementing equalizers for communication channels. In several cases the nonlinear nature of a communication channel is too severe to ignore, and at the same time no nonlinear channel model can account su ciently for the nonlinearities that are inherently present in the channel. In those cases a neural network equalizer is preferable over a conventional one. The realtime recurrent learning (RTRL) algorithm is used to train an RNN, and its performance is compared with that of a conventional equalizer based on the constant-modulus algorithm (9 Refs.) [Kennedy OE 92] R.A. Kennedy and Z. Ding,\Blind adaptive equalizers for QAM communication systems based on convex cost functions,"Optical Engineering, vol. 31, pp. 1189-1199, 1992. Abstract: The authors present a new quadrature amplitude modulated blind equalization scheme that is globally convergent in the equalizer parameter space to a compact set containing the desired ideal equalizer parameter setting. The new algorithm is based on a convex cost function and a linear constraint on the equalizer parameters. For a generic class of channels, this new algorithm results in the equalizer parameter convergence to a unique global minimum achieving intersymbol interference suppression and carrier phase error removal. Different implementation approaches are assessed and simulation results are shown to con rm the theoretical global convergence of the new algorithm (25 Refs.) [Kikuma ISAP 89] N. Kikuma et al.,\Directionally constrained adaptive array using constant modulus algorithm," in Proc. International Symposium on Antennas and Propagation (Tokyo, Japan), pp. 313-16, Aug. 22-25, 1989. Abstract: The power-minimizing adaptive array tends to cancel the desired signal with the interference, if they are correlated with each other, as is usually the case in multipath environments. On the other hand, the constant modulus (CM) adaptive array has been developed which can maintain the desired signal having constant modulus proper such as FM, PSK or FSK signals, even in the multipath environments. It is one of the advantages of the CM adaptive array that it does not need a priori knowledge of the direction of the desired signal. However, since the optimization of the CM adaptive array is based on the LMS (least mean square) algorithm using the steepest descent method, one requires such knowledge to set the best initial weights that steer a main beam in the direction of the desired signals. Therefore, assuming that one knows the direction of the desired signal, the authors propose the CM adaptive array operating under a directional constraint, and demonstrate its better transient performance by computer simulation (5 Refs.) [Kikuma ISAP 91] N. Kikuma, M. Fujimoto, N. Inagaki,\Rapid and stable optimization of CMA adaptive array by Marquardt method,"Antennas and Propagation Society Symposium Digest, vol. 1, pp. 102-105, 1991. 41 Abstract: The authors present a rapidly converging and wellconditioned algorithm based on the nonlinear least squares method referred to as the Marquardt method for the CMA (constant modulus algorithm) adaptive array. Computer simulation with a pi /4-shifted QPSK (quadrature phase-shift keying) signal veri ed its rapid and stable convergence characteristics for any radio environment. Taking into consideration the increase of computation for one weight update, the Marquardt method can reduce the convergence time by a factor of 10 to 100 compared to the steepest descent [Kikuma SWPC 94] N. Kikuma, K. Hachitori, F. Saito and N. Inagaki,\CMA adaptive array antenna using transversal lters for spatial and temporal adaptability in mobile communications," in Proc. Symposium on Wireless Personal Communications Proceedings (Blacksburg, VA, USA), pp. 4/1-10, 1-3 June 1994. Abstract: The CMA (constant modulus algorithm) was developed for the adaptive receiving systems to capture the desired signal having the constant modulus property such as FM, PSK and FSK signals. Computer simulation is carried out using an antenna array equipped with tapped delay lines for the antenna weights. A pi /4-shifted QPSK signal is generated which is transmitted over several multipath channels. Furthermore, another signal of the same modulation type is generated for the co-channel interference. For optimization of the CMA adaptive system, the steepest descent method and Marquardt method are utilized. The former has mainly been used for the CMA because the cost function is nonlinear with respect to the tap weights. However, the slow convergence of the algorithm has often limited its application in mobile communications where signals must be quickly captured. The latter, on the other hand, is one of the nonliner least squares algorithms to attain its rapidly-converging and wellconditioned adaptation. The simulation results demonstrated that the spatial and temporal adaptive system can achieve high quality of communications with low BERs. Also, it is shown that the Marquardt method can contribute to reducing the convergence time (10 Refs.) [Kikuma ISAP 94] N. Kikuma, T. Shutoku, N. Inagaki,\CMA adaptive array antenna under weight norm constraint," in Proc. International Symposium On Antennas and Propagation (Seattle, WA), pp. 1564-7 vol.3, 20-24 June 1994. Abstract: The constant modulus algorithm (CMA) adaptive array was developed to capture the desired constant envelope signals such as FM, PSK and FSK signals. For controlling the level of output noise power of the CMA system, we introduce the norm constraint of the array weight into the CMA. The constrained CMA operates to suppress the output power of the internal noise while maintaining the desired signal at the array output. As a result, it is expected that one can obtain a higher signal to interference plus noise ratio (SINR). We examine the performance of the CMA adaptive array under the weight norm constraint in the presence of multipath waves, and validate the constraint in the CMA (4 Refs.) [Kikuma PIMRC 95] N. Kikuma, K. Takai, K. Nishimori, F. Saito, N. Inagaki,\Consideration on performance of the CMA adaptive array antenna for 16 QAM signals," in Proc. International Symposium on Personal, Indoor and Mobile Radio Communications (Toronto, Ont., Canada), pp. 677-81 vol.2, 27-29 Sept. 1995. Abstract: The CMA adaptive array was developed for capture of constant modulus signals. As is well known, the CMA adaptive array can suppress interferences successfully under multipath environments. In order to investigate the availability of CMA adaptive array for other nonconstant modulus signals, this paper deals with performance of the CMA adaptive array for 16 QAM signals which have high bandwidth e ciency. Several CMA-based cost functions 42 The authors present a rapidly converging and wellconditioned algorithm based on the nonlinear least squares method referred to as the Marquardt method for the CMA (constant modulus algorithm) adaptive array. Computer simulation with a pi /4-shifted QPSK (quadrature phase-shift keying) signal veri ed its rapid and stable convergence characteristics for any radio environment. Taking into consideration the increase of computation for one weight update, the Marquardt method can reduce the convergence time by a factor of 10 to 100 compared to the steepest descent [Kikuma SWPC 94] N. Kikuma, K. Hachitori, F. Saito and N. Inagaki,\CMA adaptive array antenna using transversal lters for spatial and temporal adaptability in mobile communications," in Proc. Symposium on Wireless Personal Communications Proceedings (Blacksburg, VA, USA), pp. 4/1-10, 1-3 June 1994. Abstract: The CMA (constant modulus algorithm) was developed for the adaptive receiving systems to capture the desired signal having the constant modulus property such as FM, PSK and FSK signals. Computer simulation is carried out using an antenna array equipped with tapped delay lines for the antenna weights. A pi /4-shifted QPSK signal is generated which is transmitted over several multipath channels. Furthermore, another signal of the same modulation type is generated for the co-channel interference. For optimization of the CMA adaptive system, the steepest descent method and Marquardt method are utilized. The former has mainly been used for the CMA because the cost function is nonlinear with respect to the tap weights. However, the slow convergence of the algorithm has often limited its application in mobile communications where signals must be quickly captured. The latter, on the other hand, is one of the nonliner least squares algorithms to attain its rapidly-converging and wellconditioned adaptation. The simulation results demonstrated that the spatial and temporal adaptive system can achieve high quality of communications with low BERs. Also, it is shown that the Marquardt method can contribute to reducing the convergence time (10 Refs.) [Kikuma ISAP 94] N. Kikuma, T. Shutoku, N. Inagaki,\CMA adaptive array antenna under weight norm constraint," in Proc. International Symposium On Antennas and Propagation (Seattle, WA), pp. 1564-7 vol.3, 20-24 June 1994. Abstract: The constant modulus algorithm (CMA) adaptive array was developed to capture the desired constant envelope signals such as FM, PSK and FSK signals. For controlling the level of output noise power of the CMA system, we introduce the norm constraint of the array weight into the CMA. The constrained CMA operates to suppress the output power of the internal noise while maintaining the desired signal at the array output. As a result, it is expected that one can obtain a higher signal to interference plus noise ratio (SINR). We examine the performance of the CMA adaptive array under the weight norm constraint in the presence of multipath waves, and validate the constraint in the CMA (4 Refs.) [Kikuma PIMRC 95] N. Kikuma, K. Takai, K. Nishimori, F. Saito, N. Inagaki,\Consideration on performance of the CMA adaptive array antenna for 16 QAM signals," in Proc. International Symposium on Personal, Indoor and Mobile Radio Communications (Toronto, Ont., Canada), pp. 677-81 vol.2, 27-29 Sept. 1995. Abstract: The CMA adaptive array was developed for capture of constant modulus signals. As is well known, the CMA adaptive array can suppress interferences successfully under multipath environments. In order to investigate the availability of CMA adaptive array for other nonconstant modulus signals, this paper deals with performance of the CMA adaptive array for 16 QAM signals which have high bandwidth e ciency. Several CMA-based cost functions 42 modi ed for the QAM signals are introduced and also an algorithm of switching those cost functions in the process of adaptation is proposed. Via computer simulation, it is shown that the CMA adaptive array with the cost functions changed in adaptation has signi cantly better convergence characteristics for the 16 QAM signals. In addition, the BER performance of the cost functions is clari ed (12 Refs.) [Krishnamurthy TSP 95] V. Krishnamurthy, S. Dey and J.P.LeBlanc,\Blind equalization of IIR channels using hidden Markov models and extended least squares,"IEEE Transactions on Signal Processing, vol. 43, no.12, pp. 2994-3006, Dec. 1995. (Filed in BERG library.) Abstract: In this paper, we present a blind equalization algorithm for noisy IIR channels when the channel input is a nite state Markov chain. The algorithm yields estimates of the IIR channel coe cients, channel noise variance, transition probabilities, and state of the Markov chain. Unlike the optimal maximum likelihood estimator which is computationally infeasible since the computing cost increases exponentially with data length, our algorithm is computationally inexpensive. Our algorithm is based on combining a recursive hidden Markov model (HMM) estimator with a relaxed SPR (strictly positive real) extended least squares (ELS) scheme. In simulation studies we show that the algorithm yields satisfactory estimates even in low SNR. We also compare the performance of our scheme with a truncated FIR scheme and the constant modulus algorithm (CMA) which is currently a popular algorithm in blind equalization (24 Refs.) [Kwon SP 92] O.W. Kwon, C.K. Un, J.C. Lee,\Performance of constant modulus adaptive digital lters for interference cancellation,"Signal Processing, vol. 26 no. 2, pp. 185-196, February 1992. Abstract: The constant modulus algorithm (CMA) updates its weight vector to minimize the modulus variation of the output signal. The convergence behavior of the CMA used in interference cancellation is studied. The authors rst investigate the optimum weight vector that minimizes the performance index of the CMA which is de ned as the meansquared di erence between the estimated and true moduli. They then analyze the convergence behavior of the squared output modulus and the performance index. Based on the analysis results, several convergence properties of the CMA are discussed (12 Refs.) [Lambert MIL 95] Lambert and C. Nikias,\Optimal equalization cost functions and maximum a posteriori estimation," in Proc. 1994 IEEE MILCOM. Conference Record (Fort Monmouth, NJ), pp. 291-5 vol.1, 2-5 Oct. 1994. Abstract: A new maximum a posteriori (MAP) formulation is shown to be a straightforward and intuitive way to derive optimal blind equalization cost functions. This MAP method provides a general, systematic way to derive blind adaptation methods using the given pdf of the input and a convolutional noise model. A general blind equalization/deconvolution cost function known as Gray's Variable Norm, is shown to be derivable using the MAP formulation presented here. Gray's Variable Norm (1979) is a superset of existing blind equalization cost functions, including the Godard and Sato algorithms as special cases. The MAP method is capable of deriving cost functions needed for a wide variety of problems, including those with in nite variance pdfs and even multichannel problems (13 Refs.) [Lambert ASIL 95] R.H. Lambert and C.L. Nikias,\A sliding cost function algorithm for blind deconvolution," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c 43 Grove, CA), pp. 177-181, October 1995. Abstract: A new method for blind equalization is proposed which changes the cost function of the equalizer as the convergence proceeds. Motivation for this idea is given by tests of the new \Uniform Optimum" O2 1 cost function for blind equalization proposed in [14, 13], comparing it to the more familiarO2 4 Godard-like cost. The new cost achieves better asymptotic performance than O2 4 for communications data, but has a zero tracking ability measure, this being an example of the tracking/accuracy compromise of adaptive algorithms [5]. This suggests the use of a sliding cost function algorithm which monitors the convergence state of the equalizer. The sliding cost function algorithm is developed as a \Maximum A Posteriori (MAP) estimate of blind gradient" method for blind equalization which assumes the data ts a generalized Gaussian distribution model. The model parameters are updated at each iteration, and the algorithm adapts its cost function so as to have good tracking ability while converging, and optimal steady state performance at convergence. [Lambotharan TCS 199*] S. Lambotharan and J.A. Chambers,\A New Blind Equalization Structure for Deep Null Communication Channels,"Submitted IEEE Transactions Circuits ans Systems, Part II, March 1996, vol. xxx, pp. xxx, xxx. Abstract: We propose a new blind equalization structure which is well suited for communication channels characterized by deep spectral nulls, and estimates directly the channel coe cients. Blind equalization of communication channels has increasing importance due to the need to maximize bandwidth e ciency. Established Bussgang algorithms applied to Finite Impulse Response, FIR, structure equalizers perform poorly when one, or more, maximum phase zero of the channel is close to the unit circle. This limitation is due to the di culty of modeling the inverse of the maximum phase component of the channel with a nite length FIR lter. Theoretical analysis and simulation studies support the potential of the new structure to model entirely the required inverse, and hence to equalize such di cult channels (32 Refs.) [Lambotharan SP 97] S. Lambotharan, J. Chambers, and C.R. Johnson, Jr.,\Attraction of saddles and slow convergence in CMA adaptation,"Signal Processing, vol. 59, no. 2, pp. 335-340, June 1997. Abstract: We show that the most widely used blind equalization algorithm, the constant modulus algorithm, CMA, can be attracted during one convergence trajectory to the vicinity of more than one of the saddles in its error performance surface where it exhibits very slow convergence. We also establish bounds on the attraction and escape rates at a saddle and show that the saddles associated with lower energy levels have slower escape rates than the saddles with higher energy levels. These results highlight the need for intelligent initialization schemes for the CMA algorithm. We suggest a step normalisation technique to improve convergence speed in the vicinity of a saddle (5 Refs.) [Larimore ICASSP 83] M.G. Larimore and J.R. Treichler,\Convergence behavior of the constant modulus algorithm," in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (Boston, MA), pp. 13-16, April 14-16, 1983. (Filed in BERG library.) Abstract: An adaptive lter algorithm has been developed and introduced for use with constant envelope waveforms, e.g. FM communication signals. It has proven capable of suppressing additive interferers as well as equalization, without the need for a priori statistical information. Aspects of dynamic convergence behaviour are discussed, with conclusions supported by simulation (3 Refs.) 44 [Larimore ASIL 84] M.G. Larimore and J.R. Treichler,\The capture properties of CMA-based interference cancellers," in Proc. Conference Record Eighteenth Asilomar Conference on Circuits, Systems and Computers (Paci c Grove, CA), pp. 49-52, 5-7 Nov. 1984. Abstract: A problem which arises when using the constant modulus algorithm (CMA) to cancel narrowband interference is examined. If both the interferer and the signal have constant envelope and are spectrally nonoverlapping, then it is possible to nd two di erent lter solutions, one which suppresses the interferer and another which captures the interferer and suppresses the desired signal. It is shown how capture can occur, using an analysis of the algorithm's behavior to an input consisting of only two sinusoids. Assuming slow adaptation, the N-dimensional adaptive weight recursion is shown to compress into two-bytwo recursion in the tone output amplitudes. This simpli ed recursion is then analyzed to determine which combinations of input amplitudes (signal-to-interference ratios) and lter initial conditions lead to lock and which lead to the capture of the interferer. From these results the tendency of the algorithm to be captured is deduced for both the startup situation and for the situation when a new interferer appears suddenly. The analytic results are shown to compare favorably with computer simulations (3 Refs.) [Larimore ICASSP 85] M.G. Larimore and J.R. Treichler,\Noise capture properties of constant modulus algorithm," in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (Tampa, FL), pp. 1165-8, March 26-29, 1985. Abstract: In communication applications, the constant modulus algorithm has proved to be an e ective means for adjusting coe cients of an adaptive lter. However, under certain conditions it may generate a response that discriminates against the signal component in a preference for broadband noise for sustained periods. Work that has been done toward characterizing and modeling this phenomenon is described (5 Refs.) [Laster SWPC 94] J.D. Laster, J.H. Reed,\A survey of adaptive single channel interference rejection techniques for wireless communications," in Proc. Virginia Tech Symposium on Wireless Personal Communications (Blacksburg, VA), pp. 238, 1994. Abstract: This survey paper focuses on single-channel techniques for interference rejection (that is, techiques employing one antenna) as opposed to multi-channel techniques (which utilize arrays or cross-polarized antennas). Implementation papers are deemphasized in this survey since techniques constitute the main interest. The paper divides interference rejection techniques for digital modulation into spread spectrum techniques and nonspread spectrum techiques. Spread spectrum categories include direct sequence (DS), code division multiple access (CDMA), and frequency hopping (FH). DS techniques focus on rejection of narrowband interference and include whitening lters (i.e., adaptive notch lters), decision feedback lters, and adaptive A/D conversion. Some CDMA techniques employ interference estimation-andsubtraction from the signal-of-interest, while others exploit spectral correlation properties. FH techniques apply whitening lters and make use of the transient nature of the hopping signal. Nonspread spectrum techinques include the constant modulus algorithm, neural networks, nonlinear lters, and time-varying lters that use spectral correlation properties (81 Refs.) [LeBlanc ICASSP 94] J.P. LeBlanc, K. Dogancay, R.A. Kennedy and C.R. Johnson, Jr.,\E ects of input data correlation on the convergence of blind adaptive equalizers," in Proc. International Conference on Acoustics, Speech and Signal Processing (Adelaide, SA, Australia), pp. 313-16, 19-22 April 1994. Abstract: A variety of blind equalization algorithms exist. These algorithms, which draw on 45 some theoretical justi cation for the demonstration or analysis of their purportedly ideal convergence properties, almost invariably rely on the input data being independent and identically distributed (i.i.d.). In contrast, in this paper we show that input correlation can have a marked e ect on the character of algorithm convergence. We demonstrate that under suitable input data correlation and channels: (i) undesirable local minima present in the i.i.d. case are absent for certain correlated sources implying ideal global convergence for some situations and, (ii) the most commonly employed practical algorithm can exhibit ill-convergence to closedeye minima even under the popular single spike initialization when an eye-opening equalizer parameterization is possible (6 Refs.) [LeBlanc ICASSP 95] J.P. LeBlanc, I. Fijalkow, B. Huber, C.R. Johnson, Jr.,\Fractionally spaced CMA equalizers under periodic and correlated inputs," in Proc. International Conference on Acoustics, Speech and Signal Processing (Detroit, MI), pp. 1041-4 vol.2, 9-12 May 1995. Abstract: CMA fractionally spaced equalizers (CMA-FSEs) have been shown, under certain conditions, to be globally asymptotically convergent to a setting which provides perfect equalization. Such a result relies heavily on the assumptions of a white source and no channel noise (as is the case in much of the literature's analysis of CMA). Herein, we relax the white source assumption and examine the e ect of source correlation on CMA. Analytic results are meshed with examples showing CMA-FSE source correlation e ects. Techniques for nding all stationary and saddle points on the CMA-FSE error surface are presented using recent developments in the algebraic-geometry community (8 Refs.) [LeBlanc: Thesis] J. LeBlanc, \E ects of source distributions and correlation on fractionally spaced blind constant modulus algorithm equalizers," Ph.D. Thesis, Cornell University, Ithaca, NY, 1995. Abstract: A common assumption in blind equalization schemes using the Constant Modulus Algorithm (CMA) is that the source sequence is drawn from an independent, uniform distribution. Much of the analysis demonstrating the global convergence of CMA to an openeye setting uses such an assumption. Also, much of this analysis has been performed in the baud-spaced equalizer setting, in which an in nite order equalizer is necessarily assumed to avoid the existence of local minima. However, recent results in the literature show that