Blind Channel Equalization Research Articles

Comparison of PMD-compensation techniques at 10 Gbit/s using an optical first-order compensator and electrical transversal filter

We present, for what is believed to be the first time, a direct comparison of two basically differently working polarization mode dispersion compensation techniques, the first-order compensator working in optical domain and an adjustable transversal filter working in electrical domain. Based on numerical simulations we show the efficiency of both equalizers in presence of first- and higher order PMD applying a transmission at 10 Gbit/s. Furthermore we show the applicability of spectral filtering for penalty signal extraction in presence of higher order PMD and give an overview of adaptive blind channel equalization.

Direct blind MMSE channel equalization based on second-order statistics

A family of new MMSE blind channel equalization algorithms based on second-order statistics are proposed. Instead of estimating the channel impulse response, we directly estimate the cross-correlation function needed in Wiener-Hopf filters. We develop several different schemes to estimate the cross-correlation vector, with which different Wiener filters are derived according to minimum mean square error (MMSE). Unlike many known sub-space methods, these equalization algorithms do not rely on signal and noise subspace separation and are consequently more robust to channel order estimation errors. Their implementation requires no adjustment for either single- or multiple-user systems. They can effectively equalize single-input multiple-output (SIMO) systems and can reduce the multiple-input multiple-output (MIMO) systems into a memoryless signal mixing system for source separation. The implementations of these algorithms on SIMO system are given, and simulation examples are provided to demonstrate their superior performance over some existing algorithms.

A gradient search interpretation of the super-exponential algorithm

This article reviews the super-exponential algorithm proposed by Shalvi and Weinstein (1993) for blind channel equalization. The principle of this algorithm-Hadamard exponentiation, projection over the set of attainable combined channel-equalizer impulse responses followed by a normalization-is shown to coincide with a gradient search of an extremum of a cost function. The cost function belongs to the family of functions given as the ratio of the standard l/sub 2p/ and l/sub 2/ sequence norms, where p>1. This family is very relevant in blind channel equalization, tracing back to Donoho's (1981) work on minimum entropy deconvolution and also underlying the Godard (1980) (or constant modulus) and the earlier Shalvi-Weinstein algorithms. Using this gradient search interpretation, which is more tractable for analytical study, we give a simple proof of convergence for the super-exponential algorithm. Finally, we show that the gradient step-size choice giving rise to the super-exponential algorithm is optimal.

Optimisation problem arising in blind channel equalisation

A formula for the solution of an optimisation problem arising in blind channel equalisation is presented and proved correct.

QR factorization based blind channel identification with second-order statistics

Most eigenstructure-based blind channel identification and equalization algorithms with second-order statistics need SVD or EVD of the correlation matrix of the received signal. In this paper, we address new algorithms based on QR factorization of the received signal directly without calculating the correlation matrix. This renders the QR factorization-based algorithms more robust against ill-conditioned channels, i.e., those channels with almost common zeros among the subchannels. First, we present a block algorithm that performs the QR factorization of the received data matrix as a whole. Then, a recursive algorithm is developed based on the QR factorization by updating a rank-revealing ULV decomposition. Compared with existing algorithms in the same category, our algorithms are computationally more efficient. The computation in each recursion of the recursive algorithm is on the order of O(m/sup 2/) if only equalization is required, where m is the dimension of the received signal vector. Our recursive algorithm preserves the fast convergence property of the subspace algorithms, thus converging faster than other adaptive algorithms such as the super-exponential algorithm with comparable computational complexities. Moreover, our proposed algorithms do not require noise variance estimation. Numerical simulations demonstrate the good performance of the proposed algorithms.

Unbiased blind adaptive channel identification and equalization

The blind adaptive equalization and identification of communication channels is a problem of important current theoretical and practical concerns. Previously proposed solutions for this problem exploit the diversity induced by sensor arrays or time oversampling, leading to the so-called second-order algebraic/statistical techniques. The prediction error method is one of them, perhaps the most appealing in practice, due to its inherent robustness to ill-defined channel lengths as well as for its simple adaptive implementation. Unfortunately, the performance of prediction error methods is known to be severely limited in noisy environments, which calls for the development of noise (bias) removal techniques. We present a low-cost algorithm that solves this problem and allows the adaptive estimation of unbiased linear predictors in additive noise with arbitrary autocorrelation. This algorithm does not require the knowledge of the noise variance and relies on a new constrained prediction cost function. The technique can be applied in other noisy prediction problems. Global convergence is established analytically. The performance of the denoising technique is evaluated over GSM test channels.

