Geometric Signal Theory
where the coefficient of projection of onto is given by
and the basis signals are orthonormal:
The signal expansion (11.104) can be interpreted geometrically as a sum of orthogonal projections of onto , as illustrated for 2D in Fig.11.30.
A set of signals is said to be a biorthogonal basis set if any signal can be represented as
where is some normalizing scalar dependent only on and/or . Thus, in a biorthogonal system, we project onto the signals and resynthesize in terms of the basis .
for all integers and . The basis set is orthonormal since . The coefficient of projection of onto is given by
so that the expansion of in terms of the natural basis is simply
This expansion was used in Book II  to derive the impulse-response representation of an arbitrary linear, time-invariant filter.
Normalized DFT Basis for
and . The coefficient of projection of onto is given by
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 inner product and reconstruction of in terms of are left as exercises.
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 §7.1). Thus, the signal basis functions are naturally defined as the DFT-sinusoids multiplied by time-shifted windows, suitably normalized for unit norm:
and is the DFT length.
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
The coefficient of projection can be written
so that the signal expansion can be interpreted as
In the overcomplete case, we get a special case of weighted
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.
Given sufficient decay with , this reduces to , that is, the mother wavelet must be zero-mean.
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.
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:
which does not depend on .
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.
Figure 11.32 shows a block diagram of the discrete wavelet filter bank for (the ``dyadic'' or ``octave filter-bank'' case), and Fig.11.33 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 .
Dyadic Filter Banks
A dyadic filter bank is any octave filter bank,12.6 as illustrated qualitatively in Figure 11.34. 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
Thus, a dyadic filter bank is a special case of a constant-Q filter bank for which the is .
A generalized STFT may be defined by 
This filter bank and its reconstruction are diagrammed in Fig.11.35.
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.
Review of STFT Filterbanks