a nite fractionally-spaced equalizer allows for perfect equalization of moving average channels (under certain channel conditions known as equalizability). Futhermore, CMA has been shown to converge to a perfect equalizing setting under independent, uniformly distributed source. This thesis work investigates the e ect of source statistics on the location of CMA stationary points in the fractionally-sampled equalizer case under the conditions of equalizability. The work identi es the stationary points as the solution set of a system of multivariate polynomial equations with monomial coe cients given by the source moments. The work is divided into three main areas. First, an investigation of the properties of the CMA error surface is performed assuming source independence. This section revisits some previously known results and then extends them by quantifying aspects of the error surface based on the source kurtosis. The relevancy of the results here point to the tradeo s between coding gain (a function of the source distribution) and the error surface curvature (with its e ects on convergence). Next, features of a topological nature of the CMA error surface are discussed. Such properties, which hold independent of source statistics assumptions, form a bridge to the third section describing stationary point locations under temporal source correlation. The mathematical tools used here (e.g. Groebner bases and Homotopy Methods) are just starting to appear in applications in signal processing. The results include a Monte-Carlo study of the e ects of source correla46 tion due to periodic inputs as well as an example of source sequences resulting from Markov models. [LeBlanc CISS 96] J.P. LeBlanc and S.W. McLaughlin,\Non-equiprobable constellation shaping and blind constant modulus algorithm equalization," in Proc. Conference on Information Science and Systems (Princeton, NJ), pp. 901-903, Mar. 1996. Abstract: We consider the tradeo between constellation shaping and blind equalization. For an additive i.i.d Gaussian noise channel, capacity can be achieved if and only if the channel input is Gaussian. However, in such a case blind equalization can not be employed. Thus, there are inherently opposing goals between maximizing shaping gain and performing blind channel equalization at the receiver. This tradeo can be considerable when viewed at the system level. In this paper we charecterize the tradeo s between shaping gain and blind equalization performance using popular blind equalization Constant Modulus Algorithm (CMA) (12 Refs.) [LeBlanc ICASSP 96] J.P. Leblanc, I. Fijalkow, C.R. Johnson Jr.,\Fractionally-Spaced Constant Modulus Algorithm Blind Equalizer Error Surface Characterization: E ects Of Source Distributions," in Proc. International Conference on Acoustics, Speech and Signal Processing (Atlanta, GA), pp. 2944, May 7-9, 1996. Abstract: The Constant Modulus Algorithm (CMA) [Treichler, Agee: IEEE Trans. ASSP, April 1983] [Godard, IEEE Trans. on Commun. Nov. 1980] is a popular blind equalization algorithm. A common device used in demonstrating the convergence properties of CMA is the assumption that the source is iid (independent, identically distributed). Recent results in the literature show that a nite length fractionally-spaced equalizer allows for perfect equalization of moving average channels (under certain channel conditions known as zero-forcing criteria). CMA has been shown to converge to such perfectly equalizing settings under an independent, platykurtic distributed source. This paper investigates the e ect of the distribution from which an independent source sequence is drawn on the CMA error surface and stationary points in the perfectly-equalizable fractionally-sampled equalizer case. Results include symbolic identication of all stationary points, as well as the eigenvalues and eigenvectors associated with their Hessian matrix. Results show quantitatively the loss of error surface curvature (in both direction and magnitude) at all stationary points. Simulations included demonstrate the a ect this has on convergence speed. [LeBlanc ICDSP 97] J.P. LeBlanc and C.R. Johnson, Jr., \Global CMA error surface characteristics, source statistic e ects: Polytopes and manifolds," to appear in International Conference on Digital Signal Processing, (Santorini, Greece), 2-4 July 1997. [LeBlanc IJACSP 98] J.P. LeBlanc, I. Fijalkow, and C. Richard Johnson, Jr., \CMA fractionally spaced equalizers: Stationary points and stability under IID and temporally correlated sources," to appear in International Journal of Adaptive Control & Signal Processing, 1998. [Li CEE 92] M. Li,\Adaptive retrieval and enhancement of sinusoids in a quadrature sampling system using a decomposed complex IRR notch lter,"Computers & Electrical Engineering, vol. 18 no.3-4, pp. 195-204, May-June 1992. Abstract: The quadrature sampling method is very common in radar, sonar and communication applications. In this paper, an adaptive method for the retrieval and enhancement of sinusoids in a quadrature sampling system is developed from a decomposed notch lter structure, consisting of cascaded or parallel rst order complex notch lter modules. The proposed 47 ALE (adaptive line enhancer) can be optimized module by module individually using an identical adaptive algorithm. This adaptive algorithm is a steepest descent version designed to nd a maximum. The idea behind the proposed method is that the complex representation of a sinusoid has a constant modulus or envelope (9 Refs.) [Li TSP 94] Y. Li and Z. Ding,\ARMA system-identi cation based on 2nd-order cyclostationarity,"IEEE Transactions on Signal Processing, vol. 42,no. 12, pp. 3483-3494, 1994. Abstract: Previous work has presented novel techniques that exploit cyclostationarity for channel identi cation in data communication systems. The present authors investigate the identiability of linear time-invariant (LTI) ARMA systems based on second-order cyclic statistics. They present a parametric and a nonparametric method. The parametric method directly identi es the zeros and poles of ARMA channels with a mixed phase. The nonparametric method estimates the channel phase based on the cyclic spectra alone. They analyze the phase estimation error of the nonparametric method for nite dimensional ARMA channels. For speci c, nite dimensional ARMA channels, an improved method is given, which combines the parametric method with the nonparametric method. Computer simulation illustrates the e ectiveness of the methods in identifying ARMA system impulse responses (32 Refs.) [Li ASIL 94] Ye Li and Zhi Ding,\Global convergence of fractionally spaced Godard equalizers," in Proc. Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 617-21 vol.1, 31 Oct.-2 Nov. 1994. Abstract: We present a convergence analysis of blind fractionally spaced equalizers (FSE) utilizing the Godard (1980) algorithm. The FSE can in fact be represented as a special vector equalizer which exploits the spectral diversity (cyclostationarity) of digital channel output and also the spatial diversity (antenna array) when available. It is shown that for channels satisfying a mild length and zero condition, the Godard FSE always converges to a global minimum point. Computer simulation demonstrates the performance improvement by the adaptive Godard FSE (12 Refs.) [Li TSP 95] Ye Li and Zhi Ding,\Convergence analysis of nite length blind adaptive equalizers,"IEEE Transactions on Signal Processing, vol. 43, no.9, pp. 2120-2129, Sept. 1995. (Filed in BERG library.) Abstract: The paper presents some new analytical results on the convergence of two nite length blind adaptive channel equalizers, namely, the Godard equalizer and the ShalviWeinstein equalizer. First, a one-to-one correspondence in local minima is shown to exist between the Godard and ShalviWeinstein equalizers, hence establishing the equivalent relationship between the two algorithms. Convergence behaviors of nite length Godard and Shalvi-Weinstein equalizers are analyzed, and the potential stable equilibriumpoints are identi ed. The existence of undesirable stable equilibria for the nite length ShalviWeinstein equalizer is demonstrated through a simple example. It is proven that the points of convergence for both nite length equalizers depend on an initial kurtosis condition. It is also proven that when the length of equalizer is long enough and the initial equalizer setting satis es the kurtosis condition, the equalizer will converge to a stable equilibrium near a desired global minimum. When the kurtosis condition is not satis ed, generally the equalizer will take longer to converge to a desired equilibrium given su ciently many parameters and adequate initialization. The convergence analysis of the equalizers in PAM communication systems can be easily extended to the equalizers in QAM communication systems (20 Refs.) 48 [Li TSP 96a] Ye Li and Zhi Ding,\Global convergence of fractionally spaced Godard (CMA) adaptive equalizers,"IEEE Transactions on Signal Processing, vol. 44, no.4, pp. 818-26, April 1996. (Filed in BERG library.) Abstract: The Godard (1980) or constant modulus algorithm (CMA) equalizer is perhaps the best known and the most popular scheme for blind adaptive channel equalization. Most published works on blind equalization convergence analysis are con ned to T-spaced equalizers with real-valued inputs. The common belief is that analysis of fractionally spaced equalizers (FSEss) with complex inputs is a straightforward extension with similar results. This belief is, in fact, untrue. We present a convergence analysis of Godard/CMA FSEs that proves the important advantages provided by the FSE structure. We show that an FSE allows the exploitation of the channel diversity that supports two important conclusions of great practical signi cance: (1) a nitelength channel satisfying a length-and-zero condition allows Godard/CMA FSE to be globally convergent, and (2) the linear FSE lter length need not be longer than the channel delay spread. Computer simulation demonstrates the performance improvement provided by the adaptive Godard FSE (31 Refs.) [Li TSP 96b] Y. Li, K.J.R. Liu, Z. Ding,\Length and cost dependent local minima of unconstrained blind channel equalizers,"IEEE Transactions on Signal Processing, vol. 44, no.4, pp. 818-26, Apr. 1996. Abstract: Baud-rate linear blind equalizers may converge to undesirable stable equilibria due to di erent mechanisms. One such mechanism is the use of linear FIR lters as equalizers. In this paper, it is shown that this type of local minima exist for all unconstrained blind equalizers whose cost functions satisfy two general conditions. The local minima generated by this mechanism are thus called length dependent local minima. Another mechanism is generated by the cost function adopted by the blind algorithm itself. This type of local minima are called cost-dependent local minima. It shall be shown that several well designed algorithms do not have cost dependent local minima while other algorithms, such as the decision-directed equalizer and the Stop-and-Go algorithm, do. Unlike many existing convergence analysis the convergence of the Godard algorithms and standard cumulant algorithms under Gaussian noise is also presented here. Computer simulations are used to verify the analytical results. [Li TSP 96c] Y. Li and K.J.R. Liu,\Static and dynamic convergence behavior of adaptive blind equalizers,"IEEE Transactions on Signal Processing, vol. 44, no. 11, pp. 2736-45, Nov. 1996. Abstract: This paper presents a theoretical analysis of the static and dynamic convergence behavior for a general class of adaptive blind equalizers. We rst study the properties of prediction error functions of blind equalization algorithms, and then, we use these properties to analyze the static and dynamic convergence behavior based on the independence assumption. We prove in this paper that with a small step size, the ensemble average of equalizer coe cients will converge to the minimum of the cost function near the channel inverse. However, the convergence is not consistent. The correlation matrix or equalizer coe cients at equilibrium are determined by a Lyapunov equation. According to our analysis results, for a given channel and stepsize, there is an optimal length for an equalizer to minimize the intersymbol interference. This result implies that a longerlength blind equalizer does not necessarily outperform a shorter one, which is contrary to what is conventionally conjectured. The theoretical analysis results are con rmed by computer simulations (27 Refs.) 49 [Liu JAS 95] J.S. Liu, R. Chen,\Blind deconvolution via sequential imputations,"Journal of the American Statistical Association, vol. 90, no. 430, pp. 567-576, 1995. [Lopez de V. ASIL 91] F. Lopez de Victoria,\An adaptive blind equalization algorithm for QAM and QPR modulations: The concentric ordered modulus algorithm, with results for 16 QAM and 9 QPR," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 726-730, November 1991. Abstract: The concentric ordered modulus algorithm (COMA) represents an approach to the blind equalization of QAM (quadrature amplitude modulation) and QPR signals. COMA's error is generated as a weighted minimum di erence between a signal sample and a series of radii placed at the constellation points. COMA exhibits desired properties from the constant modulus algorithm and decision directed algorithm, such as an invariance to carrier modulation, probability distribution of constellation points, and correlation between symbols. COMA's nonlinear error generation precludes the use of classical closed-form analysis to insure an optimal weighting function design. Nevertheless, a simpli ed weighting function design algorithm is presented, which can be extended to high-order modulations (6 Refs.) [Lopez de V. ICASSP 92] F. Lopez de Victoria,\More on the concentric ordered modulus algorithm for blind equalization of QAM and QPR modulations, with results for 64 QAM, 25 QPR and 49 QPR," in Proc. International Conference on Acoustics, Speech and Signal Processing (San Francisco, CA), pp. 497-500, March 1992. Abstract: The concentric ordered modulus algorithm (COMA) exhibits desired properties from the constant modulus algorithm (CMA) and decision directed algorithm in the blind equalization of QAM and QPR signals. These properties encompass an invariance to carrier modulation, probability distribution of constellation points, and correlation between symbols. Furthermore, it is shown that COMA converges faster than CMA for severely distorted channels. The author continues the development of the algorithm, with the results for 64 QAM, 25 QPR, and 49 APR signals. COMA's nonlinear error generation precludes the use of classical closed-form analysis to insure an optimal design. Nevertheless, through the use of simulations, an algorithm and guidelines for the design are presented (6 Refs.) [Lopez de V. SPW 94] F. Lopez de Victoria, A. Bosser, I. Fijalkow, C.R. Johnson, Jr. and J.R. Treichler,\Observed (mis)behavior of CMA with periodic sources: assessment and guidelines," in Proc. IEEE Digital Signal Processing Workshop (Yosemite National Park, CA, USA), pp. 261-4, 2-5 Oct. 1994. Abstract: In the analysis of adaptive systems without training sequences, often termed "blind", a common assumption is to have a white source to excite the system. However, a condition which arises in practical systems is presented which does nor conform to the white source condition. A periodic symbol sequence at the input can steer the adaptation process to a solution which is not the desired solution. This (mis)behavior does arise when the equalizer is adapted with the constant modulus algorithm (CMA) and the periodic signal is nor constant modulus. The paper explores the conditions which lead to this (mis)behavior (8 Refs.) [Lou ICASSP 92] Y. Lou,\Comparison of adaptive blind equalizers," in Proc. International Conference on Acoustics, Speech and Signal Processing (San Francisco, CA), pp. 545-548, March 1992. Abstract: A new adaptive blind equalizer has been developed and analyzed on the basis of the 50 channel estimation standard (CES). It is also simulated for a QAM transmission system. The results are compared with the performances of other blind equalizers both theoretically and experimentally (26 Refs.) [Lou TCOM 95] Y.A. Lou,\Channel estimation standard and adaptive blind equalization,"IEEE Transactions on Communications, vol. 43, no. 2-4, pp. 182-186, 1995. Abstract: Presents a novel concept of channel estimation standard (CES) and applies a new CES error criterion to the process of an adaptive blind equalization. It is shown that the establishment of the CES contributes to the development of a practical communication scheme for approaching to the capacity of a high SNR band-limited channel without using a preamble signal training (6 Refs.) [Macchi TIT 83] O. Macchi and E. Eweda,\Converegence analysis of self-adaptive equalizers,"IEEE Transactions on Information Theory, vol. IT-30, pp. 162-176, 1983. Abstract: A theoretical analysis of self-adaptive equalization for data transmission is carried out starting from known convergence results for the corresponding trained adaptive lter. The development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel. Thanks to the boundedness of the decision function used for data recovery, it is proved that the algorithm is bounded. Strong convergence results can be reached when a perfect (noiseless) equalizer exists, and the algorithm will converge to it if the eye pattern is initially open. Otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been de ned, and some of them will result in a poor error rate. The case of a noise channel exhibits limit points for the algorithm that di er from those of the classical (trained) algorithm, the stronger the noise, the greater the di erence. One of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over (20 Refs.) [Mann ICDSP 87] R. Mann, W. Tobergte, K.D. Kammeyer,\Applications of constant modulus algorithms for adaptive equalization of time-varying multipath channels," in Proc. International Conference on Digital Signal Processing (Florence, Italy), pp. 421-5, Sept. 7-10, 1987. Abstract: In recent papers the authors have presented a class of adaptive ltering algorithms which can be applied to compensate the e ects of multiple propagation in conventional FM communication systems. The paper includes some results concerning the performance of the algorithms under the in uence of additive Gaussian noise and linear distortions caused by a nonideal intermediate frequency lter. The application of the modi ed constant algorithm for equalization of time-variant two-ray channels is illustrated through computer simulation (7 Refs.) [Mann ICASSP 89] R. Mann and K.D Kammeyer,\A pole-zero-tracking constant modulus algorithm," in Proc. International Conference on Acoustics, Speech and Signal Processing (Glasgow, UK), pp. 1227-30, May 23-26, 1989. Abstract: An approach to multipath correction in FM transmission is proposed. It is based on the factorization of the channel transfer function in terms of the roots of the underlying polynomial. A time-recursive algorithm for the identi cation of the roots of the all-zero channel transfer function at the receiver is presented. It allows an e cient adaptive design of special equalizer structures with low computational burden compared to existing solutions. The multipath correction method was tested in connection with a baseband FM receiver and a modulating signal consisting of a bandlimited stereo-multiplex signal produced by combining 51 sinusoids with random phases distributed uniformly in the interval 0-2 pi . The identi cation of the channel roots was accomplished by means of a stochastic gradient algorithm with constant step size alpha =0.0005. The trajectories of the identi ed roots in the z-plane and the corresponding modulus of the equalizer output signal are illustrated for three di erent channel con gurations (10 Refs.) [Mathur SPL 95] A. Mathur, A.V. Keerthi, J.J. Shynk,\Cochannel signal recovery using the MUSIC algorithm and the constant modulus array,"IEEE Signal Processing Letters, vol. 2, no.10, pp. 191-4, Oct. 1995. (Filed in BERG library.) Abstract: We describe an approach to cochannel signal recovery for correlated sources that is based on the the MUSIC (multiple signal classi cation) algorithm and the constant modulus (CM) array. The MUSIC algorithm provides estimates of the source angles of arrival, which are used to initialize a parallel bank of CM arrays. These adaptive beamformers are updated by the constant modulus algorithm (CMA) to capture and track the source signals, even when they are highly correlated. Computer simulations are presented to illustrate the transient behavior of the proposed algorithm (13 Refs.) [Mayrargue ICASSP 93] S. Mayrargue,\Spatial equalization of a radio-mobile channel without beamforming using the constant modulus algorithm (CMA)," in Proc. International Conference on Acoustics, Speech, and Signal Processing (Minneapolis, MN, USA), pp. 344-7, 27-30 April 1993. Abstract: The author considers the problem of signal recovery by a multisensor receiver in a multipath propagation channel. She shows that a spatial ltering can recover the transmitted signals under two conditions, namely that both the number of sensors and the number of paths be larger than the length of the intersymbol interference, which is assumed to be nite. Thereby, signal recovery is realized without beamforming, i.e. without forming a beam towards one of the paths while cancelling the others, which would require a number of sensors much larger than the number of paths. Spatial ltering is able to cancel a multipath jammer, provided that an additional number of sensors, equal to the ISI (intersymbol interference) length of the jammer, is available. The spatial ltering is obtained adaptively by a blind algorithm, the CMA, which relies on the property of the transmitted signal to have a constant modulus. Simulation results on the present spatial CMA were compared with results for a temporal single-sensor CMA. The latter failed to recover a correct signal constellation while spatial CMA succeeded, even in a noisy environment plus a jammer (9 Refs.) [Mayrargue ICASSP 94] S. Mayrargue,\A blind spatio-temporal equalizer for a radio-mobile channel using the constant modulus algorithm (CMA)," in Proc. International Conference on Acoustics, Speech and Signal Processing (Adelaide, SA, Australia), pp. 317-20, 19-22 April 1994. (Filed in BERG library.) Abstract: Discusses constant modulus signal recovery by a multi-sensor receiver in a multipath propagation channel. The author rst shows that spatio-temporal ltering can recover the transmitted signal provided that the number of sensors minus one times the length of the temporal ltering be larger than the intersymbol interference length assumed to be nite. It is also shown that spatio-temporal ltering is able to cancel jammers that have undergone multiple paths, at the expense of an increase in the number of sensors or in the temporal ltering length. The ltering is obtained adaptively by a blind algorithm, the CMA, which relies on 52 the property of the transmitted signal to have a constant modulus, which is the case of phase and frequency modulated signals, widely used in telecommunications (8 Refs.) [McLaughlin ICASSP 89] McLaughlin, S.; Mulgrew, B.; Cowan, C.F.N.,\A novel adaptive equaliser for nonstationary communication channels," in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (Glasgow, UK), pp. 944-7, 23-26 May 1989. Abstract: The performance of a Kalman decision-feedback equalizer (DFE) that uses a channel estimator based on a least-meansquares (LMS) algorithm is studied for a variety of stationary and nonstationary communications channels. This structure provides a means of model order reduction by using the residuals of the LMS to provide information on the unmodeled paths in the communication channel, which is then incorporated into the Kalman DFE structure as observation noise. The structure is compared with a conventional DFE that is trained by a Godard-Kalman algorithm with exponential windowing and adaptive Kalman structure previously reported (B. Mulgrew and C.F.N. Cowan, 1987). The results indicate that the best performance, in terms of nal MSE (mean square error), is o ered by the adaptive Kalman DFE structure, the nal MSE being lower than that achieved by the conventional DFE by some 5-10 dB (10 Refs.) [McLaughlin ICC 89] McLaughlin, S.; Mulgrew, B.; Cowan, C.F.N.,\A performance study of 3 adaptive equalisers in the mobile communication environment," in Proc. IEEE International Conference on Communications ( Boston, MA), pp. 193-7, 11-14 June 1989. Abstract: The performance of three equalizer structures are compared in terms of their meansquared error performance in a simulated mobile communications environment. The equalizers considered are: (a) a Kalman equalizer, which utilizes a least-mean-square (LMS) algorithm as a channel estimator to provide the equalizer with an estimate of the channel impulse response; and (b) a Kalman decision feedback equalizer (DFE) based on the above, but incorporating decision feedback in the structure. Both of these structures provide a means of model order reduction by using the residuals of the LMS to provide information on the unmodeled paths in the communications channel, which is incorporated in the Kalman structure as observation noise. These structures are studied and compared with a conventional decision feedback equalizer (the third equaliser) which is trained by a Godard-Kalman (1974) algorithm with exponential windowing (16 Refs.) [McLaughlin IEE-F 91] McLaughlin, S.,\Adaptive equalisation via Kalman ltering techniques,"IEE Proceedings F: Image, Radar and Signal Processing, vol. vol.138, no.4, pp. p. 388-96, Aug. 1991. Abstract: The arithmetic complexity and the mean squared error (MSE) performance of three adaptive equaliser structures are compared. The rst is a conventional decision feedback equaliser (DFE) which utilises a Godard-Kalman adaptive algorithm to carry out the tap weight update. The second is an adaptive Kalman equaliser which utilises a least mean squares (LMS) algorithm to carry out the channel estimation process and a Kalman lter structure for the data estimation. The nal, novel, structure considered utilises the performance advantage of both of the previous structures. This is achieved by using the basic structure of the adaptive Kalman equaliser but incorporating an element of decision feedback (21 Refs.) [Mendoza MIL 89] R. Mendoza, et al.,\Interference rejection using a hybrid of a constant modulus algorithm and the spectral correlation discriminator," in Proc. IEEE Military Communications Conference (Boston, MA), pp. 491-7, Oct. 15-18, 1989. 