The nonuniform discrete Fourier transform and its applications in filter design. I. 1-D

The nonuniform discrete Fourier transform (NDFT) of a sequence of length N is defined as samples of its z-transform evaluated at N distinct points located arbitrarily on the z-plane. The NDFT reduces to the conventional discrete Fourier transform (DFT) when these points are located on the unit circle at equally spaced angles. The flexibility offered by the NDFT in choosing the sampling points leads to a variable spectral resolution that can be controlled by the user. The NDFT is applied to nonuniform frequency sampling design of 1-D FIR filters. This method produces nearly optimal equiripple 1-D filters with greatly reduced design times as compared with the Parks-McClellan algorithm. Comparisons with filters designed by other methods are presented to demonstrate the effectiveness of the proposed method.