Known Input Power Spectrum in Adaptive L.M.S. and A. G. Algorithms

SIGNAL PROCESSING 111: Theories and Applications l. T. Young et al. (editors) Elsevier Science Publishers B.V. (North-Holland) © EURASIP, 1986 N2.2 KNOWN INPUT POWER SPECTRUM IN ADAPTIVE L.~1.S. AND A.G. ALGORITHMS Gregori Vázquez, Antoni Gasull and Miguel A. Lagunas. E.T.S. Ingenieros de Telecomunicación. U.P.C. C/ Jordi Girona Salgado, s/n. 08034 Barcelona - Spain. ABSTRACT. This work deals with the use of previous or colateral information to improve the behaviour of adaptive algorithms. The study is made on gradient-baseq methods due te the relatively simple and good performances that they use to exhibit. This paper shows that the complete knowledge of the data at the input of the adaptive filter (and in consequence of its autocorrelation matrix and its inverse) can be used to modify the classic L.M.S. algorithm leading to new expressions for the gradient and for the optimum 1 step size 1, alternative, in sorne cases, to the Powell expression. Finally, the description is completed with the comparison between the variation ranges and VLSI implementation cost for this two optimum 1 step size 1 values anda natural generalization set of parameter is obtained. l. INTRODUCTION For the sake of simplicity, let us focuss the classic problem of Wiener filtering. Two possible alternatives can be adopted. The first one is the direct use of the optimum Wiener equation and the other is that an adaptive approach could be better under actual situations, where finite arithmetics and no-stationary conditions are used to be imposed. If we adopt the second possibili ty, the question is how to use all the previous or colateral information available in a given adaptive algorithm. From our point of view, there would be two possible Choices to reflect these additional information in an adaptive squeme with a quadratic objective. They are the following: a) To use it to estimate better the parameters or associated functions involved in the adaptive algorithm. b) To include the colateral information as constrains or just in the minimization process. This work is driven in both senses. The first one is the most obvious and will be used only to improve the gradient estimate. On the other hand, the second ene is not so direct as the previous, and, in general, i t will try to modify the whole structure into the adaptive scheme to satisfy the constrains or the pursued error minimization criterion. 965 Thus, althrough it seems an atractive possibility, the designer will have to pay attention because, as a matter of fact, often the structure obtained will need a very intensive compu tation. In our case, only an optimum value for the step size will be searched, keeping the usual adaptive scheme. 2. REVIEW OF THE MINIMUM M.S.E. LINEAR FILTERING: 111,121,131, 141, Let's consider the general scheme given in the figure l. The objective is to minimize the mean square error (m.s.e.) between a reference signa! y(n) and an estímate of this signa! at the output of a Q arder F.I.R. filter defined by the coefficient vector W. Thus, given an input data signa! x(n), we dispose of a data vector X and the desired estimation: ..., y(n) = x T.w = wT.x (2.1) -n - - -n where: X T = (x(n),x(n-l), ••• ,x(n-Q+l) ..., T ~ = (w(O),w(1), ••• ,w(Q-1)) The weight vector will be chosen so that the M.S.E. is minimum: 2 ~ 2 T 2 € = E((y(n)-y(n)) )=E((y(n)-!. .~) ) (2,2) and developing the expression: 2 € E(y2 (n))-E(y(n)X T)W__, - - WTE(y(n)X )+WTR W - -n-xx- (2. 3)