53 Abstract: The authors present a novel time-dependent adaptive ltering technique that is formed by combining a constant modulus algorithm (CMA) with the spectral correlation discriminator (SCD). This technique is superior to the CMA or SCD alone, as it produces a lower MSE (mean square error) and bit-error rate. The hybrid CMA/SCD has improved resistance to interference and less phase roll than the CMA. Furthermore, the hybrid CMA/SCD does not distort the ltered signal, as does the SCD alone. Simulations are presented comparing the SCD, CMA, and training-sequencedirected techniques with the hybrid CMA/SCD for interference rejection (17 Refs.) [Mendoza TSP 91] R. Mendoza, J.H. Reed, T.C. Hsia, B.G. Agee,\Interference rejection using the time-dependent constant modulus algorithm (CMA) and the hybrid CMA/spectral correlation discriminator,"IEEE Transactions on Signal Processing, vol. 39 no. 9, pp. 2109-2111, September 1991. Abstract: Two new blind adaptive ltering algorithms for interference rejection using timedependent ltering structures are presented. The time-dependent structure allows the adaptive lter to outperform the conventional adaptive lter implemented with a time-independent structure for ltering of cyclostationary communication signals. At the same time, the blind adaption algorithms allow the lters to operate without the use of an external training signal. The rst algorithm applies the CMA to an unconstrained time-dependent ltering structure. The second algorithm applies the CMA to a spectral correlation discriminator, which is constrained to select signals with unique spectral correlation characteristics. Using computer simulations, it is shown that the blind time-dependent ltering algorithms can provide meansquare errors (MSEs) and bit error rates (BERs) that are signi cantly lower than the MSEs and BERs provided using conventional time-independent adaptive lters. It is also shown that these processors can outperform the nonblind training-sequence directed time-independent adaptive lter (8 Refs.) [Meyer CISS 96] W.E. Meyer and J.P. LeBlanc,\Blind adaptive fractionally-spaced CMA in the presence of channel noise," in Proc. Conference on Information Science and Systems (Princeton, NJ), pp. 373-374, Mar. 1996. [Minardi ICASSP 96] M.J. Minardi, M.A. Ingram,\Finding Misconvergences In Blind Equalizers And New Variance Constraint Cost Functions To Mitigate The Problem," in Proc. International Conference on Acoustics, Speech and Signal Processing (Atlanta, GA), pp. 1724, May 7-9, 1996. Abstract: We show that equalizer weights that cause the overall channel-equalizer response to approximate a pure delay (in a mean square error sense) are within the region of convergence of a stable local minimum of Bussgang cost functions. For any nontrivial channel the approximations for some delay values are poor. If a Bussgang algorithm is initialized with these weights it misconverges. A Monte-Carlo simulation created misconvergence as predicted. We propose the Variance Constraint (VC) algorithms that have more robust convergence properties than the Bussgang algorithms. The VC cost functions are similar to the well-known Godard cost functions. They incorporate estimates of the channel-equalizer overall gain and use a linear constraint on the weights. These two modi cations push the bad stable points farther away in weight space from the stable points with good performance (i.e., low inter-symbol interference). A Monte-Carlo simulation of nearly 2000 channels shows no misconvergence. 54 The authors present a novel time-dependent adaptive ltering technique that is formed by combining a constant modulus algorithm (CMA) with the spectral correlation discriminator (SCD). This technique is superior to the CMA or SCD alone, as it produces a lower MSE (mean square error) and bit-error rate. The hybrid CMA/SCD has improved resistance to interference and less phase roll than the CMA. Furthermore, the hybrid CMA/SCD does not distort the ltered signal, as does the SCD alone. Simulations are presented comparing the SCD, CMA, and training-sequencedirected techniques with the hybrid CMA/SCD for interference rejection (17 Refs.) [Mendoza TSP 91] R. Mendoza, J.H. Reed, T.C. Hsia, B.G. Agee,\Interference rejection using the time-dependent constant modulus algorithm (CMA) and the hybrid CMA/spectral correlation discriminator,"IEEE Transactions on Signal Processing, vol. 39 no. 9, pp. 2109-2111, September 1991. Abstract: Two new blind adaptive ltering algorithms for interference rejection using timedependent ltering structures are presented. The time-dependent structure allows the adaptive lter to outperform the conventional adaptive lter implemented with a time-independent structure for ltering of cyclostationary communication signals. At the same time, the blind adaption algorithms allow the lters to operate without the use of an external training signal. The rst algorithm applies the CMA to an unconstrained time-dependent ltering structure. The second algorithm applies the CMA to a spectral correlation discriminator, which is constrained to select signals with unique spectral correlation characteristics. Using computer simulations, it is shown that the blind time-dependent ltering algorithms can provide meansquare errors (MSEs) and bit error rates (BERs) that are signi cantly lower than the MSEs and BERs provided using conventional time-independent adaptive lters. It is also shown that these processors can outperform the nonblind training-sequence directed time-independent adaptive lter (8 Refs.) [Meyer CISS 96] W.E. Meyer and J.P. LeBlanc,\Blind adaptive fractionally-spaced CMA in the presence of channel noise," in Proc. Conference on Information Science and Systems (Princeton, NJ), pp. 373-374, Mar. 1996. [Minardi ICASSP 96] M.J. Minardi, M.A. Ingram,\Finding Misconvergences In Blind Equalizers And New Variance Constraint Cost Functions To Mitigate The Problem," in Proc. International Conference on Acoustics, Speech and Signal Processing (Atlanta, GA), pp. 1724, May 7-9, 1996. Abstract: We show that equalizer weights that cause the overall channel-equalizer response to approximate a pure delay (in a mean square error sense) are within the region of convergence of a stable local minimum of Bussgang cost functions. For any nontrivial channel the approximations for some delay values are poor. If a Bussgang algorithm is initialized with these weights it misconverges. A Monte-Carlo simulation created misconvergence as predicted. We propose the Variance Constraint (VC) algorithms that have more robust convergence properties than the Bussgang algorithms. The VC cost functions are similar to the well-known Godard cost functions. They incorporate estimates of the channel-equalizer overall gain and use a linear constraint on the weights. These two modi cations push the bad stable points farther away in weight space from the stable points with good performance (i.e., low inter-symbol interference). A Monte-Carlo simulation of nearly 2000 channels shows no misconvergence. 54 [Mulgrew SCEF 84] Mulgrew, B.,\The application of Kalman ltering to channel equalisation," in Proc. IEE Saraga Colloquium on Electronic Filters (London, UK), pp. IX/1-7, 21 May 1984. Abstract: Describes how Kalman estimation techniques have been applied to communication channel equalisation. In 1971 Lawrence and Kaufman made an exploratory study, using a Kalman lter to estimate directly the output of a channel. Unfortunately, the formulation produced a nonlinear observation equation, with the associated problems. In 1974 Godard used the Kalman lter to estimate the tap weights and produced a very powerful algorithm. In 1978 Falconer and Ljung recognised that Godard's algorithm also arose from classical least squares estimation. Using results from this eld, they produced the fast Kalman algorithm. Comparison of relative performance characteristics of the algorithms and their computational e ciency are discussed (10 Refs.) [Nishimori IEICE 95] K. Nishimori, N. Kikuma and N. Inagaki,\The di erential CMA adaptive array antenna using an eigenbeamspace system,"IEICE Transactions on Communications, vol. E78-B, no.11, pp. 1480-8, Nov. 1995. Abstract: This paper addresses approaches to enhancement of performance of the CMA (constant modulus algorithm) adaptive array antenna in multipath environments that characterize the mobile radio communications. The cost function of the CMA reveals that it has an AGC (automatic gain control) procedure of holding the array output voltage at a constant value. Therefore, if the output voltage by the initial weights is di erent from the object value, then the CMA may su er from slow convergence because suppression of the multipath waves is delayed by the AGC behavior. Our objective is to improve the convergence characteristics by adopting the di erential CMA for the adaptive array algorithm. First, the basic performance of the di erential CMA is clari ed via computer simulation. Next, the di erential CMA is incorporated into the eigen-beamspace system in which the eigenvectors of the correlation matrix of array inputs are used in the BFN (beamforming network). This BFN creates the optimum orthogonal multibeams for radio environments and works as a preprocessor of the di erential CMA. The computer simulation results have demonstrated that the di erential CMA with the eigen-beamspace system has a much better convergence characteristics than the conventional CMA with the element space system. Furthermore, a modi ed algorithm is introduced which gives the stable array output voltages after convergence, and it is con rmed that the algorithm can carry out more successful adaptation even if the radio environments are changed abruptly (14 Refs.) [Oh ICC 95] K. Oh and Y. Chin,\Modi ed constant modulus algorithm: blind equalization and carrier phase recovery algorithm," in Proc. IEEE International Conference on Communications (Seattle, WA, USA), pp. 498-502, 18-22 June 1995. Abstract: The modi ed constant modulus algorithm (MCMA) that accomplishes blind equalization and carrier phase recovery simultaneously is proposed. Since the constant modulus algorithm (CMA) converges independently of carrier recovery, at convergence an output constellation has a phase error. With modi cation of the CMA, the proposed algorithm can solve this problem. Furthermore, the new algorithm results in the performance enhancement of convergence speed and residual ISI compared to the CMA. In addition, the proposed algorithm achieves a performance as good as that of a joint CMA and decision-directed (DD) phase recovery scheme with reduction in computational complexity. Simulation results con rm the e ectiveness of the proposed algorithm in removing ISI and correcting carrier phase error at the same time (17 Refs.) 55 [Oh GLOBE 95] K. Oh and Y. Chin,\New blind equalization techniques based on constant modulus algorithm," in Proc. IEEE Global Telecommunications Conference (Singapore), pp. 865-869, 14-16 November 1995. Abstract: New blind equalization techniques based on the constant modulus algorithm (CMA) are investigated. Since CMA is phaseblind, an output constellation has a phase error at convergence. The modi ed constant modulus algorithm (MCMA) that accomplishes blind equalization and carrier phase recovery simultaneously is proposed. Then, the performance of the MCMA is enhanced by incorporating a decision-directed (DD) algorithm. The resultant scheme, dual-mode MCMA, achieves the faster convergence speed and smaller residual ISI. A blind decision feedback equalizer (DFE) using MCMA is also presented, which leads to superior performance compared to ones without the feedback path over the channel with inband spectral nulls. Simulation results con rm the e ectiveness of the proposed techniques on various channels and signals (16 Refs.) [Ohgane VTC 91] T. Ohgane, H. Sasaoka, N. Matsuzawa, K. Takeda, T. Shimura,\A development of GMSK/TDMA system with CMA adaptive array for land mobile communications,"IEEE Vehicular Technology Conference, vol. St. Louis, MO, pp. 924, May 1991. Abstract: The authors describe a hardware implementation of a GMSK/TDMA (Gaussian minimum-shift keying/time-division multiple-access) communication system and show the bit error rate (BER) performance of the system. This system employs a modulation data rate of 256 kb/s and accommodates 24 users/carrier. In order to reduce the e ect of multipath fading, an adaptive array which has four antenna elements is implemented using a digital beamforming concept. All signal processing is carried out by digital signal processors. In an AWGN (additive white Gaussian noise) channel, this system has a 6-dB gain in input power at a BER of 1.0*10/sup -3/ compared with the ordinary system with only one antenna. A eld test shows that the gain at a BER of 1.0*10/sup -3/ becomes 20 dB in a at fading channel (6 Refs.) [Ohgane ECJ 91] T. Ohgane,\Characteristics of CMA adaptive array for selective fading compensation in digital land mobile radio communications,"Electronics and Communications in Japan, vol. 74 no.9, pp. 43-53, September 1991. Abstract: The author considers the constant modulus algorithm (CMA) adaptive array for high-speed TDMA transmission using GMSK, which is the most ordinary constant-envelope modulation system in digital mobile land communication. A computer simulation is executed for the case where a four-element adaptive array is applied to cope with selective fading. CMA is employed as the array control algorithm, which is suited to the compensation of the constant-envelope modulated signal. Assuming a model with two arriving waves, the convergence and the bit error rate (BER) performances of CMA are evaluated. It is shown that the BER is improved greatly up to the limit where no more error can be reduced, compared to the case without an adaptive array. It is seen also that the error rate is improved when the delay di erence between the two arriving waves is large. Thus, the proposed method is shown to be an e ective technique to cope with selective fading (10 Refs.) [Ohgane VTC 92] T. Ohgane, H. Sasaoka, N. Matsuzawa, T. Shimura,\BER performance of CMA adaptive array for a GMSK/TDMA systema description of measurements in central Tokyo," in Proc. IEEE Vehicular Technology Conference (Denver, CO), pp. 1012-1017, May 1992. Abstract: The bit-error-rate (BER) performance of a constant modulus algorithm adaptive array for a Gaussianltered minimum shift keying/time division multiple access 56 (GMSK/TDMA) system is described. The measurement was carried out in central and suburban Tokyo. The system uses a modulation data rate of 256 kb/s. In order to reduce the e ect of multipath fading, an adaptive array of four antenna elements is implemented using a digital beam forming concept. The improvement in input power, compared with that of a single antenna system, reaches 16 dB in suburban Tokyo and 19 to 23 dB in central Tokyo at a BER of 1.0*10/sup -2/. Moreover, in central Tokyo, the irreducible error seen in the single antenna system is removed by the adaptive array (10 Refs.) [Ohgane TVT 93a] T. Ohgane, N. Matsuzawa, T. Shimura, M. Mizuno, H. Sasaoka,\BER performance of CMA adaptive array for high-speed GMSK mobile communication-a description of measurements in central Tokyo,"IEEE Transactions on Vehicular Technology, vol. 42 no.4, pp. 484-490, November 1993. Abstract: This paper describes the performance of an adaptive array as a countermeasure to multipath fading for a 256 kbps Gaussianltered minimum shift keying (GMSK) mobile communication system operating in the 1.5 GHz band. An adaptive array having four antenna elements is implemented using the digital beam forming concept. The constant modulus algorithm (CMA) is employed for the adaptation process to ease the implementation. Measurements in central Tokyo of the bit error rate (BER) performance and an array pattern arising in the multipath environment are presented. Analysis of the array pattern con rms that the array succeeds in directing nulls to the delayed signals. BER performance shows an improvement in E/sub b//N/sub 0/, compared with that of a single antenna system, of 17.5 to 22 dB at a BER of 1.0*10/sup -2/ in a frequency-selective fading channel (21 Refs.) [Ohgane TVT 93b] T. Ohgane, T. Shimura, N. Matsuzawa, H. Sasaoka,\An implementation of a CMA adaptive array for high speed GMSK transmission in mobile communications,"TVT, vol. 42 no. 3, pp. 282-288, August 1993. Abstract: The hardware implementation of an adaptive array as a technique for compensating multipath fading in mobile communications is described. The number of the antenna elements is four. The target communication system is modulated by 256 kbps Gaussianltered minimum shift keying (MSK) and has a time-division multiplexing (TDM) architecture with 24 time slots. Based on the digital beamforming concept, all of the signals and the array weights are digital-signal processed. The constant modulus algorithm (CMA) is employed for weight optimizing. In an additive white Gaussian noise channel, this system has 5.6dB gain in an energy-per-bit-to-noise-density ratio at a bit error rate (BER) of 1.0*10/sup -3/, compared with a single antenna system. The result of the basic eld test shows that the gain at a BER of 1.0*10/sup -3/ reaches 22.3 dB in a nonselective, slow Rayleigh fading channel given a 5 Hz maximum Doppler shift (18 Refs.) [Papadias ICDSP 93] C. Papadias, D.T.M. Slock,\On the convergence of normalized constant modulus algorithms for blind equalization," in Proc. International Conference on Digital Signal Processing (Nicosia-Cyprus), pp. 245-250, July 1993. [Papadias ASIL 94] C.B. Papadias and D.T.M. Slock,\On the decision-directed equalization of constant modulus signals," in Proc. 28th Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA, USA), pp. 1423-7, 31 Oct.-2 Nov. 1994. Abstract: Decision-directed (DD) equalization is the most primitive blind equalization (BE) method for the cancelling of intersymbol-interference (ISI) in data communication systems. 57 Even through DD equalizers are believed to be unable to open the channel eye when it is initially closed, this does not seem to be true in the of constant-modulus (CM) constellations (pure phase modulation). We investigate the shape of the DD cost function in this case and obtain several interesting results that indicate that the DD algorithm should be capable of opening a closed channel eye in the CM case. Based on this fact, we propose a novel hybrid CMA-DD equalization scheme that o ers an appealing alternative to the generalized Sato (GSATO) algorithm, for QAM constellations. Our theoretical claims about the performance of DD equalizers as well as the performance of our novel scheme are veri ed through computer simulations (7 Refs.) [Papadias ICASSP 94] C.B. Papadias and D.T.M. Slock,\New adaptive blind equalization algorithms for constant modulus constellations," in Proc. International Conference on Acoustics, Speech and Signal Processing (Adelaide, SA, Australia), pp. 321-4, 19-22 April 1994. Abstract: We present a new class of adaptive ltering algorithms for blind equalization of constant modulus signals. The algorithms are rst derived in a classical system identi cation context by minimizing at each iteration a deterministic criterion and then their counterpart for blind equalization is derived by modifying this criterion taking into account the constantmodulus property of the transmitted signal. The algorithms impose more constraints than the classical constant modulus algorithm (CMA) and as a result achieve faster convergence. An asymptotic analysis has provided useful parameter bounds that guarantee the algorithms' stability. A priori knowledge of these bounds helps the algorithms escape from undesirable local minima of their cost function thus giving them a potential advantage over the classical CMA. An e cient computational organization for the derived algorithms is also proposed and their behaviour has been tested by means of computer simulations (12 Refs.) [Papadias EUSIPCO 94] C.B. Papadias and D.T.M Slock,\Towards globally convergent blind equalization of constant modulus signals: A bilinear approach," in Proc. 7th European Signal Processing Conference (Edinburgh, UK), pp. 1827-30, 13-16 Sept. 1994. Abstract: We consider the problem of blind equalization of a constant modulus signal. One of the most popular classes of algorithms in this context is the Godard family of blind equalizers, which includes among others the constant modulus algorithm (CMA) (Treichler and Agee, 1983). A common drawback of these algorithms is that they may converge to undesired equalizer settings if not properly initialized. This is known as the problem of ill convergence and is primarily due to the non-convex form of the cost function of algorithms of this class with respect to the equalizer parameters. We propose a di erent approach to the problem, namely, a bilinear one, which leads to a di erent parameterization and to the construction of a convex cost function with respect to the parameters introduced. In a perfectly parameterized case (the equalizer`s order matches exactly the order of the channel inverse), the solution to the problem is unique and permits for a direct calculation of the optimal equalizer. In overparameterized cases however, there exist multiple solutions to our cost function. However, we propose a method that still allows determination of the channel inverse in this case. Di erent adaptive schemes are proposed to adaptively compute the solution of our criterion and the in uence of additive noise is also discussed (7 Refs.) [Papadias ASIL 95] C.B. Papadias, A. Paulraj,\Decision-feedback equalization and identi cation of linear channels using blind algorithms of the bussgang type," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 335-340, October 1995. Abstract: We consider the problem of blind equalization of linear communication channels. 