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

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*:

The recursion begins by setting the th reflection coefficient to . If , 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
, the polynomial order is decremented by 1
to yield
as follows (recall that
is
*monic*):

Next is set to , and the recursion continues until is reached, or is found for some .

Whenever
, 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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