Recursive total least squares algorithm for single-user blind channel equalisation

The problem of blind channel identification/equalisation using second-order statistics or equivalent deterministic properties of the oversampled channel output has attracted consider-able attention. Deterministic blind subspace algorithms are particularly attractive because of their finite sample convergence property and because their solution can be obtained in closed form. Most subspace algorithms developed up until now, however, are based on block processing and have high computational and memory requirements. In the paper, adaptive techniques are used to lower the computational burden. A single-user direct symbol estimation algorithm is presented The first step in the algorithm consists of an adaptive matrix singular value decomposition for a (virtual) channel identification-type operation. A recursive total least squares algorithm is then used to recover the input symbols. The algorithm is able to track time-varying channels.

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.

Blind channel equalization

Signals that pass through a channel undergo various forms of distortion. One very common type of distortion is intersymbol interference, or ISI. ISI is caused by interference from the symbols that were transmitted before and after the given symbol. ISI-induced errors can cause the receiver to misinterpret the received samples. The equalizer is an important part of a modern digital communications receiver. It filters the received data to minimize the linear distortion induced during the transmission of the signal. (There is no nonlinear distortion.) This distortion occurs when the signal is transmitted through a channel. The channel is the physical medium that connects the transmitter and receiver. For example, the Earth's atmosphere is the medium in a wireless communications system. By sending the signal through the equalizer, the effects of ISI are counteracted; i.e. the receiver can make accurate symbol decisions based on the equalizer output. This article discusses the constant modulus algorithm, or CMA. This is the most popular algorithm used to blindly update equalizer coefficients.

Blind receivers for nonlinearly modulated signals in multipath

Classical blind channel equalization methods may not be suitable for systems with nonlinear modulation (e.g., continuous phase-modulated signals) because the input to the channel may not be i.i.d. due to the modulator's memory. Moreover, the nonlinear characteristics of the modulator further complicate the channel estimation task. In this correspondence, we derive blind maximum likelihood channel estimation algorithms for nonlinearly modulated signals using a hidden Markov model (HMM) formulation. The proposed method implements a stochastic maximum likelihood approach and is well suited for short data records appearing in TDMA systems. Cramer-Rao bounds are derived, and some illustrative simulations are presented.

Convergence analysis of the stop-and-go blind equalization algorithm

Unlike traditional trained channel equalizers, not much work has been done to theoretically characterize the convergence properties of blind channel equalizers due to their inherent nonlinearity. It is only recently that convergence properties of some well-known algorithms such as the generalised Sato algorithm (GSA) and the constant modulus algorithm (CMA) have been analytically derived. In this paper, the convergence properties of the stop-and-go algorithm proposed by Picchi and Prati (1987) are analyzed. The derived mean squared error and the coefficient trajectories are compared with simulation results to verify the validity of the analytical results.

Soft constraint satisfaction SCS blind channel equalization algorithms

The constant modulus adaptive blind equalization algorithms presented in this paper are shown to correspond to an error performance surface which is much improved upon that of existing algorithms such as the well-known constant modulus or Godard algorithm. Many undesirable local solutions ULSs are avoided by careful derivation. We use a deterministic optimization criterion with a soft constraint to obtain an update equation which contains a normalized gradient vector and a particular continuous non-linearity. This approach is extended to multiple constraints to yield faster converging algorithms. An autoregressive AR channel model is studied to demonstrate analytically the absence of a class of ULSs. Finally, the findings are verified experimentally for various AR and moving-average MA channels. © 1998 John Wiley & Sons, Ltd.

An "SVD+Viterbi" algorithm for multiuser adaptive blind equalization of mobile radio channels

A fully adaptive algorithm for multiuser blind channel equalization is presented. The algorithm is based on an adaptive matrix singular value decomposition (SVD) for a (virtual) channel identification type operation and the Viterbi algorithm for subsequent symbol detection. Unlike other blind multiuser detection schemes that have appeared in the literature, the present algorithm removes multiple access interference (MAI) by exploiting the finite alphabet property of the input signals together with channel coding redundancy. The latter is believed to be a crucial new ingredient for performance in the context of MAI suppression.

