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Some convergence properties of godard's quartic algorithm

SIGNAL PROCESSING 1997年第3期56卷 313-320页

作者： Choi, J Song, I Park, RH DACOM R&D CTR PCS TEAM TAEJON 305350 SOUTH KOREA

Convergence analysis on godard's quartic (GQ) algorithm used for blind equalization is accomplished, The first main result is an explanation of the local behavior of the GQ algorithm around the global minimum point of the average performance function, From this result, we can determine the adaptation gain and compare the convergence rate with that of the decision directed (DD) algorithm. It is shown that the convergence rate of the GQ algorithm is faster than that of the DD equalization algorithm. The second main result is a description of the geometry of the average performance function: the region of attraction is observed to depend on the characteristics of the channel as well as the statistics of the input signal. It is shown that a good initial parameter vector of the GQ algorithm can be chosen based on the information of the geometry of the average performance function.

关键词： godard algorithm blind equalization adaptive algorithm

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A class of widely linear quaternion blind equalisation algorithms

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SIGNAL PROCESSING 2025年 230卷

作者： Zhang, Min Xiang, Min Zheng, Zhi Talebi, Sayed Pouria Mandic, Danilo P. Beihang Univ Sch Instrumentat & Optoelect Engn Beijing Peoples R China Beihang Univ Hangzhou Innovat Inst Hangzhou Peoples R China Natl Inst Extremely Weak Magnet Field Infrastruct Hangzhou Peoples R China Univ Elect Sci & Technol China Sch Informat & Commun Engn Chengdu Peoples R China Univ Roehampton Comp Sci Dept London England Imperial Coll London Dept Elect & Elect Engn London England

Quaternion adaptive filters have been applied extensively to model three- and four-dimensional phenomena in signal processing, but most of them require a known reference signal. In this paper, a class of widely linear quaternion-valued godard (WL-Qgodard) algorithms is derived, which include the widely linear quaternionvalued constant modulus algorithm (WL-QCMA) as a special case. The derived filter allows for signal recovery operations in the absence of reference signals to be performed directly in the quaternion domain, eliminating the need for transformation to real-valued vector algebras and preserving the advantages of the quaternion division algebra. Compared to state-of-the-art quaternion blind equalisation algorithms, the proposed algorithm models the signal transmission channel using the widely linear quaternion framework, which has more extensive applicability and can better represent real-world scenarios. Furthermore, aided by GHR calculus, for the first time, we present a performance analysis framework for the Qgodard algorithm and WL-Qgodard algorithms, which depicts the dynamic and their static convergence behaviours, overcoming the challenges posed by the noncommutative quaternion algebra and non-isomorphism between the quaternion equalisers and real-valued equalisers. Finally, simulation results over physically meaningful wireless communication signals indicate the effectiveness and superiority of the proposed WL-QCMA, and the validity of the theoretical performance analysis.

关键词： Quaternion blind equalisation godard algorithm Constant modulus algorithm Static convergence behaviour Dynamic convergence behaviour Widely linear model

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Characterization of the undesirable global minima of the godard cost function: Case of noncircular symmetric signals

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2006年第5期54卷 1917-1922页

作者： Houcke, S Chevreuil, A ENST Bretagne Dept Signal & Commun PRACom TAMCI F-29285 Brest France Inst Gaspard Monge UMR 8049 CNRS IGM LabInfoEquipe Signal Commun F-77454 Marne La Vallee 2 France

The deconvolution of a filtered version of a zero-mean normalized independent and identically distributed (i.i.d.) signal (s(n))(n is an element of z) having a strictly negative Kurtosis gamma(2) = E[|s(n)|(4)] - 2(E[|s(n)|(2)])(2) - |E[s(n)(2)|(2)] is addressed. This correspondence focuses on the global minimizers of the godard function. A well-known result states that these minimizers achieve deconvolution at least if the input signal shows the symmetry E[s(2)] = 0. When this constraint is relaxed, (s(n))(n is an element of Z) is said to be noncircular symmetric: It is shown that the minimizers achieve deconvolution if and only if 2|E [s(n)(2)]|(2) < -gamma(2)(s). If this condition is not met, the global minimizers are found to be finite-impulse-response filters with two taps.

