Fast transform filters, implemented in the frequency domain as shown in Fig. 8.1.1a, are described by the matrix equations

R = Tr, C = HtR, c = T^(-1)C
(same as Eq. 7.2.1). Fast transform filters use suboptimum transforms (type 1 and 2) and optimum transforms as shown in Fig. 2.8.1. Several such transforms including Walsh, Paley, Hadamard, Haar, slant, and Karhunen-Loeve were discussed in Chaps. 2.8 and 3.3. Fast Fourier transform filters implemented in the frequency domain use
R = Fr, C = HfR, c = F^(-1)C
(see Eq. 7.2.3). F and F^(-1) represent the fast Fourier and inverse fast Fourier transforms, respectively.

Fast transform filters can also be implemented in the time domain as shown in Fig. 8.1.1c. They are described by the single matrix equation

c = hr = T^(-1)HTr
which is obtained by manipulating Eq. 8.1.1. h is the impulse response of the fast transform filter. Filter gain H and filter impulse response h are related as
H = T^(-1)hT, h = ThT^(-1)

Fig. 8.1.1

© C.S. Lindquist, Adaptive and Digital Signal Processing with Digital Filtering Applications, vol. 2, pp. 512-514, 573-575, Steward & Sons, 1989.