58 Some recent results indicate that the performance of Bussgang blind equalization algorithms can be improved by using diversity such as fractional spacing or antenna array reception. In this work we examine the performance of such algorithms (especially of the popular CMA 2-2), when used in a decision-feedback setup. It turns out that such a simple structure may help avoiding the common problems of \zeros on the unit circle" (symbol rate case) and of \zeros in common" (fractioanlly-spaced case). Theoretical analysis as well as computer simulations are provided in order to demonstrate this fact. [Papadias ICASSP 97] C. Papadias,\On the existence of undesired global minima of Godard equalizers," in Proc. International Conference on Acoustics, Speech and Signal Processing (Munich, Germany), pp. 3937-3940, 20-24 Apr. 1997. Abstract: We consider the problem of global convergence of Godard (1980) equalizers in the special case of binary (2-PAM) input signals, when the channel impulse response is complex. We present a class of global minima of all Godard equalizers for this case, which do not correspond to settings free of intersymbol-interference (ISI). The equalizer output corresponding to these global minima appears as a four-point constellation in the complex plane, however it is easily shown that the decomposition in its real and imaginary part provides two ISI-free versions of the transmitted signal. In the case of multi-user constant modulus algorithms, the situation is somewhat more complicated: the real and imaginary parts of each equalizer output after convergence, may correspond to di erent user signals. These results can be extended to other types of real input signals (19 Refs.) [Parra PIMRC 96] I. Parra, G. Xu and H. Liu,\A least squares projective constant modulus approach," in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (Toronto, Ont., Canada), pp. 673-6, Sept. 27-29, 1995. Abstract: We present an iterative constant modulus algorithm using the least squares method with projection to blindly separate and estimate multiple co-channel frequency modulated (FM) signals in an antenna array system. The constant modulus property of FM signals is exploited to simultaneously determine the spatial signature and the waveform for each signal. Furthermore, the spatial response of the array and the propagation environment do not need to be known prior to using this algorithm, making it particularly useful in mobile communications. Computer simulations are presented to verify its promising performance (7 Refs.) [Peloso TCE 92] R.A. Peloso,\Adaptive equalization for advanced television,"IEEE Transactions on Consumer Electronics, vol. 38, no. 3, pp. 119-126, 1992. Abstract: The author presents an overview of the adaptive equalization process, including various equalizer topologies and error signal generation schemes, for 16 quadrature amplitude modulation (QAM) digital transmission systems. Four equalizer structures, along with the concept of blind equalization, are discussed. Simulation results for simple theoretical channels indicate that equalizer performance is a function of the topology, error generation, and underlying transmission channel (8 Refs.) [Petrus SPL 95] P. Petrus and J.H. Reed,\Time dependent adaptive arrays,"IEEE Signal Processing Letters, vol. 2, no. 12, pp. 219-222, Dec. 1995. (Filed in BERG library.) Abstract: A time dependent adaptive array (TDAA) is a combination of the time dependent optimal lter (or FRESH lter) and an adaptive array. A TDAA exploits spatial, frequency, and time diversities. The idea behind the TDAA is that additional sources of correlated data 59 can be obtained from each spatially separated array element through frequency shifting the data at each antenna element. For some signal types, the TDAA can be blindly adapted by con guring the TDAA as a spectral correlation predictor. The performance of the TDAA con gured as a spectral correlation predictor is compared with the least-squares CMA array (LSCMA), the least-squares SCORE (LSSCORE) array, the conventional array with a training signal, the TDAA with a training signal, and a nodiversity system. The test signals used are advanced mobile phone service (AMPS) signals (5 Refs.) [Petrus VTC 95] P. Petrus and J.H. Reed,\Cochannel interference rejection for AMPS signals using spectral correlation properties and an adaptive array," in Proc. IEEE Vehicular Technology Conference (Chicago, IL), pp. 30-4 vol.1, 25-28 July 1995. Abstract: Cochannel interference is a major problem that limits the cell capacity in AMPS systems. Adaptive beamforming can signi cantly reduce co-channel interference by forming a beam in the direction of the signal-of-interest (SOI). This paper shows how the supervisory audio tone (SAT) feature can be exploited by the adaptive array to distinguish the SOI from the interferers. The presence of the SAT produces a cyclostationary feature in the AMPS signals. Using this feature, a SCORE-like beamformer (spectral correlation discriminator array-SCDA) provides 15 dB improvement in the demodulated voice SNR for low SIR received signals. The performance of SCDA is compared with a least-squares CMA (LSCMA) beamformer (8 Refs.) [Picchi TCOM 87] G. Picchi and G. Prati,\Blind equalization and carrier recovery using a \stopand-go" decision directed algorithm,"IEEE Transactions on Communications, vol. COM-35, pp. 877-887, Sept. 1987. (Filed in BERG library.) Abstract: We show that the standard decision-directed estimated-gradient adaptation algorithm for joint MSE equalization and carrier recovery, normally utilized in the open-eye condition, can be turned into an algorithm providing e ective blind convergence in the MSE sense, usable in the closed-eye startup phase with no need of a known training sequence. This is obtained by means of a simple ag telling both the equalizer and the synchronizer whether the current output error with respect to the decided symbol is su ciently reliable to be used. If not, adaptation is stopped for the current iteration. In the paper, this \stop-and-go" decisiondirected algorithm is presented for both linear and decision-feedback MSE complex equalizers with joint blind carrier recovery. Simulation results demonstrate the e ectiveness of the proposed technique. [Pickholtz ASIL 93] R. Pickholtz and K. Elbarbary,\The recursive constant modulus algorithm; a new approach for real-time array processing," in Proc. 27th Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA, USA), pp. 627-32, 1-3 Nov. 1993. Abstract: The constant modulus algorithm (CMA) has proved its ability to compensate the severe e ect of the multipath interference and co-channel signals on the constant modulus signals. The application limitation of the CMA is mainly due to its slow rate of convergence. The paper presents a new algorithm which keeps the desired properties of the original CMA while it reduces the time required for convergence on the expense of some added mathematical complexity. The algorithm shows stable operations in both cases of compensating the e ect of the multipath interference by an adaptive correction lter and separation of cochannel signals by an adaptive array followed by a signal canceler. The proposed algorithm is an order of magnitude faster than the original CMA on the average (8 Refs.) 60 [Ping ACTA 95] He Ping,\The study on RG algorithm: a new blind equalization algorithm,"Acta Electronica Sinica, China, vol. 22, no.10, pp. 108-11, Oct. 1995. Abstract: In this paper, a new blind equalization algorithm, RG algorithm, has been presented. Like the Godard blind equalization algorithm, the convergence performance of the RG algorithm is very good and its symbol error performance is better than that of the Godard algorithm, especially for the MPSK modulation system. The analysis and simulation indicate that the RG algorithm is a good blind equalization algorithm for data transmission systems (4 Refs.) [Porat TSP 91] B. Porat, B. Friedlander,\Blind equalization of digital-communication channels using high-order moments,"IEEE Transactions on Signal Processing, vol. 39, no. 2, pp. 522-526, 1991. Abstract: The authors describe algorithms for blind equalization of digital communication channels of the quadrature-amplitudemodulation (QAM) type. These algorithms are based on the fourth-order statistical moments of the received data sequence. The rst of the two is a linear least-squarestype algorithm. The second algorithm is of the nonlinear least-squarestype. The algorithms use the fourth-order statistical moments of the symbol sequence to explicitly estimate the channel impulse response. The estimated impulse response is used, in turn, to construct a linear mean-square error equalizer. The performance of this equalizer is not optimal in any sense, but it is adequate for channels with mild intersymbol interference or when the number of data points available for estimating the channel response is very large (13 Refs.) [Prakriya TSP 95] S. Prakriya, D. Hatzinakos,\Blind identi cation of lti-zmnl-lti nonlinear channel models,"IEEE Transactions on Signal Processing, vol. 43, no. 12, pp. 3007-3013, 1995. Abstract: A simple method is proposed for blind identi cation of discrete-time nonlinear models consisting of two linear time invariant (LTI) subsystems separated by a polynomial-type zero memory nonlinearity (ZMNL) of order N (the LTI-ZMNL-LTI model). The linear subsystems are allowed to be of nonminimum phase (NMP), though the rst LTI can be completely identi ed only if it is of minimum phase. With a circularly symmetric Gaussian input, the linear subsystems can be identi ed using simple cepstral operations on a single 2-D slice of the N+1 th-order polyspectrum of the output signal. The linear subsystem of an LTI-ZMNL model can be identi ed using only a 1-D moment or polyspectral slice if it is of minimum phase. The ZMNL coe cients are not identi ed and need not be known. The order N of the nonlinearity can, in principle, be estimated from the output signal. The methods are analytically simple, computationally e cient, and possess noise suppression characteristics. Computer simulations are presented to support the theory (23 Refs.) [Raheli ICCO 91] R. Raheli and G. Picchi,\Synchronous and fractionally-spaced blind equalization in dually-polarized digital radio links," in Proc. International Conference on Control (Denver, CO), pp. 156-161, 1991. Abstract: Two-dimensional equalization structures with symbol-rate or fractional tap-spacing are used to counteract the combined e ects of selective fading and cross-polarization interference in dually polarized digital radio links. After generalizing the known optimum equalizers to the case of twodimensional complex signals, steady-state analysis is presented based on the evaluation of a bound on the outage probability. Finite-length structures are considered and compared to the optimum linear receiver. The blind acquisition of two-dimensional Tand 61 T/2-spaced equalizers is also shown by using a stop-and-go decision-directed algorithm in a simulation analysis (9 Refs.) [Ramesh ISCS 82] Ramesh, N.S.; Mitra, S.K.,\Block adaptive equalization," in Proc. International Symposium on Circuits and Systems (Rome, Italy), pp. 686-9, 10-12 May 1982. Abstract: High speed data transmission over voiceband (telephone) channels is limited by the slowly time-varying amplitude and phase distortion introduced by the channel. Such distortion results in intersymbol interference at the receiver. The authors describe an adaptive equalizer based on a block Kalman lter having the same performance as the Godard equalizer (1974), but requiring only 4N multiplications per sample. A tracking version of this equalizer is also described, which is capable of adapting to slowly timevarying channels. Such a tracking technique requires minimal additional e ort (6 Refs.) [Rao ASIL 95] A. Rao and R. Kumaresan,\PCMA: Parametric constant modulus algorithm for demodulating co-channel FM signals," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 473-477, October 1995. Abstract: In this paper, we propose a new approach called Parametric Constant Modulus Algorithm (PCMA) for demodulating two co-channel signals, each of which are frequency modulated. The algorithm is of interest in Cellular Communications and military/intelligence applications. Our method is based on minimizing a constan-modulus-error (CME) criterion. It is di erent from the well-known Constan Modulus Algorithm (CMA). Firstly, the CMA is traditionally employed for channel equalization problems. Secondly, the CMA minimizes the CME over the parameters of a transversal lter. On the other hand, our procedure extends te CMA concept to the co-channel signal separtation problem: it uses a parametric model ofr the underlyiing signals and minimizes the CME over the parameters of the signal model, hence the name Parametric CMA (PCMA). [Rao ICASSP 96] A. Rao, R. Kumaresan,\Separation of Cochannel Signals Using The Parametric Constant Modulus Algorithm.," in Proc. International Conference on Acoustics, Speech and Signal Processing (Atlanta, GA), pp. 2686, May 7-9, 1996. Abstract: In an earlier paper, we had proposed a new algorithm, called the Parametric Constant Modulus Algorithm (PCMA) to demodulate noiseless, synthetic as well as real-world, cochannel FM signals. Here, we extend the PCMA to develop two new procedures. The rst one addresses the problem of resolving two closely spaced noisy sinewaves (from a short data record). The second algorithm is applied to estimate phase parameters of two superimposed noisy chirp signals; both algorithm's performances are compared with the Cramer-Rao bounds. The techniques described are of interest in wireless/military communications in the context of carrier frequency and doppler estimation. [Ready ICASSP 90] M.J. Ready, R.P. Gooch,\Blind equalization based on radius directed adaptation," in Proc. International Conference on Acoustics, Speech and Signal Processing (Albuquerque, NM), pp. 1699-1702, April 1990. Abstract: A blind equalization algorithm, termed radius directed equalization (RDE), for quadrature amplitude modulation (QAM) signals based on the known modulus of the constellation symbol radii is described. For example, 16 QAM has three radii and 32 QAM has ve radii. The algorithm uses the error between the equalizer output modulus and the nearest symbol radius to update the equalizer weights. The RDE algorithm provides faster convergence than the constant modulus algorithm (CMA) for QAM signals and is independent of 62 the carrier o set. The algorithm is described in the context of blind carrier and baud clock recovery schemes (6 Refs.) [Rude ICASSP 89] M.J. Rude and L.J Gri ths,\Incorporation of linear constraints into the constant modulus algorithm," in Proc. International Conference on Acoustics, Speech and Signal Processing (Glasgow, UK), pp. 968-71, May 23-26, 1989. Abstract: A linearly constrained version of the constant modulus algorithm (CMA) is developed. It is based on the decomposition property of the generalized sidelobe canceller (GSC) lter structure. When they can be speci ed, linear constraints are an e ective means of incorporating a priori information into the constant modulus algorithm. Exploitation of this knowledge makes the linearly constrained constant modulus (LCCM) approach much less vulnerable to interfering constant-envelope signals, which can cause cancellation of the signal of interest. This is demonstrated in a simulation of the LCCM algorithm in an adaptive array application that includes multipath and constant-modulus interference (5 Refs.) [Rude EUSIPCO 90] M.J. Rude, L.J. Gri ths,\A linearly constrained adaptive algorithm for constant modulus signal processing," in Proc. European Signal Processing Conference (Barcelona, Spain), pp. 237-240, September 1990. Abstract: A new version of the linearly constrained constant modulus (LCCM) adaptive algorithm is developed that does not require a priori knowledge of the power of the signal of interest (SOI). The existing LCCM algorithm uses a xed estimate of the SOI power as a parameter in its adaptive recursion. Errors in the estimate of this parameter can adversely a ect performance. The new LCCM algorithm overcomes this problem by treating this parameter as an adaptive variable rather than a xed parameter. Simulations demonstrate the e ectiveness of this approach (6 Refs.) [Rude ASIL 90] M.J. Rude, L.J. Gri ths,\An untrained, fractionally-spaced equalizer for cochannel interference environments," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 468-472, November 1990. Abstract: A fractionally spaced adaptive equalizer is developed based on the linearly constrained constant-modulus (LCCM) algorithm. This equalizer exploits prior knowledge of synchronization, sampling strategy and pulse shape to prevent capture of the constant-modulus (CM) algorithm by narrowband, constant envelope interferers. The prior knowledge takes the form of a single linear constraint that is a replica of the received pulse. Simulations show that this approach greatly reduces the vulnerability of CMA to strong constant envelope interferers and yields a set of equalizer tap values that can be successfully used as initial conditions for follow-on decision feedback adaptation (8 Refs.) [Rude ASIL 91] M.J. Rude, L.J. Gri ths,\Sensitivity of the linearly-constrained constant-modulus cost function," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 984-988, November 1991. Abstract: The authors present results from the application of the linearly constrained constantmodulus (LCCM) adaptive algorithm to a perturbed array environment. Assuming a narrowband signal model, analytic expressions are derived for the feasible gain values on the signal in the presence of general array perturbations and ambient white noise. The feasible gain values are found to be the roots of a cubic equation that is parameterized by the number of array elements, the signal-to-noise ratio, the modulus factor, the kurtosis of the signal of interest, and the angle between the true signal steering vector, and the perturbed steering vector. A 63 principal result of the analysis is that LCCM is robust in the presence of array perturbations but is sensitive to the modulus factor value. A simulation is included that compares the sensitivity to perturbations of the techniques of linearly constrained minimum power, unconstrained constant-modulus, and LCCM. Experimental results are presented on a four-element square array with half-wavelength spacing which is used to receive a QPSK signal arriving from a presumably known direction (5 Refs.) [Rude CEE 92] M.J. Rude,\A robust algorithm for untrained adaptive signal processing,"Computers & Electrical Engineering, vol. 18 no.3-4, pp. 205-216, May-June 1992. Abstract: A new adaptive signal processing technique is described that does not require a training or reference signal for adaptation. The approach taken is to minimize variations in the complex envelope of an adaptive processor output while subjecting the processor coe cients to a set of linear constraints. This linearly-constrained constant-modulus (LCCM) method is motivated by the drawbacks associated with two existing untrained algorithms in certain signal environments. The problems speci cally addressed by LCCM are the signal cancellation problem of linearly-constrained power minimization (LCPM) and the signal ambiguity problem of the constant-modulus algorithm (CMA). A complete adaptive implementation of the LCCM algorithm is presented including its stability and convergence properties. In addition, adaptive performance of LCCM is demonstrated in a comparison with CMA and LCPM (13 Refs.) [Sato TCOM 75] Y. Sato,\A method of self-recovering equalization for multilevel amplitudemodulations systems,"IEEE Transactions on Communications, vol. ?, pp. 679-682, June 1975. Abstract: Presents a self-recovering equalization algorithm, which is employed in multilevel amplitude-modulated data transmission. Such a self-recovering equalizer has been required when TDM voice or picturephone PCM signals must by transmitted over an existing FDM transmission channel. The present self-recovering equalizer is quite simple, as is a conventional binary equalizer. The convergence processes of the present self-recovering equalizer are shown by computer simulation. Some theoretical considerations on this convergence process are also added (4 Refs.) [Sato IEICE 94] Y. Sato,\Blind equalization and blind sequence estimation,"IEICE Transactions on Communications, vol. E77B, no. 5, pp. 545-556, 1994. Abstract: The joint estimation of two unknowns, i.e. system and input sequence, is overviewed in two methodologies of equalization and identi cation. Statistical approaches such as optimizing the ensemble average of the cost function at the equalizer output have been widely researched. One is based on the principle of distribution matching that total system must be transparent when the equalizer output has the same distribution as the transmitted sequence. Several generalizations for the cost function to measure mismatching between distributions have been proposed. The other approach applies the higher order statistics like polyspectrum or cumulant, which possesses the entire information of the system. For example, the total response can be evaluated by the polyspectrum measured at equalizer output, and by zeroforcing both side of the response tail the time dependency in the equalizer output can be eliminated. This is based on the second principle that IID simultaneously at input and at output requires a transparent system. The recent progress of digital mobile communication gives an incentive to a new approach in the Viterbi algorithm. The Viterbi algorithm coupled with blind channel identi cation can be established under a nite alphabet of the transmitted symbols. In the blind algorithm, length of the candidate sequence, which decides the number 64 of trellis states, should be de ned as long enough to estimate the current channel response. The channel impairments in mobile communication, null spectrum and rapid time-variance, are solved by fast estimation techniques, for example by Kalman lters or by direct solving the short time least squared error equations. The question of what algorithm has the fastest tracking ability is discussed from algebraic view points (40 Refs.) [Satorius ASIL 88] E.H. Satorius, S. Krishnan, X. Yu, L.J. Gri ths, I.S. Reed, T. Truong,\Suppression of narrowband interference via single channel adaptive preprocessing," in Proc. Conference Record. Twenty-Second Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 270-3 vol.1, 31 Oct.