Blind channel identification and equalization with modulation-induced cyclostationarity

Recent results have pointed out the importance of inducing cyclostationarity at the transmitter for blind identification and equalization of communication channels. This paper addresses blind channel identification and equalization relying on the modulation-induced cyclostationarity, without introducing redundancy at the transmitter. It is shown that single-input single-output channels can be identified uniquely from output second-order cyclic statistics, irrespective of the location of channel zeros, color of additive stationary noise, or channel order overestimation errors, provided that the period of modulation-induced cyclostationarity is greater than half the channel length. Linear, closed-form, nonlinear correlation matching, and subspace-based approaches are developed for channel estimation and are tested using simulations. Necessary and sufficient blind channel identifiability conditions are presented. A Wiener cyclic equalizer is also proposed.

A Novel Blind Channel Identification and Equalisation AlgorithmBased on Maximum Likelihood

A novel blind non-decision directed maximum likelihood algorithm for fractionally-spaced nonminimum phase FIR channel identification and equalisation is presented. The algorithm results from using the low signal to noise approximation to the average of the likelihood function with respect to the transmitted data sequence. The channel estimation equation is derived in a closed form. The resulting algorithm has two distinct advantages. The first is that the channel estimates are asymptotically consistent, and the second is that the algorithm is computationally efficient since it only requires the calculation of one eignevector. Simulation results are presented to show the performance of the proposed algorithm.

Direct multiple-channel blind equalization

In this paper, the problem of blind channel identification and equalization is reviewed, and some recent results are presented.

Matrix outer-product decomposition method for blind multiple channel identification

Blind channel identification and equalization have attracted a great deal of attention due to their potential application in mobile communications and digital TV systems. In this paper, we present a new algorithm that utilizes second-order statistics for multichannel parameter estimation. The algorithm is simple and relies on an outer-product decomposition. Its implementation requires no adjustment for single- or multiple-user systems. This new algorithm can be viewed as a generalization of a linear prediction algorithm. It is capable of generating more accurate channel estimates and is more robust to overmodeling errors in channel order estimates. The superior performance of this new algorithm is demonstrated through simulation examples.

Transmitter induced cyclostationarity for blind channel equalization

Fractional sampling has received considerable interest recently as a means of developing blind equalization techniques without resorting to higher order statistics. Instead, cyclostationarity introduced at the receiver by fractional sampling is exploited. We show that simpler solutions are possible if cyclostationarity is introduced at the transmitter instead of the receiver. We propose specific coding and interleaving strategies at the transmitter that induce cyclostationarity and facilitate the equalization task. Novel batch and adaptive equalization algorithms are derived that make no assumptions on the channel zeros locations. Subspace methods are also proposed and, in the absence of noise, guarantee perfect estimation from finite data. Synchronization issues and bandwidth considerations are briefly discussed, and simulation examples are presented.

Filterbanks for blind channel identification and equalization

Multirate precoding using filterbanks induces cyclo-stationarity at the transmitter and guarantees blind identifiability of frequency selective communication channels with minimal decrease of information rate and without restrictions on zero locations. Finite impulse response (FIR) filterbank decoders are capable of equalizing blindly (and perfectly in the absence of noise) FIR channels without constraints on their zeros.

Least squares approach to fractionally spaced blind channel equalisation

A new approach to fractionally spaced blind equalisation of finite-duration impulse response (FIR) channels is proposed based on the method of least squares (LS) and the constant modulus algorithm (CMA). A nonlinear transformation of the equaliser parameters is formulated and solved for, so as to avoid the potential problems associated with fractionally spaced CMA (FS-CMA) when the channel input is correlated. The new algorithm has a fast convergence rate in comparison with FS-CMA and the resulting equaliser setting is invariant to channel input correlation so long as all finite-length subsequences of the channel input sequence occur with nonzero probability. Although the algorithm is based on the assumption of constant modulus channel inputs, its application to nonconstant modulus constellations such as M-ary quadrature amplitude modulation (QAM) is also illustrated. The issue of channel noise enhancement is studied in connection with subchannel zeros, and a method for reducing inflated equaliser norms is proposed by way of reduced rank matrix approximation. A modified recursive least squares implementation of the algorithm is simulated to demonstrate its superior performance vis-à-vis FS-CMA.