The poles of symmetric linear prediction models lie on the unit circle

Consequently, we can replace (20) by Then (21) permits us to substitute aY*(n)-= y*(n-k) + C a,(n) N-I ay*(n-m) $&) = y f (n-k), a a R k (n) a a R k (n-m> (22a) m = 1 $b,(n) nf(,-k), Notice that these equations are recursive in the partial derivatives since terms on the right-hand side correspond to delayed versions of the left-hand side. From (13), the component of r) corresponding to ak (which we denote 1 1 3 is Recursive expressions for the other components of r) and $, denoted by qbk, +ak, and $&, are obtained in a similar way; they are N-1 = y(n-k) + c $&-m) (25) m = 1 N-I = x(n-IC) + C a:(n) $bk(n-m). (26) Observe that (23) and (24) do not depend upon the input x or the output y. Consequently, they correspond to unforced difference equations. Since vUk and r)bk are initially zero, they will remain zero for all n. (If they were initially nonzero, then they would decay to zero because we have assumed that (1) is stable.) We will therefore assume that vak and qbk are precisely zero for all n. This leads to the result of (1 5). From the forced difference equations of (25) and (26), we can compactly write $ as m = 1 We can therefore replace (27) with $(n) = [ y f (n-1). . y f (n-N + 1) (29) x f (n) * * * x f (n-M + l)]T where, from (25), (26), and (28), we have The superscript f indicates that y and x correspond to filtered versions of y and x, respectively. The resuiting simplified GN algorithm is thus (18) and (19) coupled with (29) and (30), which clearly requires less computation and storage than that of (27). 11. CONCLUSION We have presented a Gauss-Newton (GN) algorithm for adap-tive IIR filters with complex coefficients. Although the gradient estimate appears to have two separate components [see (13) and (14)], it was shown that one component is essentially zero. Consequently , the complex-coefficient GN algorithm is a straightforward generalization of the real-coefficient GN algorithm. The algorithm is stable provided the pole polynomial o f the adaptive filter is kept minimum phase after each coefficient update. This requires the same stability-checking and pole-projection methods used by the real-coefficient GN algorithm [5]. 1 '(n) = (1-A*(n, z-') Abstract-Symmetric linear prediction models have their …