### Geometric Signal Theory

In general, signals can be expanded as a linear combination
of *orthonormal basis signals*
[264]. In the
discrete-time case, this can be expressed as

where the coefficient of projection of onto is given by

(12.105) |

and the basis signals are

*orthonormal*:

(12.106) |

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

(12.107) |

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 .

The following examples illustrate the Hilbert space point of view for various familiar cases of the Fourier transform and STFT. A more detailed introduction appears in Book I [264].

#### Natural Basis

The *natural basis* for a discrete-time signal
is the set
of shifted impulses:

(12.108) |

or,

(12.109) |

for all integers and . The basis set is orthonormal since . The coefficient of projection of onto is given by

(12.110) |

so that the expansion of in terms of the natural basis is simply

(12.111) |

*i.e.*,

This expansion was used in Book II [263] to derive the impulse-response representation of an arbitrary linear, time-invariant filter.

#### Normalized DFT Basis for

The *Normalized Discrete Fourier Transform (NDFT)* (introduced in
Book I [264]) projects the signal
onto
discrete-time *sinusoids* of length
, where the sinusoids are
normalized to have unit
norm:

(12.112) |

and . The coefficient of projection of onto is given by

and the expansion of in terms of the NDFT basis set is

for .

#### Normalized Fourier Transform Basis

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:

(12.113) |

#### Normalized DTFT Basis

The Discrete Time Fourier Transform (DTFT) is similar to the Fourier transform case:

(12.114) |

The inner product and reconstruction of in terms of are left as exercises.

#### Normalized STFT Basis

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:

(12.115) |

(12.116) |

and is the DFT length.

When successive windows overlap (*i.e.*, the hop size
is less than
the window length
), the basis functions are *not
orgthogonal*. In this case, we may say that the basis set
is *overcomplete*.

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

(12.117) |

*i.e.*,

(12.118) |

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
overlap-add* (§8.6):

#### 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:

(12.119) |

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*.^{12.5}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 E
for related discussion.

#### Discrete Wavelet Transform

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:

(12.120) |

We can show that discrete wavelet transforms are constant-Q by
defining the center frequency of the
th basis signal as the
geometric mean of its bandlimits
and
, *i.e.*,

(12.121) |

Then

(12.122) |

which does not depend on .

#### Discrete Wavelet Filterbank

In a *discrete wavelet filterbank*, each basis signal is
interpreted as the impulse response of a bandpass filter in a
constant-Q filter bank:

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).

As the channel-number increases, the channel impulse response lengthens by the factor ., while the pass-band of its frequency-response narrows by the inverse 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

(12.123) |

so that

(12.124) |

Thus, a dyadic filter bank is a special case of a

*constant-Q filter bank*for which the is .

#### Dyadic Filter Bank Design

Design of dyadic filter banks using the *window method* for FIR
digital filter design (introduced in §4.5) is described in, *e.g.*,
[226, §6.2.3b].

A ``very easy'' method suggested in [287, §11.6] is to design a two-channel paraunitary QMF bank, and repeat recursively to split the lower-half of the spectrum down to some desired depth.

#### Generalized STFT

A *generalized STFT* may be defined by [287]

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*.

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