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

The problem of implementing a high sampling rate transversal form adaptive filter is investigated. A highly pipelined systolic-type alternative to the conventional LMS adaptive filter is presented. The proposed system consists of a linear array of identical processing modules specifically suited to the computational requirements of the delayed LMS algorithm. The resulting adaptive filter structure can accommodate very high sampling rates, which are independent of the filter order. The performance of the system is analyzed in terms of computational speedup and maximum sampling rate, and the effect of adaptation delay on algorithm convergence is addressed. >

A high sampling rate delayed LMS filter architecture

Past methods for mapping the least-mean-square (LMS) adaptive finite-impulse-response (FIR) filter onto parallel and pipelined architectures either introduce delays in the coefficient updates or have excessive hardware requirements. We describe a hardware-efficient pipelined architecture for the LMS adaptive FIR filter that produces the same output and error signals as would be produced by the standard LMS adaptive filter architecture without adaptation delays. Unlike existing architectures for delayless LMS adaptation, the new architecture's throughput is independent of the filter length.

A pipelined LMS adaptive FIR filter architecture without adaptation delay

A variation of the least means squares (LMS) algorithm, called the delayed LMS (DLMS) algorithm is ideally suited for highly pipelined, adaptive digital filter implementations. In this paper, we present an efficient method to determine the delays in the DLMS filter. Furthermore, in order to achieve fully pipelined circuit architectures for FPGA implementation, we transfer these delays using retiming. The method has been used to derive a series of retimed delayed LMS (RDLMS) architectures, which allow a 66.7% reduction in delays and 5 times faster convergence time thereby giving superior performance in terms of throughput rate when compared to previous work. Three circuit architectures and three hardware shared versions are presented which have been implemented using the Virtex-II FPGA technology resulting in a throughput rate of 182 Msample/s.

High Speed FPGA-Based Implementations of Delayed-LMS Filters

A systolic array design for an adaptive filter is presented. The filter is based on the least-mean-square algorithm, but due to the problems in implementation of the systolic array, a modified algorithm, a special case of the delayed LMS (DLMS), is used. The DLMS algorithm introduces a delay in the updating of the filter coefficients. The convergence and steady-state behavior of the systolic array are analyzed. It is shown that the performance of the systolic array is similar to that of a conventional LMS implementation for a wide range of practical conditions. >

A systolic array realization of an LMS adaptive filter and the effects of delayed adaptation

Adaptive filters, employing the transversal filter structure and the least mean square (LMS) adaptation algorithm, or its variations, have found wide application in data transmission equalization, echo cancellation, prediction, spectral estimation, on-line system identification, and antenna arrays. Recently, in response to requirements of fast start-up, or fast tracking of temporal variations, fast recursive least squares (FRLS) adaptation algorithms for both transversal and lattice filter structures have been proposed. These algorithms offer faster convergence than is possible with the LMS/ transversal adaptive filters, at the price of a five-to-tenfold increase in the number of multiplications, divisions, and additions. Here we discuss architectures and implementations of the LMS/transversal, fast-converging FRLS filter, and lattice filter algorithms which minimize the required hardware speed. We show how each of these algorithms can be partitioned so as to be realizable with an architecture based on multiple parallel processors.

V. Lawrence

Papers

Multiprocessor Implementation of Adaptive Digital Filters