关键词： deconvolution constant modulus algorithm contrast function godard algorithm

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On the equivalence between the godard and Shalvi-Weinstein schemes of blind equalization

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SIGNAL PROCESSING 1999年第1-2期73卷 185-190页

作者： Regalia, PA Inst Natl Telecommun Dept Signal & Image F-91011 Evry France

Certain equivalences between the godard and Shalvi-Weinstein schemes have been previously noted under special circumstances. We present here a simple proof for real signals that an equivalence can be established assuming little more than stationarity to fourth order of the equalizer input;the exact nature of the input sequence proves otherwise irrelevant to the validity of the equivalence. The equivalence also carries over to complex signals, but subject to more restrictive circularity conditions. In a communication context, the equivalence implies that many performance issues, such as susceptibility to local minima, the ability (or lack there of) to open the eye, or mean performance degradations due to channel noise and/or source correlation properties, are common to the two, even when applied with nonlinear channels. Our equivalence also indicates a simple modification to the godard algorithm to render it applicable to leptokurtic sources. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： godard algorithm Shalvi-Weinstein algorithm blind equalization constant modulus algorithm

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Performance analysis of godard-based blind channel identification

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2001年第8期49卷 1757-1767页

作者： Schniter, P Casas, RA Touzni, A Johnson, CR Ohio State Univ Dept Elect Engn Columbus OH 43210 USA NxtWave Commun Langhorne PA 19047 USA Cornell Univ Dept Elect Engn Ithaca NY 14853 USA

We analyze a blind channel impulse response identification scheme based on the cross correlation of blind symbol estimates with the received signal, The symbol estimates specified are those minimizing the godard (or constant modulus) criterion, for which mean-squared symbol estimation error bounds have recently been derived. In this paper, we derive upper bounds for the average squared parameter estimation error (ASPE) of the blind identification scheme that depend on the mean-squared error of the Wiener equalizer, the kurtoses of the desired and interfering sources, and the channel impulse response. The effects of finite data length and stochastic gradient equalizer design on ASPE are also investigated. All results are derived in a general multiuser vector-channel context.

关键词： blind channel identification blind deconvolution constant modulus algorithm godard algorithm

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godard BLIND EQUALIZER ERROR SURFACE CHARACTERISTICS - WHITE, ZERO-MEAN, BINARY SOURCE CASE

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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING 1995年第4期9卷 301-324页

作者： JOHNSON, CR ANDERSON, BDO AUSTRALIAN NATL UNIV DEPT SYST ENGN CANBERRA ACT 0200 AUSTRALIA AUSTRALIAN NATL UNIV COOPERAT RES CTR ROBUST & ADAPT SYST CANBERRA ACT 0200 AUSTRALIA

We study the godard/CMA error surface in adapting the impulse response coefficients of a finite-length, tapped-delay-line filter to equalize a white, zero-mean, binary (+/-1) source that suffers 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 equalizer parameter space.

关键词： BLIND ADAPTIVE EQUALIZATION godard algorithm CMA algorithm MULTIMODAL COST FUNCTION STATIONARY POINT ANALYSIS

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Sufficient conditions for the local convergence of constant modulus algorithms

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2000年第10期48卷 2785-2796页

作者： Schniter, P Johnson, CR Ohio State Univ Dept Elect Engn Columbus OH 43210 USA Cornell Univ Dept Elect Engn Ithaca NY 14853 USA

The constant modulus (CM) criterion has become popular in the design of blind linear estimators of sub-Gaussian i.i.d. processes transmitted through unknown linear channels in the presence of unknown additive interference, The existence of multiple CM minima, however, makes it difficult for CM-minimizing schemes to generate estimates of the desired source (as opposed to an interferer) in multiuser environments, In this paper, we present three separate sufficient conditions under which gradient descent (GD) minimization of CIM cost will locally converge to an estimator of the desired source at a particular delay. The sufficient conditions are expressed in terms of statistical properties of the initial estimates, specifically, CM cost, kurtosis, and signal-to-interference-plus-noise ratio (SINR). Implications on CM-GD initialization methods are also discussed.

