GB1520148A - Adaptive lattice filter - Google Patents

Abstract

1520148 Approximating STANDARD TELEPHONES & CABLES Ltd 9 Oct 1975 41432/75 Heading G4A [Also in Division H4] A signal processing arrangement, referred to as a filter, includes means to construct a sequence h t , i=0, 1, . . ., of signals orthogonal under the undefined mean operator <À>, i.e. <h i h j >=0 for all i#j, from a signal x(t) and its successively delayed versions x r =x(t-rSt), r=1, 2, . . ., and means using the signals h i to construct a minimum mean-squared error approximation of a signal z(t) based upon x and its delayed versions x r , the approximation having the property that its residual #z(0, r)=z-#z(0, r) is uncorrelated with x and all its delayed versions x 1 , x 2 , . . ., x r in the sense that <#z(0,r)x t > =0, i=0, 1, ..., r, where x 0 =x(t). The filter is of particular use, Fig. 8, in eliminating echo in long distance telephony, in which application the filter would be located near a caller and would receive (i) as its x signal speech from the caller, and (ii) as its z signal speech plus echo from the far end, and would produce as its output the residual z(0,r) which would be independent of all echo produced by up to r-units delay of the original speech. The signals h r are defined in the following A manner. Define x r (k,l) to be the minimum mean-squared error approximation to x r based upon a linear combination of the signals x k , x k+1 , . . ., x l , for either r<k or r>l, then the signals h are defined as # A where x r (0,r-1) =x r -x r (0, r - 1), r#1. Assuming that the signal x satisfies the stationarity condition:- <x i x j >= <x r x s > for | i-j | = | r-s |, all i, j, r, s, #0 it then follows that the h r are orthogonal. As described, the signals h r and the residuals z(0,r) are evaluated sequentially using the following recurrence relations:- in which These equations are implemented using multipliers, subtractors and delays as indicated in Fig. 5, the delay units D being used to derive #x r+1 (1,r) from #x r (0,r-1). The coefficients α r , # r and γ r (and, in fact, with the stationarity condition (# r =γ r ) can either be derived by computing elements C, Fig. 4, stage-by-stage using the formulae quoted above directly, or they can be derived, in a simple manner, from the autocorrelation function of x and the cross-correlation function of x and z, evaluated directly from the input signals. The following observations provide some insight into the recurrence relations. The approximation #x(0,r- 1) is of the form and #z(0,r) is clearly obtained from it by adding some function of x r . A simple multiple of x r itself will not do, however, since this will in general disturb the coefficients A i already derived. However, if an estimate #x r (0,r - 1) of x r based upon the previous delayed signals x 0 , x 1 , . . ., x r-1 is formed, then its residual #x r (0,r-1)=x r -#x r (0,r-1) will be independent of all the previous signals, and a multiple of #x r (0,r-1), namely #α,x r (0,r-1), can be added to #z(0,r-1) to form #z(0,r). Thus set and choose r to minimize <#z<SP>2</SP>(0,r)>. This gives one of the above quoted recurrence relationships, and the remainder can be supported by similar arguments.