Step-Down Procedure

Let denote the th-order denominator polynomial of the recursive filter transfer function :

$\displaystyle A_N(z) \isdef 1 + a_{N,1}\,z^{-1}+ a_{N,2}\, z^{-2}+ \cdots + a_{N,N-1}\,z^{-(N-1)} + a_{N,N}\,z^{-N} \protect$

(9.4)

We have introduced the new subscript

because the step-down procedure is defined recursively in polynomial order. We will need to keep track of polynomials orders between 1 and

In addition to the denominator polynomial , we need its flip:

$\displaystyle \tilde{A}_N(z)$	$\displaystyle \isdef$	$\displaystyle z^{-N}A_N(1/z)$
	$\displaystyle =$	$\displaystyle a_{N,N} + a_{N,N-1}\,z^{-1}+ a_{N,N-2}\, z^{-2}+ \cdots$
		$\displaystyle \quad + a_{N,2}\,z^{-(N-2)} + a_{N,1}\,z^{-(N-1)} + z^{-N} \protect$	(9.5)

The recursion begins by setting the

th reflection coefficient to $k_N = a_{N,N}$ . If $\left\vert k_N\right\vert\geq 1$ , the recursion halts prematurely, and the filter is declared unstable. (Equivalently, the polynomial

is declared non-minimum phase, as defined in Chapter 11.)

Otherwise, if $\left\vert k_N\right\vert<1$ , the polynomial order is decremented by 1 to yield $A_{N-1}(z)$ as follows (recall that is monic):

$\displaystyle A_{N-1}(z) = \frac{A_N(z) - k_N \tilde{A}_N(z)}{1-k_N^2} = \frac{A_N(z) - z^{-N} k_N A_N(1/z)}{1-k_N^2} \protect$

(9.6)

Next $k_{N-1}$ is set to $a_{N-1,N-1}$ , and the recursion continues until $k_1=a_{1,1}$ is reached, or $\left\vert k_i\right\vert\geq 1$ is found for some .

Whenever $\left\vert k_m\right\vert=1$ , the recursion halts prematurely, and the filter is usually declared unstable (at best it is marginally stable, meaning that it has at least one pole on the unit circle).

Note that the reflection coefficients can also be used to implement the digital filter in what are called lattice or ladder structures [48]. Lattice/ladder filters have superior numerical properties relative to direct-form filter structures based on the difference equation. As a result, they can be very important for fixed-point implementations such as in custom VLSI or low-cost (fixed-point) signal processing chips. Lattice/ladder structures are also a good point of departure for computational physical models of acoustic systems such as vibrating strings, wind instrument bores, and the human vocal tract [81,16,48].

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Previous: Computing Reflection Coefficients to Check Filter Stability
Next: Testing Filter Stability in Matlab

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Introduction to Digital Filters
    Pole-Zero Analysis
       Stability Revisited
          Step-Down Procedure