Simplified gradient calculation in adaptive IIR lattice filters

A simplified algorithm is presented for computing the gradient in adaptive IIR lattice filters. For a filter with N zeros and N poles, this algorithm requires only order Ncomputations and it is shown to be completely equivalent to the previously proposed formulation which has a complexity of order N Z . Manuscript received May 7, 1990; revised October 4 , 1990. This work was supported by the Spanish Government under Grant PRONTIC 105/88. The authors are with ETSE Telecomunicacio, Department of Signal Theory and Communications, Universitat Politkcnica de Catalunya, 08080 Barcelona, Spain. IEEE Log Number 914471 1. I . INTRODUCTION The lattice realization of adaptive IIR filters has been shown to be an important alternative to the direct form [I]-[4]. It is well known that lattice forms have excellent finite-precision properties. Furthermore, in adaptive applications, its primary advantage is that stability can be controlled with essentially no computation, while in the direct form, stability monitoring is either computationally expensive or too restrictive in the coefficient space [3]. The complexity of the gradient has always been the major drawback of the lattice realization. In the direct form it is possible to calculate the gradient with a simple algorithm of order N , but in the lattice form, the formulation originally proposed by [ I ] requires order N 2 computations. Nevertheless, in this correspondence it is shown that the lattice form gradient can also be calculated with a simplified algorithm, thus making this structure more competitive with the direct o r other IIR realizations. A first effort to simplify this result was reported in [ 5 ] , where a simplified algorithm was also derived, but it was based on an approximation which was shown to cause convergence problems. The algorithm we have derived does not neglect any term, and, therefore, it is completely equivalent to the gradient proposed in [ l ] , although it has only a complexity of order N. 11. IIR LATTICE FILTER The adaptive IIR lattice filter (Fig. 1) consists of a feedback lattice structure characterized by the reflection coefficients { A ( n ) } and a feedforward structure characterized by the coefficients { z ) , ( n ) ) . The forward and backward residuals of each stage are calculated as ./\+I@) = x ( n ) ( l a ) f o r i = , . . . , N d o L(n) = A + , ( n ) k , (n) b,(n) (Ib) (IC) b,+,(n + 1) = b,(n) + k , ( n ) A ( n )