## EXAMPLE 7

### SELECTING FAST TRANSFORM DIGITAL FILTER

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)

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