Adaptive Signal Processing

Bandwidth-efficient data transmission over telephone and radio channels is made possible by the use of adaptive equalization to compensate for the time dispersion introduced by the channel Spurred by practical applications, a steady research effort over the last two decades has produced a rich body of literature in adaptive equalization and the related more general fields of reception of digital signals, adaptive filtering, and system identification. This tutorial paper gives an overview of the current state of the art in adaptive equalization. In the first part of the paper, the problem of intersymbol interference (ISI) and the basic concept of transversal equalizers are introduced followed by a simplified description of some practical adaptive equalizer structures and their properties. Related applications of adaptive filters and implementation approaches are discussed. Linear and nonlinear receiver structures, their steady-state performance and sensitivity to timing phase are presented in some depth in the next part. It is shown that a fractionally spaced equalizer can serve as the optimum receive filter for any receiver. Decision-feedback equalization, decision-aided ISI cancellation, and adaptive filtering for maximum-likelihood sequence estimation are presented in a common framework. The next two parts of the paper are devoted to a discussion of the convergence and steady-state properties of least mean-square (LMS) adaptation algorithms, including digital precision considerations, and three classes of rapidly converging adaptive equalization algorithms: namely, orthogonalized LMS, periodic or cyclic, and recursive least squares algorithms. An attempt is made throughout the paper to describe important principles and results in a heuristic manner, without formal proofs, using simple mathematical notation where possible.

Adaptive equalization

Fast transversal filter (FTF) implementations of recursive-least-squares (RLS) adaptive-filtering algorithms are presented in this paper. Substantial improvements in transient behavior in comparison to stochastic-gradient or LMS adaptive algorithms are efficiently achieved by the presented algorithms. The true, not approximate, solution of the RLS problem is always obtained by the FTF algorithms even during the critical initialization period (first N iterations) of the adaptive filter. This true solution is recursively calculated at a relatively modest increase in computational requirements in comparison to stochastic-gradient algorithms (factor of 1.6 to 3.5, depending upon application). Additionally, the fast transversal filter algorithms are shown to offer substantial reductions in computational requirements relative to existing, fast-RLS algorithms, such as the fast Kalman algorithms of Morf, Ljung, and Falconer (1976) and the fast ladder (lattice) algorithms of Morf and Lee (1977-1981). They are further shown to attain (steady-state unnormalized), or improve upon (first N initialization steps), the very low computational requirements of the efficient RLS solutions of Carayannis, Manolakis, and Kalouptsidis (1983). Finally, several efficient procedures are presented by which to ensure the numerical Stability of the transversal-filter algorithms, including the incorporation of soft-constraints into the performance criteria, internal bounding and rescuing procedures, and dynamic-range-increasing, square-root (normalized) variations of the transversal filters.

Fast, recursive-least-squares transversal filters for adaptive filtering

Implantable devices and methods for delivering drugs and other substances to locations within the body of a human or animal subject to treat or diagnose sinusitis and a variety of other disorders. The invention includes implantable substance delivery devices that comprise reservoirs and barriers that control the rate at which substances pass out of the reservoirs. The delivery devices may be advanced into the body using guidewires, catheters, ports, introducers and other access apparatus. In some embodiments the delivery devices may be loaded with one or more desired substance before their introduction into the body. In other embodiments the delivery devices are loaded and/or reloaded with a desired substance after the delivery device has been introduced into the body.

Implantable devices and methods for delivering drugs and other substances to treat sinusitis and other disorders

Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described. >

Adaptive polynomial filters

A sequence of vectors {x(t)} with dimension N is given, such that x(t+1) is obtained from x(t) by introducing p new elements, deleting p old ones, and shifting the others in some fashion. The sequence of vectors $ is sought. This paper presents a method of calculating these vectors with proportional-to-Np operations and memory locations, in contrast to the conventional way which requires proportional-to-N 2 operations and memory locations. A non-symmetric case is also treated.

Fast calculation of gain matrices for recursive estimation schemes

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.

Application of Fast Kalman Estimation to Adaptive Equalization

A receiver for a quadrature amplitude modulated data signal impaired by linear and nonlinear distortion, phase jitter and additive noise includes circuitry which compensates for these impairments. In particular, the receiver includes a processor (FIG. 1 44; FIG. 2, 44') which subtracts a feedback nonlinear signal (FIG. 1, D(n); FIG. 2, D'(n)) from each sample of the received signal, either prior or subsequent to demodulation, providing compensation for nonlinear intersymbol interference. The feedback nonlinear signal subtracted from each sample is comprised of a weighted sum of products of individual data decisions and/or the complex conjugates of data decisions, each such product, in turn, being multiplied by a predetermined harmonic of the carrier frequency. In an illustrative embodiment, compensation for second- and third-order intersymbol interference is provided by including two- and three-multiplicand weighted products in the feedback nonlinear signal. Weighting coefficients for each product are adaptively updated in a decision-directed manner.

Feedback nonlinear equalization of modulated data signals

An adaptive feedforward transversal equalizer acting jointly with a data decision-directed demodulating carrier recovery system to achieve substantially jitter-free passband equalization of a suppressed-carrier coherent data signal without the use of pilot tones is improved by adding a feedback transversal filter section operating on past demodulated data symbols. The weighted output of the feedback section is subtracted from the demodulated output output of the feedforward section to provide signals from which data decisions are made and common error signals are derived. Tap-gain coefficients for both feedforward and feedback filter sections and phase angle for the demodulating carrier recovery system are adaptively updated in accordance with the common error signal. Severe frequency offset and phase jitter are simultaneously compensated.

David Duncan Falconer

Papers

Fast calculation of gain matrices for recursive estimation schemes

Application of Fast Kalman Estimation to Adaptive Equalization

Application of Fast Kalman Estimation to Adaptive Equalization

Feedback nonlinear equalization of modulated data signals

Joint decision feedback equalization and carrier recovery adaptation in data transmission systems