Block adaptive IIR filters using preconditioned conjugate gradients for orthogonalization

The method of preconditioned conjugate gradients (PCG) is introduced as an accelerator for simple IIR algorithms to significantly increase their rate of convergence without dramatically adding to their complexity. This paper develops a block Newton-type adaptive IIR digital filter with complexity O(log(N)). The proposed algorithm exploits a structured approximation to the Hessian which permits the application of a fast preconditioned conjugate gradient optimization method. In this novel formulation, the identification problem of the IIR coefficients separates into two subproblems, each of which may be solved by application of fast adaptive FIR techniques. Present IIR algorithms require greater computational cost, or converge more slowly. It is the adoption of fast PCG which permits the development of an O(log(N)) adaptive algorithm. The PCG method manipulates an approximation of the Hessian matrix to form an orthogonalizing update term for the IIR LMS algorithm. Rapid convergence follows, and the method is robust with respect to fixed-point instability. The use of preconditioned conjugate gradients in the Gauss-Newton update leads naturally to the application of the planar least squares inverse to bound the poles of the adaptive system by projecting an unstable denominator onto a stable polynomial. This technique is invoked whenever the output of the adaptive filter exceeds a certain threshold. This approach provides a computationally efficient means to ensure robust IIR adaptive behavior