Continuous Wavelet Transform
In the present (Hilbert space) setting, we can now easily define the continuous wavelet transform in terms of its signal basis set:
The parameter is called a scale parameter (analogous to frequency). The normalization by maintains energy invariance as a function of scale. We call the wavelet coefficient at scale and time . The kernel of the wavelet transform is called the mother wavelet, and it typically has a bandpass spectrum. A qualitative example is shown in Fig.11.31.
The so-called admissibility condition for a mother wavelet is
Given sufficient decay with , this reduces to , that is, the mother wavelet must be zero-mean.
The Morlet wavelet is simply a Gaussian-windowed complex sinusoid:
The scale factor is chosen so that . The center frequency is typically chosen so that second peak is half of first:
In this case, we have , which is close enough to zero-mean for most practical purposes.
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 §7.2).
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.12.5Before 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 E for related discussion.
Discrete Wavelet Transform
Normalized STFT Basis