The nonuniform discrete Fourier transform and its applications in filter design .2. 2-D

The concept of the nonuniform discrete Fourier transform (NDFT) is extended to two dimensions to provide a basic framework for nonuniform sampling of 2-D sequences in the frequency domain. The 2-D NDFT of a sequence of size N-1 x N-2 is defined as samples of its 2-D z-transform evaluated at N1N2 distinct points located in the 4-D (z(1), z(2)) space. These points are chosen appropriately so that the inverse transform exists. We discuss two special cases in which the choice of the sampling points is constrained so that the 2-D NDFT matric is guaranteed to be nonsingular, and the number of operations required for computing its inverse is reduced, The 2-D NDFT is applied to nonuniform frequency sampling design of 2-D finite-impulse-response (FIR) filters. Nonseparable biters with good passband shapes and low peak ripples are obtained. This is illustrated by design examples, in which 2-D filters with various shapes are designed and compared with those obtained by other existing methods.