-2 Nov. 1988. Abstract: The authors discuss several promising temporal processing techniques for suppressing narrowband, cochannel interference: (1) adaptive linear prediction (ALP) lters, (2) constant modulus adaptive (CMA) lters, and (3) cross-coupled phase-locked loops (CCPLLs). Both the ALP and the CMA lters consist of tapped delay lines with adaptable coe cients. The only fundamental di erence between these two approaches lies in the algorithm used to update the tapped delay line coe cients. The ALP algorithm minimizes output lter power subject to the constraint that one of the lter taps is unity. In doing so, the ALP lter suppresses the narrowband interference. The CMA algorithm minimizes a more complicated error function of the lter output such that the output amplitude remains as close to unity as possible. The CMA algorithm thus restores the constant modulus property to the desired signal, thereby e ectively suppressing the interference. In contrast to these two approaches, the CCPLL suppresses interference by incorporating a pair of phase-locked loops, one to track the desired signal component and a secondary loop to track the interference, thereby providing a coherent phase reference for canceling the interference. The authors also consider hybrid architectures which combine the di erent techniques in novel ways (9 Refs.) [Satorius ELET 92] E. Satorius, J. Mulligan,\Minimum entropy deconvolution and blind equalization,"Electronics Letters, vol. 28, no. 16, pp. 1534-1535, 1992. Abstract: Relationships between minimum entropy deconvolution developed primarily for geophysics applications, and blind equalisation are pointed out. It is seen that a large class of existing blind equalisation algorithms are directly related to the scale-invariant cost functions used in minimum entropy deconvolution. Thus the extensive analyses of these cost functions can be directly applied to blind equalisation, including the important asymptotic results of Donoho (1981) (7 Refs.) [Schell TSP 95] S.V. Schell, W.A. Gardner,\Programmable canonical correlation-analysis{a fexible framework for blind adaptive spatial ltering,"IEEE Transactions on Signal Processing, vol. 43, no. 12, pp. 2898-2908, 1995. Abstract: Relationships between minimum entropy deconvolution developed primarily for geophysics applications, and blind equalisation are pointed out. It is seen that a large class of existing blind equalisation algorithms are directly related to the scale-invariant cost functions used in minimum entropy deconvolution. Thus the extensive analyses of these cost functions can be directly applied to blind equalisation, including the important asymptotic results of Donoho (1981) (7 Refs.) [Schirtzinger ICASSP 95] T. Schirtzinger, X. Li and W.K. Jenkins,\A comparison of three algorithms for blind equalization based on the constant modulus error criterion," in Proc. International Conference on Acoustics, Speech, and Signal Processing (Detroit, MI, USA), pp. 1049-52, 9-12 May 1995. 65 Abstract: Three constant modulus algorithms (CMA), the fast quasiNewton CMA, the transform domain CMA, and the genetic search based CMA are proposed in this paper. The performances of these three algorithms are compared with each other via computer simulation. It is shown that the fast quasi-Newton CMA and the transform domain CMA achieve much faster convergence rate than the constant modulus algorithm based on the LMS algorithm. This fact shows that the whitening technique is not only useful but also necessary for the CMA (9 Refs.) [Schirtzinger ISCS 95] T.A. Schirtzinger, W.K. Jenkins,\Designing adaptive equalizers based on the constant modulus error criterion," in Proc. 1995 IEEE Symposium on Circuits and Systems (Seattle, WA), pp. 1094-7 vol.2, 28 April-3 May 1995. Abstract: This paper presents two new design techniques for adaptive equalizers based on a constant modulus error criterion: (1) the fast quasi-Newton CMA and (2) the conjugate gradient CMA. The algorithms for these new equalization techniques are summarized and their performance is demonstrated through computer simulation (5 Refs.) [Sethares ICASSP 89] W. Sethares, et al.,\Approaches to blind equalization of signals with multiple modulus," in Proc. International Conference on Acoustics, Speech and Signal Processing (Glasgow, UK), pp. 972-5, May 23-26, 1989. Abstract: The constant modulus algorithm (CMA) and decision-directed (DD) equalizer are two ways to approach blind equalization of signals that are known to lie on a circle of xed radius, but where speci c values at any given time are unknown. In m-ary quadrature amplitude modulation, the signals lie on n circles of known radius. The authors present two possible approaches to the n-modulus problem, both in the spirit of feature reconstruction algorithms. The multiple-modulus algorithm uses a straightforward generalization of the CMA cost function to derive its update, whereas the decision-adjusted-modulus algorithm is a hybrid of the CMA and DD approaches (7 Refs.) [Sethares TSP 91] W.A. Sethares,\Adaptive algorithms with nonlinear data and error functions,"IEEE Transactions on Communications, vol. 40 no. 9, pp. 2199-2206, September 1992. (Filed in BERG library.) Abstract: The tools of nonlinear system theory are used to examine several common nonlinear variants of the LMS algorithm and derive a persistence of excitation criterion for local exponential stability. The condition is tight when the inputs are periodic, and a generic counterexample is demonstrated which gives (local) instability for a large class of such nonlinear versions of LMS, speci cally, those which utilize a nonlinear data function. The presence of a nonlinear error function is found to be relatively benign in that it does not a ect the stability of the error system. Rather, it de nes the cost function the algorithm tends to minimize. Speci c examples include the dead zone modi cation, the cubed data nonlinearity, the cubed error nonlinearity, the signed regressor algorithm, and a singlelayer version of the backpropagation algorithm (18 Refs.) [Shalvi TIT 90] O. Shalvi and E. Weinstein,\New criteria for blind deconvolution of nonminimum phase systems (channels),"IEEE Transactions on Information Theory, vol. 36, no. 2, pp. 312-321, March 1990. (Filed in BERG library.) 66 Three constant modulus algorithms (CMA), the fast quasiNewton CMA, the transform domain CMA, and the genetic search based CMA are proposed in this paper. The performances of these three algorithms are compared with each other via computer simulation. It is shown that the fast quasi-Newton CMA and the transform domain CMA achieve much faster convergence rate than the constant modulus algorithm based on the LMS algorithm. This fact shows that the whitening technique is not only useful but also necessary for the CMA (9 Refs.) [Schirtzinger ISCS 95] T.A. Schirtzinger, W.K. Jenkins,\Designing adaptive equalizers based on the constant modulus error criterion," in Proc. 1995 IEEE Symposium on Circuits and Systems (Seattle, WA), pp. 1094-7 vol.2, 28 April-3 May 1995. Abstract: This paper presents two new design techniques for adaptive equalizers based on a constant modulus error criterion: (1) the fast quasi-Newton CMA and (2) the conjugate gradient CMA. The algorithms for these new equalization techniques are summarized and their performance is demonstrated through computer simulation (5 Refs.) [Sethares ICASSP 89] W. Sethares, et al.,\Approaches to blind equalization of signals with multiple modulus," in Proc. International Conference on Acoustics, Speech and Signal Processing (Glasgow, UK), pp. 972-5, May 23-26, 1989. Abstract: The constant modulus algorithm (CMA) and decision-directed (DD) equalizer are two ways to approach blind equalization of signals that are known to lie on a circle of xed radius, but where speci c values at any given time are unknown. In m-ary quadrature amplitude modulation, the signals lie on n circles of known radius. The authors present two possible approaches to the n-modulus problem, both in the spirit of feature reconstruction algorithms. The multiple-modulus algorithm uses a straightforward generalization of the CMA cost function to derive its update, whereas the decision-adjusted-modulus algorithm is a hybrid of the CMA and DD approaches (7 Refs.) [Sethares TSP 91] W.A. Sethares,\Adaptive algorithms with nonlinear data and error functions,"IEEE Transactions on Communications, vol. 40 no. 9, pp. 2199-2206, September 1992. (Filed in BERG library.) Abstract: The tools of nonlinear system theory are used to examine several common nonlinear variants of the LMS algorithm and derive a persistence of excitation criterion for local exponential stability. The condition is tight when the inputs are periodic, and a generic counterexample is demonstrated which gives (local) instability for a large class of such nonlinear versions of LMS, speci cally, those which utilize a nonlinear data function. The presence of a nonlinear error function is found to be relatively benign in that it does not a ect the stability of the error system. Rather, it de nes the cost function the algorithm tends to minimize. Speci c examples include the dead zone modi cation, the cubed data nonlinearity, the cubed error nonlinearity, the signed regressor algorithm, and a singlelayer version of the backpropagation algorithm (18 Refs.) [Shalvi TIT 90] O. Shalvi and E. Weinstein,\New criteria for blind deconvolution of nonminimum phase systems (channels),"IEEE Transactions on Information Theory, vol. 36, no. 2, pp. 312-321, March 1990. (Filed in BERG library.) 66 [Shalvi TIT 94] O. Shalvi, E. Weinstein,\Maximum likelihood and lower bounds in systemidenti cation with non-gaissian inputs,"IEEE Transactions on Information Theory, vol. 40, no. 2, pp. 328-339, March 1994. Abstract: We consider the problem of estimating the parameters of an unknown discrete linear system driven by a sequence of independent identically distributed (i.i.d.) random variables whose probability density function (PDF) may be non-Gaussian. We assume a general system structure that may contain causal and noncausal poles and zeros. The parameters characterizing the input PDF may also be unknown. We derive an asymptotic expression for the Cramer-Rao lower bound, and show that it is the highest (worst) in the Gaussian case, indicating that the estimation accuracy can only be improved when the input PDF is non-Gaussian. It is further shown that the asymptotic error variance in estimating the system parameters is una ected by lack of knowledge of the PDF parameters, and vice verse. Computationally e cient gradient-based algorithms for nding the maximum likelihood estimate of the unknown system and PDF parameters, which incorporate backward ltering for the identi cation of noncausal parameters, are presented. The dual problem of blind deconvolution/equalization is considered, and asymptotically attainable lower bounds on the equalization performance are derived. These bounds imply that it is preferable to work with compact equalizer structures characterized by a small number of parameters as the attainable performance depend only on the total number of equalizer parameters (43 Refs.) [Shuangtian ICCT 92] Li Shuangtian; Wu Deben; Li Changli,\Further study and TMS320C30 realtime implementation of Kalman/Godard adaptive equalization algorithm (KGA) used for highspeed serial HF modem," in Proc. International Conference on Communication Technology (Beijing, China), pp. 13.01/1-4 vol.1, 16-18 Sept. 1992. Abstract: The authors focused on the equalization performance of the KGA for communication over 3 kHz bandwidth HF channels. They studied and modi ed the KGA. The study and modi cation have been veri ed through computer simulations. The results indicate that KGA modi cation yields superior performance to KGA, in which there are higher stability and faster convergence rate than KGA. Finally, the TMS320C30 hardware realization is reported. The real-time processing performance of the algorithm is evaluated in terms of the equalizing rate and the memory (words) occupied by program and data (9 Refs.) [Shukla IEE-I 92] P.K. Shukla, L.F. Turner,\Examination of an adaptive DFE and MLSE nearMLSE for fading multipath radio channels,"IEE Proceedings-I (Communications, Speech and Vison), vol. 139, no. 4, pp. 418-428, 1992. Abstract: The authors examine the performance over an HF channel of two adaptive equalisers; the decision feedback equaliser (DFE) and the maximum likelihood sequence estimator (MLSE), or an approximation to it (near-MLSE). It is shown that the superiority of the more complex MLSE/near-MLSE over the DFE is not great, and that this advantage is only due to error propagation in the DFE. In the absence of any special arrangement by which the receiver can re-align itself (e.g. periodic training), 16-QAM appears to be the highest feasible signal constellation size. With the periodic insertion of known symbols into the data stream, it is shown that performance can be maintained near the level for which decision errors do not a ect the tracking algorithm. With the facility of a request-for-training link to the transmitter, it is shown that performance can be signi cantly improved for all constellation sizes, with only a small loss in the useful data rate (38 Refs.) [Shynk ISCS 90a] J.J. Shynk, C.K. Chan, M.R. Petraglia,\Blind adaptive ltering in the frequency domain," in Proc. IEEE International Symposium on Circuits and Systems (New Orleans, LA), 67 pp. 275-278, May 1990. Abstract: E cient block implementations which are based on frequencydomain techniques of the constant modulus and the P-vector least-mean-square blind algorithms are described. These realizations have much less computational complexity than nonblock time-domain methods, and they can have improved convergence properties. Although several implementations are possible, including lter-bank and multirate structures, only linear convolution methods are used. A frequency-domain realization of a T/2-spaced constant modulus algorithm equalizer is examined, and computer simulations are presented (16 Refs.) [Shynk ISCS 90b] J.J. Shynk, C.K. Chan,\Error surfaces of the constant modulus algorithm," in Proc. IEEE International Symposium on Circuits and Systems (New Orleans, LA), pp. 13351338, May 1990. Abstract: The authors derive closed-form expressions for the performance function of the constant modulus algorithm (CMA) that are based on two di erent Gaussian assumptions, and they present examples of the corresponding error surfaces for a binary transmitted signal. In one case, they assume that the received signal of the equalizer is unconditionally Gaussian with zero mean. This approach leads to an in nity of stationary points, each of which corresponds to a minimum of the error surface and which depend only on the correlation properties of the received signal. In the second case, the authors assume that the received signal conditioned on one of the transmitted symbols is Gaussian with nonzero mean. The resulting stationary points are not so easily characterized, and they depend on the channel impulse response as well as the correlation of the received signal (10 Refs.) [Shynk ICASSP 90] J.J. Shynk, C.K. Chan,\A comparative analysis of the stationary points of the constant modulus algorithm based on Gaussian assumptions," in Proc. International Conference on Acoustics, Speech and Signal Processing (Albuquerque, NM), pp. 1249-1252, April 1990. Abstract: A comparative analysis of the stationary points of the constant modulus algorithm (CMA) performance function that is based on two di erent Gaussian assumptions is presented. In one case, it is assumed that the received signal is unconditionally Gaussian. This approach leads to an in nity of solutions, all of which correspond to a minimum of the performance function. In the second case, it is assumed that the received signal conditioned in one of the transmitted symbols in Gaussian. This approach leads to several possible solutions that depend not only on the channel characteristics, but also on the symbol on which the conditioning is performed (8 Refs.) [Shynk SPIE 91] J.J. Shynk, R.P.Gooch, G. Krishnamurthy, C.K. Chan,\A comparitive performance study of several blind equalization algorithms,"The International Society for Optical Engineering, vol. San Diego, CA, pp. 102-117, July 1991. (Filed in BERG library.) Abstract: This paper examines the transient and steady-state characteristics of several Bussgang-type blind equalization algorithms. A combination of computer simulations and analysis is used to assess their relative performance. The computer simulations involve channel characteristics typical of those found in an urban multipath environment, and include the e ects of frequency o set. The equalizer structures considered comprise of a T/2 fractionallyspaced linear nite impulse response lter. The analysis of misadjustment is based on an approximate Gaussian model of the data (26 Refs.) 68 [Shynk TSP 93] J.J. Shynk, C.K. Chan,\Performance surfaces of the constant modulus algorithm based on a conditional Gaussian model,"IEEE Transactions on Signal Processing, vol. 41 no. 5, pp. 1965-1969, May 1993. (Filed in BERG library.) Abstract: A stochastic analysis of the performance surfaces for four versions of the constant modulus algorithm is presented. By conditioning on the transmitted data symbols, the equalizer output is modeled as a Gaussian process, from which closedform expressions of the performance functions are derived. The resulting conditional expectations are then evaluated according to the probability distribution of the conditioning symbols. This approach leads to analytical results that might otherwise be di cult to derive without the immediate conditioning step (10 Refs.) [Shynk ASIL 93] J.J. Shynk, R.P. Gooch,\Convergence properties of the multistage CMA adaptive beamformer," in Proc. Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 622-6 vol.1, 1-3 Nov. 1993. Abstract: The multistage CMA adaptive beamformer is capable of separating multiple narrowband sources without pilot or training signals. It is comprised of a cascade of CM (constant modulus) array subsections, each of which captures one of the signals impinging an the array. An adaptive signal canceller follows each CM array to remove captured signals from the input before processing by subsequent sections. Based an a stochastic analysis, we derive the steady-state convergence properties of the system, including its directionnding capabilities. For mutually uncorrelated sources and noise, it is shown that the canceller exactly removes a captured signal, thereby reducing the rank of the e ective array matrix of the next subsection by one (11 Refs.) [Shynk MIL 94] J.J. Shynk, R.P. Gooch,\Performance analysis of the multistage CMA adaptive beamformer," in Proc. 1994 IEEE MILCOM. Conference Record (Fort Monmouth, NJ), pp. 316-20 vol.2, 2-5 Oct. 1994. Abstract: The multistage CMA adaptive beamformer is a blind adaptive antenna system capable of recovering several narrowband cochannel signals. Each stage of the system captures one source (without using a training signal), removes it before processing by subsequent stages, and provides an estimate of its direction of arrival. This system would be useful in frequency reuse applications, such as cellular radio or personal communication networks, where cochannel interference is an important consideration. We present the steady-state convergence properties of the rst stage of the multistage CMA adaptive beamformer. Computer simulations are presented to verify the analytical results (12 Refs.) [Shynk TSP 96a] J.J. Shynk, A.V. Keerthi, A. Mathur,\Steady state analysis of the multistage constant modulus array,"IEEE Transactions on Signal Processing, vol. 44, no. 4, pp. 948-962, 1996. Abstract: The multistage constant modulus (CM) array is a cascade adaptive beamforming system that can recover several narrowband cochannel signals without training. We examine its steady-state properties at convergence using a stochastic analysis and computer simulations. Based on a Wiener model of convergence for the gradient adaptive algorithms, closed-form expressions are derived for the CM array and canceller weight vectors, as well as the e ective source direction vectors at all stages along the cascade system. The signal-capture and directionnding capabilities of the system are also discussed. Computer simulations for sta69 tionary and fading sources are presented to con rm the theoretical results and to illustrate the rapid convergence behavior of the adaptive algorithms (25 Refs.) [Shynk TSP 96b] J.J. Shynk, R.P. Gooch,\The constant modulus array for cochannel signal copy and direction fading,"IEEE Transactions on Signal Processing, vol. 44, no. 3, pp. 652-660, 1996. Abstract: The constant modulus (CM) array is a blind adaptive beamformer capable of recovering a narrowband signal among several cochannel sources without using a pilot or training signal. It is a conventional weight-and-sum adaptive beamformer whose weights are updated by the constant modulus algorithm. An adaptive signal canceller follows the beamformer to remove the captured signal from the array input and to provide an estimate of its direction vector. Based on a Wiener model, we investigate the steady-state properties of the CM array and the signal canceller. For mutually uncorrelated sources and noise, it is shown that the signal canceller exactly removes the source captured by the array. Thus, identical stages of the CM array and signal canceller may be used in a multistage system to recover several cochannel sources. Computer simulations are presented to verify the analytical results and to illustrate the transient behavior of the system (24 Refs.) [Slock ICASSP 94] D.T.M. Slock,\Blind Fractionally-Spaced Equalization, Perfect-Reconstruction Filter Banks and Multichannel Linear Prediction," in Proc. International Conference on Acoustics, Speech, and Signal Processing (Adelaide, SA, Australia), pp. 585-588, 19-22 April 1994. Abstract: Equalization for digital communications constitutes a very particular blind deconvolution problem in that the received signal is cyclostationary. Oversampling (OS) (w.r.t the symbol rate) of the cyclostationary received signal leads to a stationary vector-valued signal (polyphase representation (PR)). OS also leads to a fractionally spaced channel model and equalizer. In the PR, channel and equalizer can be considered as an analysis and synthesis bank. Zero forcing (ZF) equalization corresponds to a perfect reconstruction lter bank. We show that in the OS case FIR ZF equalizers exist for a FIR channel. In the PR, the multichannel linear prediction of the noiseless received signal becomes singular eventually, reminiscent of the single channel prediction of a sum of sinusoids. As a result, the channel can be identi ed from the received signal second order statistics by linear prediction in the noise free case, and by using the Pisarenco method when there is additive noise. In the given data case, Music (subspace) or ML techniques can be applied. Comments : A method to identify the channel with second order statistics by linear prediction method was proposed. A correspondence between the linear prediction coe cients and the null space of the covariance matrix (of the received signal) was established. Since the covariance matrix is estimated with nite length data, a better way to estimate the channel through sub space tting and deterministic maximum likelihood was also proposed. In all of his methods, the length of the equalizer should be at least as long as that of the channel. {lambo [Smith ASIL 84] J.O. Smith and B. Friedlander,\Extensions of the constant modulus algorithm," in Proc. Asilomar Conference on Circuits, Systems and Computers (Paci c Grove, CA), pp. 378-82, Nov. 5-7, 1984. Abstract: The constant modulus algorithm (CMA) computes and applies an adaptive channel equalizer for constant amplitude signals such as frequencyand phase-modulation, Several extensions to the CMA are presented, including IIR equalization, a real signal version having properties as good as the complex version, use of the Gauss-Newton method in place of gradient descent, interference rejection, and more (12 Refs.) 70 [Smith ICASSP 85] J.O. Smith and B. Freidlander,\Global convergence of the constant modulus algorithm," in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (Tampa, FL), pp. 1161-4, March 26-29, 1985. Abstract: Global convergence of the constant modulus algorithm (CMA) is proved for the case of a real channel when the model order is equal to or greater than that of the channel (the so-called model-complete case). The analysis is based on an exact fourth-order Taylor series representation of the cost function minimized asymptotically by the CMA (13 Refs.) [Stockham PROC 75] T. Stockham, T. Cannon, and R. Ingerbretsen,\Blind deconvolution through digital signal processing,"Proceedings of the IEEE, vol. 63, pp. 678-692, 1975. Abstract: Addresses the problem of deconvolving two signals when both are unknown. The authors call this problem blind deconvolution. The discussion develops two related solutions which can be applied through digital signal processing in certain practical cases. The case of reverberated and resonated sound forms the center of the development. The speci c problem of restoring old acoustic recordings provides an experimental test. The important e ects of noise and nonstationary signals lead to the detailed part of the presentation. In addition, the paper presents results for the case of images degraded by some common forms of blur (25 Refs.) [Swaminathan ICASSP 93] R. Swaminathan and J.K. Tugnait,\On improving the convergence of constant modulus algorithm adaptive lters," in Proc. International Conference on Acoustics, Speech, and Signal Processing (Minneapolis, MN, USA), pp. 340-3, 27-30 April 1993. Abstract: An FIR ( nite impulse response) communications channel is considered. Two computationally simple algorithms are proposed for globally convergent, linear estimation of the FIR channel parameters where the FIR model order is not necessarily known. The estimated channel impulse response is used to derive a reliable initialization for the CMA (constant modulus algorithm) adaptive lter. So initialized, the CMA lter avoids undesirable equilibria and saddle points. A simulation example is presented to illustrate this advantage. Previous approaches either propose di erent criteria leading to computationally much more expensive schemes, or propose complex modi cations without alleviating the above-mentioned shortcomings of the traditional approach (18 Refs.) [Swindlehurst TSP 95] A.L. Swindldehurst, S. Daas, J.K. Yang,\Analysis of a decision directed beamformer,"IEEE Transactions on Signal Processing, vol. 43, no. 12, pp. 2920-2927, 1995. Abstract: Studies a technique for using decision direction to extract digital signals from antenna array data. The algorithm alternates between 1) estimating and demodulating the received signals and 2) using the resulting bit decisions to regenerate the signal waveforms and recompute the beamformer weights. An analysis of the (asymptotic) symbol error rate performance of the algorithm for the case of M-ary PSK signals is included, along with several representative simulation examples (20 Refs.) [Takao IEE-F 92] K. Takao C.S. Boon,\Importance of the exclusion of the desired signal from the control of a generalized sidelobe canceller,"IEE Proc.