Heuristic Optimal Design of Multiplier-less Digital Filter - Advances in Heuristic Signal Processing and Applications - page 252

filters and the desired frequency response is minimal. Moreover, all the poles of
H(z) will be placed in the disk D(α,r) , i.e., the IIR filters are D(α,r) -stable.

11.5.2 Stability Criterion for an IIR Filter

The main difference between the design of an FIR filter and an IIR filter is that
the stability problem should be considered in the design of an IIR filter while it is
unnecessary in the design of an FIR filter, since an FIR filter is always stable as we
have mentioned in Sect. 11.4 . Hence, this section will introduce a stability criterion
for the IIR filter design. Before proceeding, the following definitions are introduced
to help the description of this section.

Definition 11.1 A polynomial d(z) is P -stable if all solutions of the equation
d(z) = 0 lie inside the unit circle [ 15 ].

Definition 11.2 A polynomial d(z) is PD(α,r) -stable if all solutions of the equa-
tion d(z) = 0 are within the disk D(α,r) centered at α with radius r , in which r> 0

and | α |+ r< 1[ 15 ].

Definition 11.3 Let the transfer function of an IIR filter be described as H(z) =

k = 0 b k z − k

= k = 0 a k z − k ,is P -stable, then
the IIR filter is stable. Moreover, if a(z) is PD(α,r) -stable, then the IIR filter is
D(α,r) -stable [ 15 ].

k = 0 a k z − k . If the denominator of the H(z) , a(z)

In the following paragraph, a theorem which is useful in the evolution of GA for
design of a robust stable IIR filter is introduced. First, we introduce a useful theorem
as follows:

Theorem 11.1 If f(z)is analytic in a bounded domain ψ and continuous in the
closure of ψ , then

| f(z) | takes its maximum on the boundary of ψ [ 16 ].

The following theorem will provide a boundary test criterion for the PD(α,r) -
stability of a polynomial.

= k = 0 a k z − k . If the following in-
equality ( 11.20 ) is satisfied , then all the solutions of d(z)

Theorem 11.2 Consider the polynomial d(z)

0 will lie inside a disk
D(α,r) , i . e ., the polynomial d(z) will be PD(α,r)-stable with

=

| α |+ r< 1 and

| α |≤ r [ 15 ]

α − k

n

a 0 re − jθ

a k

+ < 1 , ∀ θ ∈[ 0 , 2 π ] . (11.20)

k = 1