The wavelet transform
The wavelet transform provides additional flexibility on two levels when compared with the Fourier transform. (1) The transformed quantity is local in both frequency (or wavenumber) and time (or space); (2) many different kinds of basis function are available, and indeed it is possible to create new functions, provided certain mathematical constraints are satisfied.
The continuous wavelet transform of a signal q(a) is written:
This amounts to the convolution of a signal of interest with a set of wavelet functions ф. This set of functions is generated by dilation and translation of a basic form known as the mother wavelet: dilation is achieved by varying the scale, s, translation being effected by means of the variable, a, which could be a space or time coordinate, for example. The mother wavelet function must satisfy the mathematical constraints of admissibility and regularity; however, provided these constraints are satisfied a good deal of flexibility remains for the design of new mother wavelet functions.
The main difference between the wavelet transform and the Fourier transform is that the former allows space – or time-localised characteristics of a signal to be more clearly identified: the transformed signal is local in both space (and/or time) and scale, whereas its Fourier transformed counterpart is local only in frequency, being infinitely extended in space (and/or time).
The following are some relations between the fluctuation energy of a signal, its wavelet transform and its Fourier transform.
1. The relationship between the fluctuation energy, E of the signal q(a) and the wavelet transform of the signal is given by:
E = f q(a)2da = C— f ( |</(s, a)|.|</*(s, a)| dsdda
Jr ф Jr+ J r s2
where Сф is a constant associated with the mother wavelet function used.
2. A global wavelet energy spectrum can be defined as:
egiobai(s) = / e(s, a) da (131)
where e(s, a) is the energy density as a function of scale, s and the space or time dimension, a.
3. This can also be expressed in terms of the Fourier energy spectrum
E(f ) = W )|2:
egiobai(s) = f E(f)^(sf)|2df (132)
where ^(sf) is the Fourier transform of the wavelet. This shows that the global wavelet energy spectrum corresponds to the Fourier energy spectrum smoothed by the wavelet spectrum at each scale.
4. The total fluctuation energy of the signal can be obtained by
E = C-1 f eglobal(s)— (133)
5.1 Proper Orthogonal Decomposition
The Proper Orthogonal Decomposition is a data processing technique which is known by this name when used in the field of turbulence analysis, following its introduction for such usage by Lumley (1967). It can also be found referred to as Karhunen-Loeve decomposition, principal component analysis (Jolliffe (1986)) and singular value decomposition (Golub and Van Loan (1996)). The presentation of POD given here follows that of Delville (1995).
Consider a flow system for which we possess the information q(a, b). The vector q could contain, for example, the values of the three components of velocity on the four-dimensional grid, (x, t); in this case a would represent three-dimensional cartesian space, and b the time direction. We retain the notation a and b in order to keep the derivation as general as possible, because different variants of the POD can be derived from different specific choices of a and b, and associated definitions of the inner product and averaging operations that are applied, respectively, with respect to these coordinates.
POD consists in searching for the function, 0(a), that is best aligned, on average, with the field q(a, b), the averaging operation being with respect to the coordinate b. Both q(a, b) and ф(a) are indefinitely differentiable, have compact support, and belong to the space of square integrable functions. The problem is considered in Hibert space, and so it is possible to define the inner product (q, ф)а with respect to a:
л nc /»
(^Ф)а = q(a, b^*(a) da / qi(a, b)0*(a) da (134)
Л i=i ■’a
where nc denotes the number of components of the vector q (the three components of velocity for example).
The search for the function ф amounts to a search, over the ensemble of realisations of q, for the ф that most closely resembles q on average. This means maximising the projection q(a, b) on the function ф^) with respect to the inner product defined above: we must find the function ф
an equation known as the Fredholm integral.
Solution of the integral eigenvalue problem is obtained by means of the theory of Hilbert-Schmidt Lovitt (1950). The details are not given here, but we recall some of the main results:
1. As with most eigenvalue problems, rather than admitting a unique solution, the equation yields a set of solutions:
J Щ(a, a’^jn)(a’) da’ = A (п)Ф(п) (a) n = 1, 2, 3,
2. The ensemble of solutions can be chosen such that the eigenfunctions are orthonormal:
[ Ф(Рa)49)(a) da = Spq (144)
3. Any field, qi(a, b), can be expanded in terms of these eigenfunctions, ф(п) (a):
qi(a, b) = £ a(n)(b^n)(a) (145)
where the coefficients, a(n) (b), are obtained by the projection of qi(a, b) onto ф(п)(a):
a(n) (b) = f qi(a, Ь)Ф(п) (a) da (146)
4. The series converges in a least mean square sense and the coefficients, a(n)(b), are mutually uncorrelated:
< a(n) ■ a(m) >= SmnA(n) (147)
5. The eigenvalues are real, positive, their sum finite and they form a convergent series:
A(1) >A(2) > A(3),… (148)
The most common experimental implementation of POD involves spacetime velocity or pressure fields: q(a, b) = u(x, t) = u(x, y, z, t) or q(a, b) = p(x, t) = p(x, y,z, t) in which case expansion of the data in terms of the POD eigenfunctions reads
Ui(x, y,z, t) = ^2 a(n)(t)^nx, y,z) (149)
The space-time structure of the measured field is thus separated into spatial (topos) and temporal (chronos) functions. Example implementations are provided in section §4.