FIR Digital Filter Design

FFT processors implement long FIR filters more efficiently than any other method, thanks to the speed of the FFT algorithm and the Fourier convolution theorem (§2.3.5). The convolution theorem states that the convolution of an input signal $x$ with a filter impulse response $h$ is given by the inverse DTFT of the signal's spectrum $X(\omega)$ and the filter's frequency response $H(\omega)$. Of course, in practice the DTFT is used in sampled form, replacing it with a (zero-padded) FFT. To make the most of FFT processors for FIR filter implementation, we need flexible ways to design all kinds of FIR filters for use in such FFT processors.

This appendix provides a starting point in the area of FIR digital filter design. The so-called ``window method'' for FIR filter design is discussed in some detail, and it is compared with the optimal Chebyshev method. Other methods, such as least-squares, are discussed briefly to provide some perspective. Tools for FIR filter design in both Octave and the Matlab Signal Processing Tool Box are listed where applicable. For further information on digital filter design, see the documentation for the Matlab Signal Processing Toolbox and/or [182,244,200,176,228].