-F Radar and Signal Processing, vol. 139, no. 4, pp. 265-272, 1992. Abstract: A generalised sidelobe canceller (GSC), which operates by intentionally excluding the desired signal, is proposed. By excluding the desired signal from the primary input, the convergence rate is greatly improved. The response of the system can further be accelerated by using a forgetting factor of small value, without serious loss in the output signal to interference plus noise ratio (SINR). The system is also robust, in the sense that it can satisfactorily protect 71 the desired signal from suppression in the presence of pointing errors and coherent interferences as well. An angular domain notch lter is incorporated into the conventional GSC to exclude the desired signal component (12 Refs.) [Takao ECJ 1993] K. Takao,\Overview of the theory of adaptive antennas,"Electronics and Communications in Japan, vol. 76, no.7, pp. 110-118, July 1993. Abstract: The problem of control methods as the basics of the adaptive antenna is discussed. First, the general concept of the guiding principle and control algorithm is introduced. Next, the relationships between the guiding principles and corresponding control algorithms are explained for the LMS method, H-A method, DCMP method, and CMA method. With respect to the rst three methods, the control characteristics of the optimization systems with feedback based on the steepest descent method are analyzed. Several factors governing the convergence speed are pointed out. The use of several algorithms intended to improve the convergence characteristics, where they came from and where they will develop are discussed (21 Refs.) [Takao IEICE 95] K. Takao and H. Matsuda,\The choice of the initial condition of CMA adaptive arrays,"IEICE Transactions on Communications, vol. E78-B, no.11, pp. 1474-9, Nov. 1995. Abstract: We analyze the convergence behavior of the CMA (constant modulus algorithm) adaptive array working under the steepest decent method, and investigate how to achieve the highest possible output SINR (signal to interference plus noise ratio). In multipath radio environments, CMA sometimes suppresses the desired signal (stronger one) and selects to receive the interference (the weaker one) resulting in a low output SINR. This is one of the problems met in an optimization system under a nonlinear control equation where there are two or more local minima of the cost function and the nal state depends on the initial condition. The study can be done only numerically by starting from various initial values. In our analysis of the CMA adaptive array in multipath radio environments, we assume that there are two waves which are radiated from the same source and try to nd out what conditions may a ect the convergence behavior. In this process, we discover the e ect of a factor neglected in previous papers and revise the initial condition to guarantee the reception of the desirable wave. In conclusion, we propose the prescription of the initial weights of the array elements as well as the choice of preferable arrays (7 Refs.) [Tanaka IEICE 95] T. Tanaka, R. Miura, I. Chiba and Y. Karasawa,\An ASIC implementation scheme to realize a beam space CMA adaptive array antenna,"IEICE Transactions on Communications, vol. E78-B, no.11, pp. 1467-73, Nov. 1995. Abstract: We demonstrate the feasibility of a beam space CMA (constant modulus algorithm) adaptive array antenna by implementing a digital signal processor (DSP) in ASICs using eld programmable gate arrays (FPGA). The DSP can synthesize 16 multibeams and eliminate interference signals by CMA adaptive processing. The whole function was implemented in about 127000 equivalent gates. Simple experimental results in a radio anechoic chamber have con rmed the basic function of the BSCMA adaptive array antenna (12 Refs.) [Tanaka ISAP 95] T. Tanaka, R. Miura, I. Chiba and Y. Karasawa,\ASIC implementation of DSP for beam space CMA adaptive array antenna in mobile communications," in Proc. IEEE Antennas and Propagation Society International Symposium (Newport Beach, CA), pp. 98101 vol.1, 18-23 June 1995. Abstract: A digital signal processor (DSP) for a beam space constant modulus algorithm (BSCMA) adaptive array antenna has been implemented using 10 eld programmable gate 72 arrays. The DSP provides multibeam synthesis, beam selection, vector rotation, and CMA adaptive processing for a BSCMA array antenna that has 16 element antennas. The DSP has 16 processors, corresponding to 16 multibeams, and 2 processors for vector rotation and CMA calculations (2 Refs.) [Tong ASIL 92] L. Tong,\Blind equalization of fading channels," in Proc. The Twenty-Sixth Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 324-8 vol.1, 26-28 Oct. 1992. Abstract: Two blind equalization schemes using an antenna array are proposed for multipath fading channels. The methods exploit second-order statistics and spatial diversity to provide fast convergence rate and better performance in low SNRs. A special orthogonalization technique is used in conjunction with the Viterbi algorithm to achieve optimal sequence detection without channel identi cation. The proposed approach also has a special structure particularly suitable for time varying channels (16 Refs.) [Tong TIT 94] L. Tong, G. Xu, and T. Kailath,\Blind channel identi cation based on second-order statistics: A time domain approach,"IEEE Transactions on Information Theory, vol. 40, no. 2, pp. 340-349, March 1994. (Filed in BERG library.) Abstract: A new blind channel identi cation and equalization mehtod is proposed that exploits the cyclostationarity of oversampled communication signals to achieve identi cation and equalization of possibly nonminimum phase (multipath) channels without using training signals. Unlike most adaptive blind equalization methods for which the convergence properties are often problematic, the channel estimation algorithm proposed here is asymptocially exact. Moreover, since it is based on second-order statistics, the new approach may achieve equalization with fewer symbols than most techniques based only on higher-order statistics. Simulations have demonstrated promising performance of the proposed alogrithm for the blind equalization of a three-ray multipath channel. [Tong TIT 95] L. Tong, G. Xu, B. Hassibi, and T. Kailath,\Blind channel identi cation based on second-order statistics: A frequency domain approach,"IEEE Transactions on Information Theory, vol. 41, pp. 329-334, January 1995. (Filed in BERG library.) Comments : Blind, linear, possibly non-minimum-phase channel identi cation using fractionally sampled received second-order-statistics. The candidate matrix for decomposition consists of weighted, shifted copies of vectors which are comprised of elements taken from subchannel contributions to the received autocorrelation sequence. SVD of this matrix yields the channel estimate. Algorithm assumes noise power is known (or estimated) at the receiver. This paper reads more easily than the time-domain-approach paper. {tje Abstract: In this communication, necessary and su cient conditions are presented for the unique blind identi cation of possibly nonminimum phase channels driven by cyclostationary processes. Using a frequency domain formulation, it is rst shown that a channel can be identi ed by the second-order statistics of the observation if and only if the channel transfer function does not have special uniformly spaced zeros. This condition leads to several necessary and su cient conditions on the observation spectra and the channel impulse response. Based on the frequency-domain formulation, a new identi cation algorithm is proposed (13 Refs.) 73 [Tong TCOM 95] L. Tong,\Blind sequence estimation,"IEEE Transactions on Communications, vol. 43, no. 12, pp. 2986-2994, 1995. Abstract: Estimating the data sequence from the received signal without knowing the transmission channel is referred to as blind sequence estimation. A new blind sequence estimation scheme is proposed by exploiting the second order statistical properties of the source and the algebraic structure of the data sequence. An optimal source (deterministic) correlation estimator and the Viterbi algorithm are used to achieve blind sequence estimation. The proposed approach also has a special structure particularly attractive for time varying channels (26 Refs.) [Tong SPL 96] L. Tong and H.H. Zeng,\Channel-Sur ng Re-initialization for the Constant Modulus Algorithm,"Signal Processing Letters (Submitted to...), vol. ???, pp. ???, ???. (Filed in BERG library.) Abstract: The proper initialization of the constant modulus algorithm (CMA) is a critical problem that has not been solved. This paper presents a new re-initialization scheme for CMA. The proposed scheme searches in the channel space for the constant modulus equalizer with minimum mean square error by exploiting the connection between the constant modulus and the minimum mean square (MMSE) equalizers, and the structures of the MMSE equalizer. [Tong SPL 97] L. Tong and H. Zeng,\Channel sur ng re-initialization for the constant modulus algorithm,"Signal Processing Letters, vol. 4, no. 3, pp. 85-87, Mar. 1997. Abstract: The proper initialization of the constant modulus algorithm (CMA) is a critical problem that has not been solved. This paper presents a new reinitialization scheme for CMA. The proposed scheme searches in the channel space for the constant modulus equalizer with minimum mean-square error by exploiting the structure of the minimum mean square error (MMSE) equalizer and connections between the constant modulus and the MMSE equalizers (6 Refs.) [Touzni ICASSP 96] A. Touzni, I. Fijalkow, J.R. Treichler,\Fractionally-Spaced CMA Under Channel Noise," in Proc. International Conference on Acoustics, Speech and Signal Processing (Atlanta, GA), pp. 2674, May 7-9, 1996. Abstract: In the noise-free case, Fractionally-Spaced Equalization by Constant Modulus Algorithm (FSE-CMA) was shown to be robust to the channel identi ability condition. In this paper, the e ect of additive channel noise on FSE-CMA steady-state behavior is analyzed in terms of perturbation of the equalizer convergence settings with respect to the convergence settings of the noise-free case. The loss of perfornance is characterized by the additional residual mean square error induced by the noise. In this contribution we will show that, in noisy conditions, FSE-CMA exhibits an interesting \smoothing e ect" that leads to a trade-o between Zero Forcing (ZF) equalization and output Noise Enhancement (NE). We show that the main loss is due to NE and not to residual intersymbol interference. [Touzni SPW 96] A. Touzni, I. Fijalkow and J.R. Treichler,\Robustness of fractionally-spaced equalization by CMA to lack of channel disparity," in Proc. IEEE Signal Processing Workshop on Statistical Signal and Array Processing (Corfu, Greece), pp. 144-147, June 24-26, 1996. (Filed in BERG library.) Abstract: We study the Fractionally-spaced equalization by CMA (FSE-CMA) robustness to channel noise and lack of disparity. When there is lack of disparity, we will show that, whereas other recent technics as linear prediction or subspace like methods fail, FSE-CMA can still 74 equalize. In particular for long enough equalizers FSE-CMA exhibits a \smoothing e ect" which leads to an interesting trade-o between achieving zero-forcing equalization and noise enhancement. [Touzni EUSIPCO 96] A. Touzni and I. Fijalkow,\Does fractionally-spaced CMA converge faster than LMS?," in Proc. EUSIPCO 96 (Trieste, Italy), pp. 1227-1230, Sep. 1996. Abstract: This paper addresses the convergence rate study of the Fractionally Spaced Equalizer updated by Constant Modulus Algorithm (FSE-CMA). By analyzing the average algorithm behavior we copmare the FSE-CMA to the FSE-LMS. Although the FSE-CMA is based on a fourth order statistics criterion, we will show for constant modulus input that the algorithm has the amazing property to converge locally twice as fast as FSE-LMS (which requires a training sequence). Furthermore, we will show that the global FSE-CMA transient behavior convergence is accomplished in two steps. [Touzni SPAWC 97] A. Touzni and I. Fijalkow,\Channel robust blind fractionally-spaced equalization," in Proc. IEEE Signal Processing Advances in Wireless Communications (Paris, France), pp. 33-36, Apr. 1997. Abstract: We present a new adaptive robust equalization method aimed to overcome the issue of lack of disparity of most recent spatio-temporal diversity based equalization algorithms, and to pick the equalizer minimizing input/output errors in the presence of additive noise. It combines fractionally-spaced equalization by constant modulus algorithm (FSE-CMA) robustness to lack of disparity and delay diversity to nd several equalizer settings in di erent basins of attraction of the FSE-CM cost-function. We prove that the algorithm allows one to choose the best equalizer's output when there is disparity. Moreover, in the worst possible channel case, i.e. when there is such a lack of disparity that even the FSE-CMA fails, some of the equalizers may escape from converging to undesired settings and come very close to MMSE (8 Refs.) [Touzni ICASSP 97] A. Touzni and I. Fijalkow,\Robustness of blind fractionally-spaced identi cation/equalization to loss of channel disparity," in Proc. International Conference on Acoustics, Speech and Signal Processing (Munich, Germany), pp. 3937-3940, Apr. 1997. Abstract: We address the comparison of sub-space (SS), linear prediction (LP) and constant modulus (CM) identi cation/equalization algorithms in terms of robustness to loss of fractionallyspaced channel disparity. We show that SS procedure leads to an inconsistent channel estimation. Investigating a leftinverse channel estimation, we show that LP results in the estimation of the so-called minimum-phase multivariate channel factorization. We show that the CM criterion still perform reasonable channel estimation, even if proper algorithm initialization is still a critical subject (9 Refs.) [Treichler TASSP 83] J.R. Treichler and B.G. Agee,\A new approach to multipath correction of constant modulus signals,"IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-31, no.2, pp. 459-72, April 1983. (Filed in BERG library.) Abstract: An adaptive digital ltering algorithm that can compensate for both frequencyselective multipath and interference on constant envelope modulated signals is presented. The method exploits the fact that multipath reception and various interference sources generate incidental amplitude modulation on the received signal. A class of so-called constant modulus performance functions is developed which sense this AM term but are insensitive to the angle modulation. Simple adaptive algorithms for nite-impulseresponse (FIR) digital lters are 75 developed which employ a gradient search of the performance function. One of the resulting algorithms is simulated for the example of an FM signal degraded by specular multipath propagation. Substantial improvements in noise power ratio (NPR) are observed (e.g., 25 dB) with moderately rapid convergence time. These results are then extended to include tonal interference on a FM signal and intersymbol interference on a QPSK data signal (29 Refs.) [Treichler ICASSP 84] J.R. Treichler, M.G. Larimore,\A real-arithmetic implementation of the constant modulus algorithm," in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (San Diego, CA), pp. 1-4, March 19-21, 1984. Abstract: A version of the constant modulus algorithm (CMA) is presented which uses real signals and real arithmetic. The algorithm is developed in an evolutionary form from the version based on complex arithmetic. A key technical result is that certain error terms can be highly simpli ed due to the smoothing intrinsic to a gradient search algorithm. Computer simulations indicate excellent agreement with the analytically predicted performance (3 Refs.) [Treicher ICASSP 85] J.R. Treicher and M.G. Larimore,\Convergence rates for the constant modulus algorithm with sinusoidal inputs," in Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (Tampa, FL), pp. 1157-60, March 26-29, 1985. Abstract: A class of adaptive ltering algorithms has recently been developed that appears to be very useful for removing multipath and interference from communication signals. Practical application of this class of algorithms requires that the approximate convergence time of the algorithms be predicted, given signal properties and lters parameters. A set of formulas that predict the lter convergence rate for one or more sinusoidal inputs is developed here. Simulation work veri es the accuracy of the formulas. The formulas are shown to provide an interesting contrast with the performance of an LMS-based adaptive lter (5 Refs.) [Treichler TASSP 85a] J.R. Treichler and M.G. Larimore,\The tone capture properties of CMAbased interference suppressors,"IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-33, no.4, pp. 946-58, Aug. 1985. (Filed in BERG library.) Abstract: Examines a problem that arises when using the constant modulus algorithm (CMA), an adaptive ltering technique for correcting multipath and interference-induced degradations in constant envelope waveforms such as FM and QPSK signals, to suppress narrowband interference. If both the interferer and the signal have constant envelope and are spectrally nonoverlapping, then it is possible to nd two di erent lter solutions, one which suppresses the interferer and another which 'captures' the interferer and suppresses the desired signal. How 'capture' can occur and how it may be prevented are examined. This problem is studied by characterizing the algorithm's behavior to an input consisting of only two sinusoids. The results are then broadened to include multiple input tones and signals with nonzero bandwidth (4 Refs.) [Treichler TASSP 85b] J.R. Treichler, M.G. Larimore,\New processing techniques based on the constant modulus adaptive algorithm,"IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-33, no.2, pp. 420-31, April 1985. (Filed in BERG library.) Abstract: The authors present three extensions of the constant modulus algorithm (CMA), introduced in an earlier paper (J.R. Treichler and B.G. Agee, 1983) as a means of correcting degradation in constant envelope waveforms. As originally formulated, the CMA uses an FIR 76 lter with complex coe cients and accepts complex (quadrature) input data. In this paper, a real input, real coe cient version of the algorithm is shown to perform arbitrarily closed to the fully complex version. The algorithm is then extended for the enhancement of signals having a nonconstant but known envelope, a case that might arise in data signals with pulse shaping. Lastly, a multichannel version of CMA, wherein several observations are linearly combined, is presented for joint adaptation of multiple lters. This approach can be used, for example, as a means of spatial or polarization 'beamsteering' to reject additive interferers and compensate for channel-induced polarization rotation (10 Refs.) [Treichler MIL 86] J.R. Treichler, M.G. Larimore,\CMA-based techniques for adaptive interference rejection," in Proc. IEEE Military Communications Conference (Monterey, CA), pp. 3, Oct. 5-9, 1986. Abstract: Several recent developments in the area of applying the constant modulus algorithm (CMA) to the problem of rejecting interference from a signal deemed to be of interest are surveyed. This algorithm was originally developed for removing the dispersive e ects of multipath propagation from constant envelope communications signals, but has proven useful in a variety of other applications, such as suppressing additive narrow band interference, combining diversity receiving channels to reject crosspolarized signal components, and steering nulls in a sensor array. The algorithm itself and its original application are reviewed. From this departure point, several applications, including those listed, are examined and performance results cited. Some of these applications have encountered and stimulated the development of modi ed algorithms, of which several are described, including one using real-valued data and another which allows fast convergence rather than the slower convergence characteristics of the gradient-search-based algorithms (15 Refs.) [Treichler ICASSP 89] J.R. Treichler, S.L Wood, and M.G. Larimore,\Convergence rate limitations in certain frequency-domain adaptive lters," in Proc. International Conference on Acoustics, Speech and Signal Processing (Glasgow, UK), pp. 960-3, May 23-26, 1989. Abstract: It is shown that the transport delay present in the feedback path of a frequencydomain adaptive lter consisting of the combination of a CMA (constant modulus algorithm) with a transmultiplexer