## 12.4 DERIVATION OF ONE-MULTIPLIER LATTICE FILTER

By choosing *s*_{i} = 1 − _{i}k_{i}in (12.36), the Schur algorithm can be expressed as

*Fig. 12.8* (a) Implementation of (12.52) and (b) implementation of *H*_{N}(*z*) = **Ψ**_{N}(*z*).

*Fig. 12.9* A 3rd-order FIR lattice filter in Example 12.3.3.

where _{i} is called a sign parameter and is either 1 or −1. Consider an *N*-th order IIR transfer function *H*_{N}(*z*) = *N*_{N}(*z*)*/D*_{N}(*z*). Then, initialize the *N*-th order Schur polynomial ∧_{N}(*z*) as ∧_{N}(*z*) = *D*_{N}(*z*). From ∧_{N}(*z*), form the polynomial ∧_{N−1}(*z*) by the Schur algorithm as

where

With *i = N*,

From (12.56),

Using (12.54) and (12.57 ...