ABSTRACT

D(z1, z2) 6¼ 0 for zij j 1, i ¼ 1, 2 (23:2)

The amplitude response of the 2-D filter is expressed as

M(v1,v2) ¼ H(e jv1T, e jv2T) (23:3)

the phase response as

f(v1,v2) ¼ argH(e jv1T, e jv2T ) (23:4)

and the two group delay functions as

ti(v1,v2) ¼ df(v1,v2)dvi , i ¼ 1, 2 (23:5)

Equation 23.1 is the general form of transfer functions of the nonseparable numerator and denominator 2-D IIR filters. It can involve two subclasses, namely, the separable product transfer function

H(z1, z2) ¼ H1(z1)H2(z2)

¼ PN2

PM2 j¼0 a2iz

PM1 j¼0 b2iz

(23:6)

and the separable denominator, nonseparable numerator transfer function given by

H(z1, z2) ¼ PN2

PM1 j¼0 b2jz

(23:7)

The stability constraints for the above two transfer functions are the same as those for the individual two 1-D cases. These are easy to check and correspondingly the transfer function is easy to stabilize if the designed filter is found to be unstable. Therefore, in the design of the above two classes, in order to reduce the stability problem to that of the 1-D case, the denominator of the 2-D transfer function is chosen to have two 1-D polynomials in z1 and z2 variables in cascade. However, in the general formulation of nonseparable numerator and denominator filters, this oversimplification is removed. The filters of this type are generally designed either through transformation of 1-D filters, or through optimization approaches, as is discussed in the following.