关键词： blind beamforming blind deconvolution blind equalization blind multiuser detection constant modulus algorithm godard algorithm

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CMA fractionally spaced equalizers: Stationary points and stability under iid and temporally correlated sources

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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING 1998年第2期12卷 135-155页

作者： Leblanc, JP Fijalkow, I Johnson, CR New Mexico State Univ Klipsch Sch ECE Las Cruces NM 88003 USA ENSEA ETIS F-95014 Cergy Pontoise France Cornell Univ Dept Elect Engn Ithaca NY 14853 USA

A common assumption in blind equalization schemes using the Constant Modulus algorithm (CMA) is that the source sequence is an independent identically distributed (i.i.d.) sequence with equiprobable symbols. Much of the analysis demonstrating the global convergence of CMA in a noiseless channel to an open-eye setting uses this assumption. This work investigates the effect of source statistics (distributions and correlations) on the location of CMA stationary points in the fractionally sampled equalizer case under the conditions of equalizability. The work identifies the stationary points as the solution set of a system of multivariate polynomial equations with monomial coefficients given by the source moments. (C) 1998 John Wiley & Sons, Ltd.

关键词： blind equalization godard algorithm CMA stationary point analysis

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Bounds for the MSE performance of constant modulus estimators

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IEEE TRANSACTIONS ON INFORMATION THEORY 2000年第7期46卷 2544-2560页

作者： Schniter, P Johnson, CR Ohio State Univ Dept Elect Engn Columbus OH 43210 USA Cornell Univ Dept Elect Engn Ithaca NY 14853 USA

The constant modulus (CM) criterion has become popular in the design of blind linear estimators of sub-Gaussian independent and identically distributed (i.i.d.) processes transmitted through unknown linear channels in the presence of unknown additive interference. In this paper, me present an upper bound for the conditionally unbiased mean-squared error (USME) of CM-minimizing estimators that depends only on the source kurtoses and the UMSE of Wiener estimators. Further analysis reveals that the extra UMSE of CM estimators can be upper-bounded by approximately the square of the Wiener (i.e., minimum) UMSE. Since our results hold for vector-valued finite-impulse response/infinite-impulse response (FIR/IIR) linear channels, vector-valued FIR/IIR estimators with a possibly constrained number of adjustable parameters, and multiple interferers with arbitrary distribution, they confirm the longstanding conjecture regarding the general mean-square error (MSE) robustness of CM estimators.

关键词： blind beamforming blind deconvolution blind equalization blind multiuser detection constant modulus algorithm godard algorithm

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Stationary points of the finite length constant modulus optimization

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SIGNAL PROCESSING 2002年第4期82卷 625-641页

作者： Jamali, H Ogunfunmi, T Canon R&D Ctr Amer Inc San Jose CA 95134 USA

Existing studies of the stationary points of the constant modulus adaptive algorithm, typically, assume a channel equalization setting and several restrictive conditions on the source sequence, the communications channel and the noise environment. Often, results are presented in terms of the combined channel and equalizer impulse response parameters, the source vector and the channel convolution matrix. As such, these results are not easily applicable in other areas. Modeling after Wiener filtering theory, this paper derives new expressions for the basic equations of the constant modulus (CM) optimization method. In particular, new formulas for the index function, its gradient vector and its Hessian matrix are obtained. These formulas involve the unknown parameters and the statistics of the sample sequence only. Consequently, they serve to convert the CM minimization problem into that of solving a system of cubic equations in the parameter vector. This new formulation of the CM optimization problem leads to a characterization of the critical points that is not exclusive to channel equalization. It also makes it possible to derive new equations for the minimum value of the cost function, the conditions for perfect signal recovery, and a tighter lower bound for the approach. Computer simulation examples are also presented to illustrate our findings. (C) 2002 Elsevier Science B.V. All lights reserved.

关键词： godard algorithm constant modulus adaptive algorithm (CMA) blind deconvolution blind equalizers optimum filters higher order statistics output only based performance criterion intersymbol interference (ISI)

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