On the Use of Discrete Prolate Spheroidal Windows for Frequency Selective Filter Design

The FIR filter design problem is considered with respect to a combined integrated squared error-Chebyshev error criterion. Truncation of the DTFT of the desired frequency response with tapered windows is introduced and the discrete prolate spheroidal window is shown to be a good window for this purpose, due to its optimal mainlobe width-sidelobe energy tradeoff. The discrete prolate spheroidal window is shown to be a scalar multiple of the zeroorder discrete prolate spheroidal sequence, and a commutable matrix reformulation for numerically well-conditioned computation is given. Discrete prolate spheroidal window parameters are related to frequency selective filter parameters through empirical study. Integrated squared error-Chebyshev error tradeoffs for filters designed with discrete prolate spheroidal windows are compared with other window methods, with the optimal constrained least squares method, and with the optimal Chebyshev method. It is found that the discrete prolate spheroidal window method is suboptimal, but is best among window methods. Finally, a suggestion for the use of discrete prolate spheroidal windows for spectral analysis is provided. MATLAB code is included in the appendices.