On the nonvanishing stability of undesirable equilibria for FIRGodard blind equalizers

The existence of undesirable equilibria of Godard (1980) or constant modulus algorithms (CMA) is demonstrated for all finite-dimensional FIR equalizers. Sufficient stability conditions are presented for these equilibria. It is shown that for the Godard-2 algorithm, these undesirable equilibria are locally stable as long as the equalizer length remains finite. Results indicate that merely increasing the equalizer length does not necessarily nullify the potential for local convergence by Godard algorithms     

On the (non)existence of undesirable equilibria of Godard blindequalizers

Existing results in the literature have proved that particular blind equalization algorithms, including Godard algorithms, are globally convergent in an ideal and nonimplementable setting where a doubly infinite dimensional equalizer is available for adaptation. Contrary to popular conjectures, it is shown that implementable finite dimensional equalizers which attempt to approximate the ideal setting generally fail to have global convergence to acceptable equalizer parameter settings without the use of special remedial measures. A theory based on the channel convolution matrix nullspace is proposed to explain the failure of Godard algorithms for such practical blind equalization situations. This nullspace theory is supported by a simple example showing ill convergence of the Godard algorithm     

Application of Fast Kalman Estimation to Adaptive Equalization

Very rapid initial convergence of the equalizer tap coefficients is a requirement of many data communication systems which employ adaptive equalizers to minimize intersymbol interference. As shown in recent papers by Godard, and by Gitlin and Magee, a recursive least squares estimation algorithm, which is a special case of the Kalman estimation algorithm, is applicable to the estimation of the optimal (minimum MSE) set of tap coefficients. It was furthermore shown to yield much faster equalizer convergence than that achieved by the simple estimated gradient algorithm, especially for severely distorted channels. We show how certain "fast recursive estimation" techniques, originally introduced by Morf and Ljung, can be adapted to the equalizer adjustment problem, resulting in the same fast convergence as the conventional Kalman implementation, but with far fewer operations per iteration (proportional to the number of equalizer taps, rather than the square of the number of equalizer taps). These fast algorithms, applicable to both linear and decision feedback equalizers, exploit a certain shift-invariance property of successive equalizer contents. The rapid convergence properties of the "fast Kalman" adaptation algorithm are confirmed by simulation.     

Length- and cost-dependent local minima of unconstrained blindchannel equalizers

Baud-rate linear blind equalizers may converge to undesirable stable equilibria due to different mechanisms. One such mechanism is the use of linear FIR filters as equalizers. 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 is shown that several well-designed algorithms do not have cost-dependent local minimum, whereas other algorithms, such as the decision-directed equalizer and the stop-and-go algorithm (SGA), do. Unlike many existing convergence analysis, the convergence of the Godard (1980) algorithms (GAs) and standard cumulant algorithms (SCAs) under Gaussian noise is also presented. Computer simulations are used to verify the analytical results     

Global convergence of fractionally spaced Godard (CMA) adaptiveequalizers

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 confined 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 significance: (1) a finite-length channel satisfying a length-and-zero condition allows Godard/CMA FSE to be globally convergent, and (2) the linear FSE filter length need not be longer than the channel delay spread. Computer simulation demonstrates the performance improvement provided by the adaptive Godard FSE     

Undermodeled equalization: a characterization of stationary pointsfor a family of blind criteria

We attack specific problems related to equalizer performance in undermodeled cases in which assumptions of perfect equalizability are dismissed in favor of a more realistic situation in which no equalizer setting may achieve perfect channel equalization. We derive a characterization of candidate convergent points for a family of blind criteria which appeal, tacitly or wittingly, to maximizing the ratio of different sequence norms of the combined channel-equalizer impulse response. This may be accomplished in a practical implementation by using equalizer output cumulants of different orders. The popular Godard and Shalvi-Weinstein schemes are accommodated at one extreme of the family of criteria. We also show that each maximum at the other extreme of the family, involving progressively higher order output cumulants, yields, precisely, a Wiener response. This suggests that blind algorithms using progressively higher order statistics may converge more closely to a Wiener response than those using more modest order statistics. We show, moreover, that the superexponential family of algorithms is also included and establish a convergence proof for undermodeled cases that appeals to no approximation. Finally, some apparently novel bounds on attainable open-eye measures in undermodeled cases are also derived     

Static and dynamic convergence behavior of adaptive blindequalizers

This paper presents a theoretical analysis of the static and dynamic convergence behavior for a general class of adaptive blind equalizers. We first 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 coefficients will converge to the minimum of the cost function near the channel inverse. However, the convergence is not consistent. The correlation matrix or equalizer coefficients 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 longer-length blind equalizer does not necessarily outperform a shorter one, which is contrary to what is conventionally conjectured. The theoretical analysis results are confirmed by computer simulations     