reduces the lter's maximum attainable convergence rate. An upper bound for the convergence rate is developed in terms of the amount of group delay and hardware pipeline delay present in the adaptive system under analysis. These limitations are shown to be fundamental in nature. They relate directly to the amount of spectral resolution desired in the adaptive lter and not to the particular nonlinear error function employed (5 Refs.) [Treichler ASIL 89] J.R. Treichler, S.L. Wood, M.G. Larimore,\Some dynamic properties of transmux-based adaptive lters," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 682-686, October 1989. Abstract: Two aspects of the dynamic behavior of a constant modulus algorithm (CMA) directed, transmux-based, frequency-domain adaptive lter used to suppress additive, narrowband interferers of disparate powers are examined. Analysis shows that the ability to powernormalize the bin-level adaptive gains o ers the promise of signi cant improvements in overall lter convergence rate. It is shown, however, that the transport delay present in the feedback path of the frequency-domain adaptive lter reduces the lter's maximum attainable convergence rate. An upper bound for the convergence rate was developed in terms of the amount of group delay and hardware pipeline delay present in the adaptive system under analysis. These limitations are shown to be fundamental in nature; they relate directly to the amount 77 of spectral resolution desired in the adaptive lter and not to the particular nonlinear error function used (7 Refs.) [Treichler ASIL 91] J.R. Treichler, V. Wol , C.R. Johnson, Jr.,\Observed misconvergence in the constant modulus adaptive algorithm," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 663-667, November 1991. Abstract: Practical application of the constant modulus algorithm (CMA) has demonstrated a number of circumstances in which CMA fails to converge, or, equally bad from a practical standpoint, converges to a solution which fails to equalize the input signal. The authors describe several situations in which misconvergence occurs, suggesting that a rmer analytical understanding is needed of the behavior of blind algorithms in the presence of cyclostationary and/or quasiperiodic, nonwhite inputs. While this analytical understanding is not yet established, the practical experience reported should be directly useful to those designing new digital communications systems. An example presented, is the use of CMA to equalize quadrature amplitude modulation (QAM) signals used to broadcast highde nition television signals (8 Refs.) [Treichler SPM 96] J.R. Treichler, I. Fijalkow, and C.R. Johnson, Jr.,\Fractionally-spaced equalizers:how long should they really be?,"IEEE Signal Processing Magazine, vol. 13, No. 3, pp. 65-81, May 1996. (Filed in BERG library.) Abstract: Modern digital transmission systems commonly use an adaptive equalizer as a key part of the receiver. The design of this equalizer is important since it determines the maximum quality attainable from the system, and represents a high fraction of the computation used to implement the demodulator. Recent analytical results o er a new way of looking at fractionally spaced equalizers and have some surprising practical implications. This article describes the data communications problem, the rationale for introducing fractionally spaced equalizers, the new results, and their implications. We then apply those results to actual transmission channels. [Treichler SPAWC 97] J.R. Treichler, L. Tong, I. Fijalkow, C.R. Johnson, Jr., and C.U. Berg,\On the current shape of FSE-CMA behavior theory," in Proc. IEEE Signal Processing Advances in Wireless Communications (Paris, France), pp. 105-108, Apr. 1997. Abstract: Recent work has begun to reveal fundamental similarities between the cost functions associated with adaptation of a fractionally-spaced equalizer via (i) LMS with training and (ii) CMA without training. The bulk of this paper describes recent research that relates the local minima of the FSE-CMA cost function to Wiener receiver parameterizations for various delays. Our sense is that these relationships can be exploited to extract a set of design guidelines from a suitably interpreted emerging FSE-CMA behavior theory (20 Refs.) [Tsakalides ASIL 95] P. Tsakalides and C.L. Nikias,\A new criterion for blind deconvolution of colored input signals," in Proc. Conference Record of The Twenty-Seventh Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA), pp. 746-50 vol.1, Nov. 1-3, 1993. Abstract: In this paper, a new criterion with memory nonlinearity is introduced for blind deconvolution problems when the input signals are colored. The basic idea is to make use of the autocorrelation of the input sequence as the only statistical knowledge about the data. An adaptive weight algorithm is presented and tested with simulation examples of signals of known autocorrelation function. It is shown that the optimum memory size is directly related 78 to the signi cant values of the autocorrelation function, and that the new algorithm converges faster than the Godard algorithm (7 Refs.) [Tsatsanis TCOM 96] M.K. Tsatsanis and G.B. Giannakis,\Equalization of rapidly fading channels{self-recovering methods,"IEEE transactions on Communications, vol. 44, pp. 619630, 1996. Abstract: Blind estimation of time-varying (TV), rapidly fading channels is addressed, using a basis expansion approach. Each TV coe cient is expanded onto a basis and the expansion parameters are estimated for subsequent use in Viterbi or decision-feedback equalizers. Blind estimation of the expansion parameters is accomplished using higher order statistics. Identiability of the channel is shown from second and fourth-order TV cumulants of the received signal. A cumulant matching criterion is adopted and instantaneous approximations are proposed in place of the nonstationary ensembles. Linear methods are also derived to initialize the nonlinear optimization procedure. Strong convergence of the proposed method is established. Finally, the method is tested on a simulated mobile radio channel with multipath (32 Refs.) [Tugnait SPIE 91] J.K. Tugnait,\Adaptive lters and blind equalizers for mixed phase channels," in Proc. The International Society for Optical Engineering (209-220), pp. 209-220, July 1991. Abstract: A new approach is proposed to recursive estimation of the parameters of nite and in nite impulse response nonGaussian signals assumed to be generated by driving a nitedimensional channel (system) by an IID non-Gaussian sequence. The signal model is allowed to be nonminimum phase, and hence applicable to blind channel equalization in data communication systems. The proposed recursive parameter estimator is shown to be globally convergent to the true model regardless of its initialization, and is based upon a two model decomposition approach: spectrally equivalent minimum phase system in cascade with an allpass system (37 Refs.) [Tugnait ICC 91] J.K. Tugnait,\Blind channel estimation and adaptive blind equalizer initialization," in Proc. International Conference on Communications (xxx), pp. 1388-1392, 1991. Abstract: The authors propose approaches to the problem of adaptive blind equalizer initialization and channel impulse response estimation for data communication systems, including telephone channels as well as digital radio channels. They propose two schemes for channel impulse response estimation that can be used for reliable initialization of the conventional decision-directed equalizers. Only batch (nonrecursive) methods are considered. The proposed methods yield (globally) optimal solutions (32 Refs.) [Tugnait IEE-I 94] J.K. Tugnait,\Globally convergent adaptive blind lters and blind equalisers for mixed-phase channels,"IEE Proceedings I Communications, vol. 141, no.6, pp. 390-395, Dec. 1994. Abstract: A new approach is proposed for recursive estimation of the parameters of niteimpulse response and in nite-impulseresponse nongaussian signals which are assumed to be generated by driving a nite-dimensional channel (system) by an IID nongaussian sequence. The problem is addressed in a blind setting, i.e. only the channel output is observed, not the input to it. The signal model is allowed to be nonminimum phase; hence the model is applicable to the problem of blind channel equalisation in data communication systems. The proposed recursive parameter estimator is shown to be globally convergent, i.e. the parameter estimator converges to the true model regardless of its initialisation. Therefore, the blind-equaliser design based on the estimated channel parameters is also globally convergent. The proposed parameter 79 estimator is based on a two-model decomposition approach: a spectrally equivalent minimumphase system in cascade with an allpass system. An illustrative computer-simulation example using a 16-QAM signal is presented where the proposed approach is compared with a Godard blind equaliser (20 Refs.) [Tugnait TCOM 94] J.K. Tugnait,\Blind estimation of digital-communication channel impulseresponse,"IEEE Transactions on Communications, vol. 42, no.. 2-4, pp. 1606-1616, 1994. Abstract: The article propose novel approaches to the problem of blind channel impulse response estimation for data communication systems, including telephone channels as well as digital radio channels. No training sequence is assumed to be available. Two novel schemes are proposed for channel impulse response estimation. Only batch (nonrecursive) methods are considered. The higher order cumulant statistics is exploited, in addition to the usual second-order statistics, of the data and of appropriately de ned error signals. The proposed methods yield (globally) optimal solutions. The estimated channel impulse response can be used for channel equalization, either for reliable ("open eye") initialization of the conventional decision-directed equalizers or as a channel estimator for a Viterbi algorithm based equalizer. Two illustrative examples one for a telephone channel and the other for a multipath channel, are given using an 8-level PAM signal (41 Refs.) [Tugnait ICC 94] J.K. Tugnait,\A parallel multimodel: CMA/Godard adaptive lter bank approach to fractionally-spaced blind adaptive equalization," in Proc. International Conference on Communications (New Orleans, LA), pp. 549-53 vol.1, 1-5 May 1994. Abstract: We consider the problem of blind equalization of digital communication channels using fractionally spaced samples. Fractionally sampled data are cyclo-stationary rather than stationary. The problem is cast into a mathematical framework of a vector stationary process with single input (information sequence) and multiple outputs, by using a timeseries representation of a cyclostationary process. This leads to number of subchannels equal to number of samples per symbol. Each subchannel is rst equalized independently using a baud-rate CMA/Godard (1980) blind equalizer. Several approaches are presented for selection and fusion of the equalized outputs of the various subchannels in order to achieve a performance that is much less sensitive to the symbol-timing-phase errors when compared with a band-rate equalizer. An illustrative simulation example 16-QAM (V22 source) signal is presented where e ect of symbol-timingphase o set is studied via computer simulations (9 Refs.) [Tugnait TCOM 95] J.K. Tugnait,\Blind equalization and estimation of digital communication FIR channels using cumulant matching,"IEEE Transactions on Communications, vol. 43, no. 2-4, pp. 1240-1245, 1995. Abstract: We consider the problem of blind estimation and equalization of digital communication FIR ( nite impulse response) channels. The channel parameters are estimated by nonlinear batch optimization of a quadratic cumulant matching criterion involving second and fourth order cumulants of the received data. A new algorithm is proposed for (asymptotically) globally convergent, linear estimation of the FIR channel parameters where the FIR model order is not necessarily known. The nonlinear cumulant matching algorithm is initialized by the linear parameter estimator. The estimated channel impulse response is then used to construct a linear equalizer. Two illustrative simulation examples using 4, 16 and 64-QAM signals are presented (11 Refs.) [Tugnait TIT 95] J.K. Tugnait,\On blind identi ablity of multipath channels using fractional sampling and 2nd-order cyclostaionarity statistics,"IEEE Transactions on Information Theory, vol. 80 41, no. 1, pp. 308-311, 1995. Abstract: The problem of blind identi ability of digital communication multipath channels using fractionally spaced samples is considered. Fractionally sampled data are cyclostationary rather than stationary. The problem is cast into a mathematical framework of parameter estimation for a vector stationary process with single input (information sequence) and multiple outputs, by using a time-series representation of a cyclostationary process. A necessary and su cient condition for channel identi ability from the correlation function of the vector stationary process is derived. This result provides an alternative but equivalent statement of an existing result. Using this result, it is shown that certain class of multipath channels cannot be identi ed from the second-order statistics irrespective of how the sampling rate is chosen (16 Refs.) [Tugnait ICASSP 95] J.K. Tugnait,\On fractionally-spaced blind adaptive equalization under symbol timing o sets using Godard and related equalizers," in Proc. International Conference on Acoustics, Speech, and Signal Processing (Detroit, MI), pp. 1976-1979 vol.3, 9-12 May 1995. Abstract: The problem of fractionally-spaced (FS) blind adaptive equalization under symboltiming-phase o sets is considered. It is well-known that in the case of trained (non-blind) equalizers, the performance of FS equalizers is independent of the timing-phase unlike that of baud-rate equalizers. Moreover, trained FS equalizers synthesize optimal lters in the MMSE sense, and hence are superior to baud-rate trained equalizers. These advantages of trained FS equalizers have not been shown to be true for blind equalizers, rather they have been simply assumed. The authors present a simulation example where such advantages do not materialize. Then they present a solution based upon a parallel, multimodel Godard adaptive lter bank approach which yields a performance almost invariant w.r.t. symboltiming-phase. An illustrative simulation example 16-QAM (V22 source) signal is presented where the e ect of symboltiming-phase o set is studied via computer simulations (10 Refs.) [Tugnait TSP 96] Tugnait, J.K.,\On fractionally spaced blind adaptive equalization under symbol timing o sets using Godard and related equalizers," in Proc. IEEE Transactions on Signal Processing (vol.44, no.7), pp. p. 1817-21, July 1996. (Filed in BERG library.) Abstract: The problem of fractionally spaced (FS) blind adaptive equalization under symboltiming-phase o sets is considered. In the case of trained (nonblind) equalizers, the performance of FS equalizers is known to be independent of the timing-phase unlike that of baud-rate equalizers. Moreover, trained FS equalizers synthesize optimal lters in the minimum meansquare error sense, and hence are superior to baud-rate trained equalizers. It is an open question whether such properties hold true for FS Godard blind equalizers. We present a simulation example where such advantages do not materialize. We also present a solution based upon a parallel multimodel Godard baud-rate lter bank approach which yields a performance almost invariant w.r.t. symbol-timing-phase and superior to that of baud-rate equalizers (7 Refs.) [Tugnait TCOM 96] J.K. Tugnait,\Blind equalization and estimation of FIR communications channels using fractional sampling,"IEEE Transactions on Communications, vol. 44, no. 3, pp. 324-336, 1996. Abstract: We consider the problem of blind estimation and equalization of digital communication nite impulse response (FIR) channels using fractionally spaced samples. Fractionally sampled data are cyclostationary rather than stationary. The problem is cast into a mathematical framework of parameter estimation for a vector stationary process with single input 81 (information sequence) and multiple outputs, by using a timeseries representation of a cyclostationary process. The channel parameters are estimated by rst estimating various subchannels using the secondand the fourth-order cumulant function of the received data, and then appropriately aligning and scaling them. The estimated channel impulse response is then used to construct a linear equalizer. Two illustrative simulation examples using fourand 16-QAM signals are presented where e ect of symbol-timing-phase o set is studied via simulations (32 Refs.) [Ueng ELET 95] F.B. Ueng, Y.T. Su,\Adaptive IIR blind algorithms,"Electronics Letters, vol. 31, no. 12, pp. 942-943, 1995. Abstract: In nite impulse response (IIR) ltering, when compared with nite impulse response (FIR) ltering, can result in a substantial computational saving and small mean-squared error (MSE). Two IIR blind algorithms based on the second and fourth order cumulants are presented. Simulation results indicate that the proposed IIR blind algorithms not only have faster convergence rates but also lower MSEs than their FIR counterparts (0 Refs.) [Ueng SAC 95] F.B. Ueng, Y.T. Su,\Adaptive blind equalization using 2nd-order and higher-order statistics,"IEEE journal on Selected Areas in Communications, vol. 13, no. 1, pp. 132-140, 1995. Abstract: This paper presents two classes of adaptive blind algorithms based on secondand higher order statistics. The rst class contains fast recursive algorithms whose cost functions involve second and thirdor fourth-order cumulants. These algorithms are stochastic gradientbased but have structures similar to the fast transversal lters (FTF) algorithms. The second class is composed of two stages: the rst stage uses a gradient adaptive lattice (GAL) while the second stage employs a higher order-cumulant (HOC) based least mean squares (LMS) lter. The computational loads for these algorithms are all linearly proportional to the number of taps used. Furthermore, the second class, as various numerical examples indicate, yields very fast convergence rates and low steady state mean square errors (MSE) and intersymbol interference (ISI). MSE convergence analyses for the proposed algorithms are also provided and compared with simulation results (20 Refs.) [Ungerboeck TCOM 82] Ungerboeck, G.,\Comments on 'Self-recovering equalization and carrier tracking in two-dimensional data communication systems',"IEEE Transactions on Communications, vol. vol.COM-30, no.3, pp. p. 557, March 1982. Abstract: The author comments on the above paper by Godard (1980) and draws attention to similar published works (4 Refs.) [Van der Veen SPIE 94] A.J. van der Veen and A. Paulraj,\A constant modulus factorization technique for smart antenna applications in mobile communications," in Proc. SPIE The International Society for Optical Engineering (San Diego, CA, USA), pp. 230-41, 24-27 July 1994. Abstract: A fundamental problem in sensor array signal processing is to separate and retrieve all independent co-channel signals that arrive at the antenna array. Such problems arise in smart antenna applications for mobile wireless communication, such as interference reduction and in-cell frequency reuse. In a mobile environment, the presence of large delay multipath makes the array manifold poorly de ned, and spatial model methods are not applicable. However, in case the signals have a constant modulus property (as in FDMA/FM systems like AMPS or TAGS), iterative algorithms such as Godard and CMA have been used to retrieve the signals. Because of a non-convex optimization criterion, these algorithms su er from local 82 minima and random convergence behavior, with no satisfactory remedy known as yet. In this paper, we present an algorithm to compute the exact solution to the underlying constant modulus (CM) factorization problem. With this new approach, it is possible to detect the number of CM signals present at the array, and to retrieve all of them exactly, rejecting other, non-CM signals. Only a modest amount of samples are required. The algorithm is robust in the presence of noise, and is tested on real data, collected from an experimental set-up (21 Refs.) [Van der Veen ASIL 95] A.J. Van der Veen and A. Paulraj,\Analytical solution to the constant modulus factorization problem," in Proc. 28th Asilomar Conference on Signals, Systems and Computers (Paci c Grove, CA, USA), pp. 1433-7, 31 Oct.