Geometric Signal Theory
where the coefficient of projection of onto is given by
A set of signals is said to be a biorthogonal basis set if any signal can be represented as
This expansion was used in Book II  to derive the impulse-response representation of an arbitrary linear, time-invariant filter.
and the expansion of in terms of the NDFT basis set is
The Fourier transform projects a continuous-time signal onto an infinite set of continuous-time complex sinusoids , for . These sinusoids all have infinite norm, but a simple normalization by can be chosen so that the inverse Fourier transform has the desired form of a superposition of projections:
The Short Time Fourier Transform (STFT) is defined as a time-ordered sequence of DTFTs, and implemented in practice as a sequence of FFTs (see §6.1). Thus, the signal basis functions are naturally defined as the DFT-sinusoids multiplied by time-shifted windows, suitably normalized for unit norm:
The basis signals are orthonormal when and the rectangular window is used (). That is, two rectangularly windowed DFT sinusoids are orthogonal when either the frequency bin-numbers or the time frame-numbers differ, provided that the window length equals the number of DFT frequencies (no zero padding). In other words, we obtain an orthogonal basis set in the STFT when the hop size, window length, and DFT length are all equal (in which case the rectangular window must be used to retain the perfect-reconstruction property). In this case, we can write
so that the signal expansion can be interpreted as
In the overcomplete case, we get a special case of weighted
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.10.32.
The scale factor is chosen so that . The center frequency is typically chosen so that second peak is half of first:
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.11.4Before 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.
The discrete wavelet transform is a discrete-time,
discrete-frequency counterpart of the continuous wavelet transform of
the previous section:
where and range over the integers, and is the mother wavelet, interpreted here as a (continuous) filter impulse response.
The inverse transform is, as always, the signal expansion in terms of the orthonormal basis set:
Thus, the th channel-filter is obtained by frequency-scaling (and normalizing for unit energy) the zeroth channel filter . The frequency scale-factor is of course equal to the inverse of the time-scale factor.
Recall that in the STFT, channel filter is a shift of the zeroth channel-filter (which corresponds to ``cosine modulation'' in the time domain).
Figure 10.33 shows a block diagram of the discrete wavelet filter bank for (the ``dyadic'' or ``octave filter-bank'' case), and Fig.10.34 shows its time-frequency tiling as compared to that of the STFT. The synthesis filters may be used to make a biorthogonal filter bank. If the are orthonormal, then .
A dyadic filter bank is any octave filter bank,11.5 as illustrated qualitatively in Figure 10.35. Note that is the top-octave bandpass filter, is the bandpass filter for next octave down, is the octave bandpass below that, and so on. The optional scale factors result in the same sum-of-squares for each channel-filter impulse response.
A dyadic filter bank may be derived from the discrete wavelet filter bank by setting and relaxing the exact orthonormality requirement on the channel-filter impulse responses. If they do happen to be orthonormal, we may call it a dyadic wavelet filter bank.
For a dyadic filter bank, the center-frequency of the th channel-filter impulse response can be defined as
This filter bank and its reconstruction are diagrammed in Fig.10.36.
The analysis filter is typically complex bandpass (as in the STFT case). The integers give the downsampling factor for the output of the th channel filter: For critical sampling without aliasing, we set . The impulse response of synthesis filter can be regarded as the th basis signal in the reconstruction. If the are orthonormal, then we have . More generally, form a biorthogonal basis.
Radians versus Cycles
Sliding FFT (Maximum Overlap), Any Window, Zero-Padded by 5