Multichannel Algorithms for Simultaneous Equalization and Interference Suppression

This paper proposes techniques for simultaneous cancellation of intersymbol and interchannel or multi-access interference (ISI and ICI) that shows up in several multi-input, multi-output (MIMO) communication channels. Correlation and kurtosis based optimization criteria are derived for multi-channel decision feedback equalizers (MC-DFE) and compared with the popular Godard algorithm (CMA) and the minimum mean-square error in a decision directed mode (MMSE-DD). The proposed adaptive algorithms are easily extended to a scenario with more than two users with the computational complexity increasing linearly with the number of inputs. Simulation results show that the algorithms converge to the global minimum in a blind environment with channels that introduce moderate distortion.     

A Blind Equalizer Based on Unsupervised Gaussian Cluster Formation with an Adaptive Non—Linearity

In this paper we present a blind equalizer algorithm based on an unsupervised Gaussian cluster formation technique with an optimized gradient adaptive step-size to update the equalizer coefficients.The novelty of this work lies in the optimization of the nonlinearity of the cluster formation used to achieve an optimal soft decision.The proposed iterative procedure combined with the variable step-size gradient-based adaptation,significantly accelerates the convergence speed of the blind equalization.The advantages of the proposed equalization techniques are illustrated by simulation.Simulation results obtained are compared with the Sato and Godard blind equalizers.     

A novel expression for the achievable MSE performance obtained by blind adaptive equalizers

In this paper, I propose for the noisy, real, and two independent quadrature carrier case, an approximated closed-form expression for the achievable minimum mean square error (MSE) performance obtained by blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer’s taps can be expressed as a polynomial function of the equalized output of order three like in Godard’s algorithm. The proposed closed-form expression for the achievable MSE is based on the step-size parameter, on the equalizer’s tap length, on the channel power, on the signal to noise ratio (SNR), on the nature of the chosen equalizer, and on the input signal statistics. Since the channel power is measurable or can be calculated if the channel coefficients are given, there is no need anymore to carry out any simulation with various step-size parameters, different values for the signal to noise ratio (SNR) and equalizer’s tap length for a given equalization method, and input signal statistics in order to find the MSE performance in the convergence state.     

A family of low-complexity blind equalizers

Two important topics in equalizer design are its complexity and its training. We present a family of blind equalizers which, by incorporating a decomposition finite-impulse response filtering technique, can reduce the complexity of the convolution operation therein by about one half. The prototype algorithm in this equalizer family employs the prevalent Godard cost function. Several simplified algorithms are proposed, including a sign algorithm which eliminates multiplications in coefficient adaptation and a few delayed versions. We also study the convergence properties of the algorithms. For the prototype algorithm, we show that, in the limit of an infinitely long equalizer and under mild conditions on signal constellations and channel characteristics, there are only two sets of local minima on the performance surface. One of the sets is undesirable and is characterized by a equalized channel response. The other corresponds to perfect equalization, which can be reached with proper equalizer initialization. For the simplified algorithms, corresponding cost functions may not exist. Some understanding of their convergence behaviors are obtained via examination of their adaptation equations. Simulation results are presented to demonstrate the performance of the algorithms.     

一种用于水声通信的快速自适应均衡器设计

信道均衡是现代水声通信系统中克服码间干扰的重要手段,根据时变水声信道需进行信道均衡的要求,设计了一种判决反馈盲均衡器。针对恒模算法在固定步长下存在收敛速度与剩余误差的矛盾缺陷,提出了一种基于剩余误差的变步长恒模算法,并对改进算法进行了计算机仿真及试验测试,结果表明,改进算法无论是在收敛性能还是在均衡效果上都有大幅提高,在实际水声通信中具有较好的应用价值。     

Chip-Level MMSE Equalization for CDMA Downlink

1IntroductionNext generation standards support high data rate ser-vices,in which users are only assigned a smaller-orderspreading code . This leads to a necessity for the tech-niques that suppress Inter-Chip Interference (ICI) aswell as Multi-Access Interference ( MAI) . The tradi-tional RAKE receiver experiences difficulty because theorthogonality decays when signal transmits over multi-path frequency selective fading channels . The RAKEreceiver cannot overcome the serious MAI and I…     

Adaptive decision feedback equalization: can you skip the trainingperiod?

