VLSI Digital Signal Processing Systems: Design and Implementation

12.2    SCHUR ALGORITHM

The Schur algorithm was originally used to test if a power series is analytic and bounded in the unit disk [3]. If an N-th order polynomial Φ_N(z) has all zeros inside the unit circle, N +1 polynomials {Φ_i(z), i = N, N − 1, ···, 0} can be generated by the Schur algorithm. One of the most important properties of the Schur algorithm is that these N + 1 polynomials are orthonormal to each other and can be used as orthonormal basis functions to expand any N-th order polynomial. This orthonormality of the Schur algorithm has been exploited to synthesize various types of lattice filters.

To illustrate the orthonormality of the Schur algorithm, we propose an inner product formulation, which is based on the power computation at an internal node of a filter structure. Although the inner product formulation includes complex integration in the definition, no actual complex integration is needed for the evaluation of the inner product due to some useful properties of the inner product.

12.2.1    Computation of Schur Polynomials

The Schur polynomial is a polynomial that does not have zeros on or outside the unit circle. Therefore, the denominator of a stable IIR filter is a Schur polynomial. Let a real polynomial with all zeros inside the unit circle be defined by

Then, initialize the N-th order Schur polynomial Φ_N(z) as

where

From Φ_N(z), form the polynomial Φ_N−1(z