What is the canonical representation of a causal wavelet? Let be the minimum-phase wavelet with the same amplitude spectrum as that of the causal wavelet w. Then is called the minimum-phase counterpart of w. The canonical representation' states that any causal wavelet w can be represented as the convolution of its minimum-phase counterpart and a causal all-pass wavelet p; that is,
Because the inverse of a minimum-phase wavelet is minimum phase, it follows that is minimum phase and hence is causal. From the canonical representation, we see that the inverse wavelet is given by
Two cases can occur. In the first case, the causal wavelet w is itself minimum phase. Then the causal all-pass wavelet is trivial, so the inverse wavelet is simply . In this case, the inverse wavelet is minimum phase and causal.
In the second case, the causal wavelet w is not minimum phase. Then the causal all-pass wavelet p is not trivial and therefore its inverse is a one-sided noncausal wavelet. In this case, the inverse wavelet is a two-sided noncausal wavelet.
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Also in this chapter
- Fourier transform
- Delay: Minimum, mixed, and maximum
- Two-length wavelets
- Illustrations of spectra
- Delay in general
- Zero-phase wavelets
- Symmetric wavelets
- Ricker wavelet
- Appendix G: Exercises