This paper presents a novel unsupervised (blind) adaptive decision feedback equalizer (DFE). It can be thought of as the cascade of four devices, whose main components are a purely recursive filter (ℛ) and a transversal filter (𝒯). Its major feature is the ability to deal with severe quickly time-varying channels, unlike the conventional adaptive DFE. This result is obtained by allowing the new equalizer to modify, in a reversible way, both its structure and its adaptation according to some measure of performance such as the mean-square error (MSE). In the starting mode, ℛ comes first and whitens its own output by means of a prediction principle, while 𝒯 removes the remaining intersymbol interference (ISI) thanks to the Godard (1980) (or Shalvi-Weinstein (1990)) algorithm. In the tracking mode the equalizer becomes the classical DFE controlled by the decision-directed (DD) least-mean-square (LMS) algorithm. With the same computational complexity, the new unsupervised equalizer exhibits the same convergence speed, steady-state MSE, and bit-error rate (BER) as the trained conventional DFE, but it requires no training. It has been implemented on a digital signal processor (DSP) and tested on underwater communications signals-its performances are really convincing     

Performance analysis of Godard-based blind channel identification

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 (1980) (or constant modulus) criterion, for which mean-squared symbol estimation error bounds have been derived. 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     

Least squares approach to blind channel equalization

We present a least squares (LS) algorithm for blind channel equalization based on a reformulation of the Godard algorithm. A transformation for the equalizer parameters is considered to convert the nonlinear LS problem inherent in the Godard algorithm to a linear LS problem. Unlike the Godard (1980) algorithm, the proposed LS approach does not suffer from ill-convergence to closed-eye local minima. Methods for extracting the equalizer parameters from their transformed version are developed. Offline and recursive implementations of the LS algorithm are presented. The algorithm requires only a small number of channel output observations to estimate the equalizer parameters and is therefore fast vis-a-vis the Godard algorithm. The channel input correlation does not impose any restriction on the application of the algorithm, as long as a weak sufficient-excitation condition is satisfied. Simulation examples are presented to demonstrate the LS approach and to compare it with the Godard algorithm     

Grouped signed power-of-two algorithms for low-complexity adaptive equalization

With increasing demand for higher data rate, modern communication systems have grown more complex. Equalization has become more and more important as it is effective in mitigating the multipath fading often occurred in high-data-rate communication systems. However, the implementation complexity of adaptive equalizers is usually too high for mobile communication applications. In this paper, a novel adaptive equalization algorithm and its low-complexity architecture are proposed. This algorithm employs a new grouped signed power-of-two (GSPT) number representation. The GSPT algorithm and several enhanced versions are simulated as adaptive equalizers in a phase-shift keying communication receiver for several practical channels and the GSPT-based equalizers perform as well as the least mean square (LMS) equalizer. Moreover, for comparison, two GSPT-based equalizers and two other equalizers are implemented in field-programmable gate arrays. The GSPT-based equalizers require only about 25%-30% of the hardware resources needed in the LMS equalizer. Also the GSPT-based equalizers are more than twice as fast as the LMS equalizer.     

Fast recovery equalization techniques for DTV signals

Blind equalizers do not require any training sequence for the initial startup period but rather perform equalization blindly on the data directly. In this paper, a general approach for designing such equalizers for 8-VSB transmission systems is presented. The candidate algorithms considered here include Godard's algorithm (Godard 1980), Sato's algorithm (Sato 1975), the G-pseudo error algorithm using Sato's function and the G-pseudo error algorithm using Godard's function. This paper analyzes these algorithms for the 8-VSB transmission format and recommends the most suitable algorithm. The performance comparison parameters include MSE, convergence characteristics, computational load and accuracy of estimating the channel     

Elimination of DC Offset by MMSE Adaptive Equalizers

Data communication systems are often beset with the problem of dc offset. The questions addressed and answered in this concise paper are: 1) can minimum mean-squared error (MMSE) equalizers correct dc offset? 2) what are the exact effects of the equalizer on offset, and offset on residual MSE? We conclude that MMSE equalizers can reduce the effects of dc offset but cannot completely eliminate them. Quantitative results give residual MSE as a function of offset magintude, equalizer length, and signal energy.     

Bussgang盲均衡器

文中描述了基带均衡通信系统的信号模型,研究了基于此模型的三个典型的Bussgang均衡器：恒模算法（CMA）,Sato和直接判决（DD）均衡器。给出了与算法相关的性能仿真结果。仿真结果展示了CMA均衡器可以提供更好的BER性能及比Sato和DD均衡器更快的收敛速度。