-2 Nov. 1994. Abstract: Iterative constant modulus algorithms have been used to blindly separate and retrieve interfering constant modulus signals impinging on an antenna array. These algorithms have several well-known but basically unsolved de ciencies. In this paper, we present an algorithm to analytically compute the solution to the underlying constant modulus (CRI) factorization problem. With this new approach, if is possible to detect the number of CM signals present in the channel, and to retrieve all of them exactly, rejecting other, non-CM signals. Only a modest amount of samples are required. The algorithm is robust in the presence of noise, and is tested on real data, collected from an experimental set-up (6 Refs.) [Van der Veen TSP 96] A.J. Van der Veen and A. Paulraj,\An analytical constant modulus algorithm,"IEEE Transactions on Signal Processing, vol. 44, no.5, pp. 1136-55, May 1996. (Filed in BERG library.) Abstract: Iterative constant modulus algorithms such as Godard (1980) and CMA have been used to blindly separate a superposition of cochannel constant modulus (CM) signals impinging on an antenna array. These algorithms have certain de ciencies in the context of convergence to local minima and the retrieval of all individual CM signals that are present in the channel. We show that the underlying constant modulus factorization problem is, in fact, a generalized eigenvalue problem, and may be solved via a simultaneous diagonalization of a set of matrices. With this new analytical approach, it is possible to detect the number of CM signals present in the channel, and to retrieve all of them exactly, rejecting other, non-CM signals. Only a modest amount of samples is required. The algorithm is robust in the presence of noise and is tested on measured data collected from an experimental set-up (55 Refs.) [Vembu CISS 91] S. Vembu, S. Verdu, R.A. Kennedy, and W.A. Sethares,\Convex cost functions in blind equalization," in Proc. 25th Conf. on Info. Sci. and Systems (Baltimore, MD), pp. 792-797, 1991. [Vembu TSP 94] S. Vembu, S. Verdu, R.A. Kennedy, and W.A. Sethares,\Convex cost functions in blind equalization,"IEEE Transactions on Signal Processing, vol. 42, no. 8, pp. 1952-1960, Aug. 1994. (Filed in BERG library.) Abstract: Existing blind adaptive equalizers that use nonconvex cost functions and stochastic gradient descent su er form lack of global convergence to an equalizer setup that removes su cient ISI when an FIR equalizer is used. In this paper, we impose convexity on the cost function and anchoring of the equalizer away from the all-zero setup. We establish that there exists a globally convergent blind equalization strategy for 1-D pulse amplitude modulation (PAM) systems which bounded input data (discrete or continous) even when the equalizer is 83 truncated. The resulting cost function is a constrained l1 norm of the joint impulse response of the channel and the equalizer. Our results apply to arbitrary linear channels (provided there are no unit circle zeros) and apply regardless of the initial ISI (that is whether the eye is initially open or closed). We also show a globally convergent stochastic gradient scheme based on an implementable approximation of the l1 cost function. [Verdu AOS 84] S. Verdu,\On the selection of memoryless adaptive laws for blind equalization in binary communications," in Proc. 6th Intl. Conf. on Analysis and Optimization of Systems (Nice, France), pp. 239-249, 1984. [Verdu TIT 93] S. Verdu, B.D.O. Anderson, and R.A. Kennedy,\Blind equalization without gain identi cation,"IEEE Transactions on Information Theory, vol. 39, pp. 292-297, 1993. Abstract: Blind equalization up to a constant gain of linear timeinvariant channels is studied. Dropping the requirement of gain identi cation allows equalizer anchoring. This results in the elimination of a degree of freedom that causes illconvergence of conventional blind equalizers, and a ords the possibility of using simple update rules based on the stochastic approximation of output energy. Unlike conventional blind equalizers, truncations of the nonrecursive in nitedimensional realizations of those equalizers inherit the convergence properties of their in nitely parametrized counterparts. A globally convergent blind recursive equalizer for channels without all-pass sections is obtained based on the exact equalization of the minimum-phase part of the channel and the identi cation of its nonminimum-phase zeros (19 Refs.) [Walden SP 88] A.T. Walden,\A comparison of stochastic gradient and minimum entropy deconvolution algorithms,"Signal Processing, vol. 15, pp. 203-211, 1988. Abstract: The author examines the connection between a stochastic gradient method for deconvolution, originally formulated in the electrical engineering literature, and minimum entropy type (ME-type) deconvolution methods, well known in geophysics. It is shown that Gray's favoured ME-type deconvolution algorithm has a direct stochastic gradient equivalent which di ers from the method examined here only in the presence of the inverse autocovariance matrix in Gray's method. The two procedures can be made more alike if the stochastic gradient algorithm is used merely for phase estimation, and is preceded by standard whitening deconvolution. In any event, the stochastic gradient approach is de cient in two respects, namely the assumption of an independent and identically distributed input sequence, and knowledge of its rst and second absolute moments, assumptions which might be reasonable in the setting of data transmission problems, but not for seismic re ection coe cient sequences (18 Refs.) [Wang ICASSP:96] Z. Wang and E.M. Dowling,\Block Shannon Constant Modulus Algorithm for Wireless Equalizations," in Proc. International Conference on Acoustics, Speech and Signal Processing (Atlanta, GA), pp. 2678, May 7-9, 1996. Abstract: In this paper, we formulate a block processing objective function for the constant modulus algorithm (CMA) and present a nonlinear optimization strategy for its rapid minimization. The resulting algorithm is called the Block Shanno Constant Modulus Algorithm (BSCMA), which is applied in the simulation to equalize multipath fading channel models with a multi-antenna system. The results show that the algorithm can converge considerably faster than the steepest descent based CMA and the RLSCMA con gured similarly. [Weerackody TCOM 91] V. Weerackody, S.A. Kassam, and K.R. Laker,\Convergence analysis of an algorithm for blind equalization,"IEEE Transactions on Communications, vol. 39, no. 6, 84 pp. 856-865, June 1991. (Filed in BERG library.) Abstract: Adaptive equalization of communication channels without training sequences is a research area that has recently found wide interest. The primary algorithms available for these blind equalizers are based on minimizing a cost functional, which incorporates knowledge of the transmitted data constellation statistics, using a stochastic gradient approach. Unlike conventional equalizers, relatively little is known about the detailed convergence properties of these blind equalizers. In this paper we consider a key algorithm for blind equalization and derive expressions for the evolution of the equalizer output error trajectory. We introduce suitable approximations to overcome the problems introduced by the nonlinearity of the algorithm. We demonstrate that these approximations are justi able in typical situations where the algorithm is employed. [Weerackody TCOM:94] V. Weerackody and S.A Kassam,\Dual-mode type algorithms for blind equalization,"IEEE Transactions on Communications, vol. 42, no.1, pp. 22-28, Jan. 1994. (Filed in BERG library.) Abstract: Adaptive channel equalization accomplished without resorting to a training sequence is known as blind equalization. The Godard algorithm and the generalized Sato algorithm are two widely referenced algorithms for blind equalization of a QAM system. These algorithms exhibit very slow convergence rates when compared to algorithms employed in conventional dataaided equalization schemes. In order to speed up the convergence process, these algorithms may be switched over to a decision-directed equalization scheme once the error level is reasonably low. The authors present a scheme which is capable of operating in two modes: blind equalization mode and a mode similar to the decision-directed equalization mode. In this proposed scheme, the dominant mode of operation changes from the blind equalization mode at higher error levels to the mode similar to the decisiondirected equalization mode at lower error levels. Manual switch-over to the decision-directed mode from the blind equalization mode, or vice-versa, is not necessary since transitions between the two modes take place smoothly and automatically (17 Refs.) [Weerackody TCS 92] V. Weerackody, S.A. Kassam, K.R. Laker,\A simple hard-limited adaptive algorithm for blind equalization,"IEEE Trans. on Circuits and Systems II, vol. 39, no. 7, pp. 482-487, 1992. Abstract: An algorithm for adaptive blind equalization is proposed. It is interpreted as a sign version of a widely referenced algorithm for blind equalization, and it is demonstrated by way of simulations that the performance of this new sign way of simulations that the performance of this new sign algorithm is comparable to that of its unsigned version. A simple variable step-size scheme to accelerate the convergence process is also proposed. Simulation results are presented to demonstrate the signi cant improvements in the convergence rates that are obtained using this variable stepsize scheme (13 Refs.) [Wesolowski ETTRT 92] K. Wesolowski,\Analysis and properties of the modi ed constant modulus algorithm for blind equalization,"European Transactions on Telecommunications and Related Technologies, vol. 3 no. 3, pp. 225-230, May-June 1992. Abstract: A modi cation of the Godard blind equalization criterion and algorithm (see IEEE Trans. Commun., vol.COM-28, no.11, p.1867-75, 1980) is presented. The analysis of the proposed criterion is performed, showing that the corresponding cost function has four equivalent 85 global minima. The original Godard algorithm and the proposed algorithm are compared by simulation of data transmission through telephone channels (11 Refs.) [Weslowski TCOM 95] K. Wesolowski,\Adaptive blind equalizers with automatically controlled parameters,"IEEE Transactions on Communications, vol. 43,no 2-4, pp. 170-172, 1995. Abstract: Automatic control of the step size and lter length is presented for a transversal blind equalizer which is driven by the "stop-and-go" decision directed algorithm. The proposed method results in substantial shortening of the convergence time (6 Refs.) [White TCOM 96] L. B. White,\Blind equalization of constant modulus signals using an adaptive observer approach,"IEEE Transactions on Communications, vol. 44, no.2, pp. 134-6, Feb. 1996. (Filed in BERG library.) Abstract: The paper addresses the problem of blind equalization of constant modulus signals which are degraded by frequency selective multipath propagation and additive white noise. An adaptive observer is used to update the weights of an FIR equalizer in order to restore the signal`s constant modulus property. The observer gain is selected using fake algebraic Riccati methods in order to guarantee local stability. The performance of this method is compared to the constant modulus algorithm for simulated FM-FDM signals and exhibits signi cantly better convergence properties, particularly for heavy-tailed noise (4 Refs.) [Won TCE 94] Y.K. Won, G.H. Lee, R.H. Park, J.H. Park, B.U. Lee,\Channel equalization techniques for HDTV systems,"IEEE Transactions on Consumer Electronics, vol. 40, no. 4, pp. 903-912, 1994. Abstract: Channel equalization techniques for full-digital high de nition television (HDTV) systems are investigated. Conventional equalization methods are surveyed and a variable step size least mean square (VS-LMS) algorithm using the simple time constant concept is proposed. Several equalization techniques for HDTV systems are simulated for various channel models, and their characteristics are analyzed. Also the simulation results of the equalizer using xed-point operations are shown (22 Refs.) [Wong ICNN 96] C.F. Wong, T.L. Fine,\Adaptive blind equalization using arti cial neural networks," in Proc. International Conference on Neural Networks (Washington, DC), pp. 19741979, June 1996. Abstract: We attempt to use a neural network to solve the channel blind equalization problem. An equalizer is a device which by observing the channel outputs recovers the channel inputs. A blind equalizer does not require any known training sequence for the startup period. We have implemented a blind equalizer using a neural network for channel inputs of (-1, 1). The key to our approach is a three-component error/loss function which controls the hidden layer node output, the nal network output and the output layer weight parameters. The neural network is trained using a scaled conjugate gradient method which is faster than the steepest descent algorithms and is free from user-de ned parameters. Our method is robust. It makes no assumption about the channel input distribution of channel frequency response and needs fewer taps than conventional blind equalizers. Compared to the popular CMA blind equalizers, our network achieves a signi cantly lower BER but takes longer to train (13 Refs.) [Wu SSTT 95] M. Wu and F. Cornett,\Discrete-time and continuous-time constant modulus algorithm analysis," in Proc. 27th Southeastern Symposium on System Theory (Starkville, MS, 86 USA), pp. 504-8, 12-14 March 1995. Abstract: The purpose of this study is to analyze the properties of one of the most common blind adaptive techniques, the Constant Modulus Algorithm, in a mathematical sense. In this paper, the discrete-time and continuous-time Constant Modulus Algorithm is presented. Convergence behavior of two adaptation algorithm, the steepest descent algorithm and the stochastic gradient algorithm, is discussed in both discretetime and continuous-time cases. The stochastic gradient algorithm performance of each case is characterized and terms of steady state error. The results of computer simulations are presented. Finally, conclusions and recommendations for future research are given (5 Refs.) [Yang MSCS 95] Yoon Gi Yang, Nam Ik Cho, and Sang Uk Lee,\Fast blind equalization by using frequency domain block constant modulus algorithm," in Proc. Midwest Symposium on Circuits and Systems (Rio de Janeiro, Brazil), pp. 1003-6, Aug. 13-16, 1995. Abstract: This paper presents fast algorithms for the CMA (constant modulus algorithm), which is one of the widely used blind equalization algorithms. To derive the fast algorithm of the CMA for high-speed equalization, we rst introduce BCMA (block CMA) which adjusts the equalizer coe cients by block processing of the received symbols in the time domain. Based on the BCMA, we propose the FBCMA (frequency domain block CMA) which employs fast linear convolution in the DFT domain by using the overlap save method. In this paper, a non-linear error function in the frequency domain is derived using the Parseval's relation. Also, an adaptive algorithm in the DFT domain is developed to adjust the DFT domain lter coe cients. If the block size and lter length is N, the multiplications required for the conventional CMA and proposed FBCMA are in the order of O(N/sup 2/) and O(N log N), respectively. The computer simulations show that the proposed FBCMA presents comparable performance to the conventional CMA, while requiring less computations (7 Refs.) [Yellin TSP 96] D. Yellin, E. Weinstein,\Multichannel signal separation,"IEEE Transactions on Signal Processing, vol. 44, no. 1, pp. 106-118, 1996. Abstract: The problem of multichannel signal separation has attracted considerable interest in recent literature. A variety of methods and criteria have been proposed to solve the problem based on statistical independence between the source signals. Most of these criteria involve the computation of high-order statistics of the observed signals. We present a uni ed framework for many of these criteria and analyze their statistical variance (23 Refs.) [Youngkyun ETRI 96] Youngkyun Kim; Sungjo Kim; Mintaig Kim,\The derivation of a new blind equalization algorithm,"ETRI Journal, vol. vol.18, no.2, pp. p. 53-60, July 1996. Abstract: Blind equalization is a technique for adaptive equalization of a communication channel without the aid of training sequences. This paper proposes a new blind equalization algorithm. The advantage of the new algorithm is that it has the lower residual error than the GA (proposed by Godard (1980)) and sign GA (proposed by Weerackody et al. (1992)). The superior performance of the proposed algorithm is illustrated for the 16-QAM signal constellation. A Rummler (1986) channel model is assumed as a transmission medium. The performance of the proposed algorithm is compared to the GA, sign GA and stop and go algorithm (SGA). The simulation results demonstrate that an improvement in performance is achieved with the proposed equalization algorithm (7 Refs.) [H.Zeng CISS 96] H. Zeng and L. Tong,\On the performance of CMA in the presence of noise," in Proc. Conference on Information Science and Systems (Princeton, NJ), pp. 890-894, March, 87 1996. Abstract: Most existing literature address the performance of CMA in the absence of noise and predict the similar behavior when noise is present. This is true when the noise is small and the CMA equalizer remains nearby. In this paper, an analytical performance of CMA in the presence of noise is investigated. The region containing CMA equalizer is given and the mean square error (MSE) bounds of the CMA equalizer are derived. As a consequence, we also obtain the upper bound of MSE for the optimum CMA equalizer which has the minimum MSE. It is also shown that there may exist other CMA equalizers which have sign cantly large MSE. This indicates the ill-convergence of CMA. [H.Zeng ASIL 96] H. Zeng, L. Tong, C.R. Johnson Jr.,\Mean square error performance of CMA receivers," in Proc. Asilomar Conference on Signals, Systems and Computers (Paci c grove, CA), pp. 305-309, October, 1996. Abstract: The constant modulus algorithm (CMA) proposed by Godard and Treichler is an e ective technique for blind receiver design in practice. In this paper, a geometrical approach, generalized from a recent approach in [?] is presented that relates the real CMA with the well-known minimum mean square error (MMSE) receivers. The e ect of constellation scheme to CMA equalization is also investigated. Given the dispersion constant of the transmitted signal, the MSE and the intersymbol interference of a MMSE receiver, a CMA local minimum is located in a neighborhood nearby. The MSE upper bound of the CMA receiver is derived and shown to be tight in simulations using an empirically measured channel. It is shown that the theorems are robust to the model mismatch and channel disparity. [H.Zeng ICASSP 97] H.H. Zeng and L. Tong,\The MSE performance of constant modulus receivers," in Proc. International Conference on Acoustics, Speech and Signal Processing (Munich, Germany), pp. 3577-3580, Apr. 1997. Abstract: The constant modulus algorithm (CMA) is an e ective technique for blind receiver design in practice. Treating CMA as a linear estimation problem, e ects of noise and channel conditions are investigated. For the class of channels with arbitrary nite impulse responses, an analytical description of locations of constant modulus receivers and an upper bound of their mean squared errors (MSE) are derived. We show that, with proper initializations, CMA can achieve almost the same performance as the (nonblind) minimum mean square error (MMSE) receiver. Our analysis reveals a strong relationship between the (blind) constant modulus and the (nonblind) MMSE receivers. It also highlights the signi cance of initialization/reinitialization schemes. The approach developed in this paper also applies to CMA blind beamforming in array signal processing (11 Refs.) [H.Zeng TSP tbd] H. Zeng, L. Tong, C.R. Johnson, Jr.,\Relationships between the constant modulus and Wiener receivers,"submitted to IEEE Transactions on Signal Processing, vol. xxx, pp. xxx, xxx. Abstract: The Godard or constant modulus algorithm (CAM) is an e ective technique for blind receiver design in communications. However, due to the complexity of the cost function, the performance of CMA receivers has primarily been evaluated using simulations. Theoretical analysis is typically based on either the noiseless case or approximations of the cost function. The following question, while resolvable numerically for a speci c example, remains unanswered in a generic manner: In the presence of channel noise, where are the CMA local minima and what are their mean-squared errors (MSE)? In this paper, a geometrical approach is presented 88 that relates CMA with Wiener (or minimum MSE) receivers. Given the MSE and the intersymbol/user interference of a Wiener receiver, a su cient condition is given for the existence of a CMA local minimum in the neighborhood of the Wiener receiver. MSE bounds on CMA receiver performance are derived and shown to be tight in simulations. The analysis shows that, while in some cases the CMA receiver performs almost as well as the (nonblind) Wiener receiver, it is also possible that, due to its blind nature, CMA may perform considerably worse than a (nonblind) Wiener receiver. [S.Zeng CISS 96] S. Zeng, H. Zeng, L. Tong,\A blind channel estimator using the constant modulus criterion with subspace constraints," in Proc. Conference on Information Science and Systems (Princeton, NJ), pp. xxx, March, 1996. Abstract: A new blind channel estimation approach is proposed to exploit simultaneously the constant modulus property of the communication signal and the subspace constraints of the channel. The advantages of the new algorithm are (i) fast convergence, (ii) achieving asymptotically minimum mean-square error (MMSE) equalization. In simulations, the new approach o ers considerable improvement over the CMA and the equalization based on the blind channel estimation using the subspace algorithm. [S.Zeng ICC 96] S. Zeng, H.H. Zeng, and L. Tong,\Blind equalization using CMA: Performance analysis and a new algorithm," in Proc. IEEE Conference on Communications (Dallas, TX), pp. 847-51, June 1996. Abstract: The performance of the fractionally-spaced CMA (FSCMA) is investigated. The mean square error (MSE) of the FSCMA equalizer is compared with that of the optimal minimum mean square error (MMSE) equalizer. It has been demonstrated that FSCMA may converge to an undesirable minimum which has an MSE considerably larger than the MMSE. To avoid this drawback, a new algorithm is proposed. The new algorithm, referred to as the constant modulus-correlation tting (CM-CF) algorithm, simultaneously exploits the constant modulus property of the source and the cyclostationary properties of the fractionally sampled signal. In simulations, the CM-CF o ers substantial improvement over the least-squares CMA (11 Refs.) [S.Zeng SPW 96] S. Zeng, H. Zeng, L. Tong,\Blind channel equalization via multiobjective optimization," in Proc. 8th IEEE Sig. Proc. Workshop on Statistical Signal and Array Processing (Corfu, Greece), pp. 160-163, June 24-26, 1996. Abstract: We present in this paper a multiple objective optimization approach to fast blind channel equalization. By investigating rst the performance (mean-square error) o the standard fractionally spaced CMA equalizer in the presence of noise, we show that CMA local minima exist near the minimum mean square error equalizers. Consequently, CMA may converge to a local minimum corresponding to a poorly designed MMSE receiver with considerably large mean square error. Based on the mulitple objective optimization techniques, we propose next a blind channel estimator by exploiting simultaneously the second-order cyclostationary statistics and the constant moulus of QAM-type communication signals. Such a channel estimation-based blind equalization scheme has the advantage of designing FIR minimummean square error equalizers with the optimal delay. [Zervas SPIE 91] E. Zervas, J.G. Proakis, and V. Eyuboglu,\E ect of constellation shaping on blind equalization," in Proc. The International Society for Optical Engineering (San Diego), pp. 1393-1397, 1991. 89 Abstract: The authors address the problem of blind equalization where the input of the unknown system is a shaped data signal. They demonstrate the e ects of shaping on some of the widely known schemes. Results point to the need for more powerful blind equalization algorithms for shaped signals (12 Refs.) [Zheng TSP 93] F.C. Zheng, S. Mclaughlin, B. Mulgrew,\Blind equalization of nonminimum phase channels{higher order cumulant based algorithm,"IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 681-691, 1993. Abstract: Higher order cumulant analysis is applied to the blind equalization of linear timeinvariant (LTI) nonminimum-phase channels. The channel model is moving-average based. To identify the moving average parameters of channels, a higher-order cumulant tting approach is adopted in which a novel relay algorithm is proposed to obtain the global solution. In addition, the technique incorporates model order determination. The transmitted data are considered as independently identically distributed random variables over some discrete nite set (e.g., set (+or-1, +or-3)). A transformation scheme is suggested so that third-order cumulant analysis can be applied to this type of data. Simulation examples verify the feasibility and potential of the algorithm. Performance is compared with that of the noncumulant-based Sato scheme in terms of the steady state MSE and convergence rate (22 Refs.) The authors address the problem of blind equalization where the input of the unknown system is a shaped data signal. They demonstrate the e ects of shaping on some of the widely known schemes. Results point to the need for more powerful blind equalization algorithms for shaped signals (12 Refs.) [Zheng TSP 93] F.C. Zheng, S. Mclaughlin, B. Mulgrew,\Blind equalization of nonminimum phase channels{higher order cumulant based algorithm,"IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 681-691, 1993. Abstract: Higher order cumulant analysis is applied to the blind equalization of linear timeinvariant (LTI) nonminimum-phase channels. The channel model is moving-average based. To identify the moving average parameters of channels, a higher-order cumulant tting approach is adopted in which a novel relay algorithm is proposed to obtain the global solution. In addition, the technique incorporates model order determination. The transmitted data are considered as independently identically distributed random variables over some discrete nite set (e.g., set (+or-1, +or-3)). A transformation scheme is suggested so that third-order cumulant analysis can be applied to this type of data. Simulation examples verify the feasibility and potential of the algorithm. Performance is compared with that of the noncumulant-based Sato scheme in terms of the steady state MSE and convergence rate (22 Refs.)