Two-dimensional IIR filters

A linear 2-D IIR digital filter can be characterized by its transfer function [Formula] where the sampling period T_i = 2π/ω_si for i = 1, 2 with ω_si and the sampling frequencies a_ij and b_ij are real numbers known as the coefficients of the filter. Without loss of generality we can assume M₁ = M₂ = N₁ = N₂ = M and T₁ = T₂ = T. Designing a 2-D filter is to calculate the filter coefficients a_ij and b_ij in such a way that the amplitude response and/or the phase response (group delay) of the designed filter approximates to some ideal responses while maintaining the stability of the designed filter. The latter requires that [Formula] The amplitude response of the 2-D filter is expressed as [Formula] the phase response as [Formula] and the two group delay functions as [Formula] 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 [Formula] and the separable denominator, nonseparable numerator transfer function given by [Formula] 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.