### Step-Down Procedure

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].

**Next Section:**

Testing Filter Stability in Matlab

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Computing Reflection Coefficients to Check Filter Stability