In this chapter, expansions of signals into orthogonal or biorthogonal functions, as well as into frames, are formulated as generalized transforms, where the expansion coefficients provide a discrete representation of a signal (globally seen as a generalized transform). The second topic is that of filter banks and subband decomposition, which implement generalized transforms and signal expansions with multirate components, such as decimators and interpolators. It will be shown that such an implementation is possible for generalized transforms that satisfy the condition of periodic shift invariance.
All these topics are preliminary to multiresolution analysis and wavelets, which will be developed in the next chapter. In fact, wavelets may be viewed both as generalized transforms and as signal expansions, and their practical implementation is obtained by filter banks.
Impulse Response Filter Bank Signal Expansion Perfect Reconstruction Symmetric Component
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