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Sysotolic Array Essay

8075 words - 33 pages

IEEE T R A N S A C T I O N S O N I N F O R M A T I O N T H E O R Y . VOL. 37, N O . I , J A N U A R Y 1991


A Class of Least-Squares Filtering and Identification Algorithms with Systolic Array Architectures
Seth Z. Kalson, Member, IEEE, and Kung Yao,
Member, IEEE
Abstract -A unified approach is presented for deriving a large class of new and previously known time and order recursive least-squares algorithms with systolic array architectures, suitable for high throughput rate and VLSl implementations of space-time filtering and system identification problems. The geometrical derivation given here is unique in that no assumption is made concerning the rank of the sample data ...view middle of the document...

Furthermore, we have restricted T to be finite so as to [2] problem have found widespread applications to such avoid dealing with infinite dimensional Hilbert spaces. This topics as channel equalization, adaptive antenna arrays, noise is not really restrictive from a practical engineering point of canceling, spectrum estimation, and system identification, to view since T can be thought of as some arbitrarily large but name just a few. In the classical discrete time Wiener filter- finite integer. ing problem, given two random discrete time sequences The filtering and identification problems are intimately denoted by y ( t ) and x ( t ) , we desire the optimum finite connected. This can be made precise by constructing an impulse linear filter G that minimizes E { l y ( t > - G * x ( t ) 1 2 ) , isomorphism between suitably chosen spaces so that time where E denotes the statistical ensemble averaging operator recursive algorithms can be easily derived for the filter and G . x ( t ) denotes time-domain filtering. In this paper, we weights g f ( t ) [3]-[5]. However, such an approach will not be will be interested in generalizing this problem where y ( t ) taken here, but rather, we shall primarily be concerned with and x , ( t ) , i = 1,. . . ,n are deterministic complex-valued dis- filtering algorithms since these possess a highly concurrent crete time sequences (or realizations of random sequences) architecture. Once such filtering algorithms are obtained, it with time indices t = 1,2;. * , T . For each time index, let will be shown that the filter weights may be found with g ; ( t ) , i = 1 . .,ti denote a set of n time varying filter weights additional processing, thus allowing one to use the same ; that satisfies the exponentially weighted, growing memory systolic array for both filtering and identification. least-squares criterion As old as the least-squares problem is, new algorithms continue to be developed. This is in part due to the pressing P'- Y ( 7 ) gj'(t)Xr(T) needs of modern communication, radar, and sonar systems in 7=1 which real-time signal processing is important. As a result, the signal processing community has been interested in least-squares algorithms that are either fast [31, [41, [61-[131, in the sense that the total number of numerical computations where p is a real-valued weight factor satisfying 0 < p I 1. performed each time update is in some sense minimized, or For convenience, we shall make the following two classifica- that can take advantage of concurrent processing, such as systolic arrays [13]-[21]. Henceforth, algorithms for systolic arrays will be referred Manuscript received September 16, 1986; revised July 17, 1989. This work was supported in part by the NASA/AMES Contract No. NAG-2- to as systolic algorithms. The differences between fast and 304 and the NSF grant NCR-8814407. systolic algorithms are not always clearly defined. For examS . Kalson is with MIT Lincoln Laboratory, ANI, P.O. Box 73, Lexing- ple,...

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