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Complex Digital Signal Processing in Telecommunications
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Search Spectral Audio Signal Processing
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In the present (Hilbert space) setting, we can now easily define the continuous wavelet transform in terms of its signal basis set:
The so-called admissibility condition for a mother wavelet is
The Morlet wavelet is simply a Gaussian-windowed complex sinusoid:
Since the scale parameter of a wavelet transform is analogous to frequency in a Fourier transform, a wavelet transform display is often called a scalogram, in analogy with an STFT ``spectrogram'' (discussed in §6.2).
When the mother wavelet can be interpreted as a windowed sinusoid (such as the Morlet wavelet), the wavelet transform can be interpreted as a constant-Q Fourier transform.^{11.4}Before the theory of wavelets, constant-Q Fourier transforms (such as obtained from a classic third-octave filter bank) were not easy to invert, because the basis signals were not orthogonal. See Appendix F for related discussion.