文章目录

Basis Vectors
Inner Products, Orthogonality, and Orthonormality
EXAMPLES
THE WAVELET SYNTHESIS
Discretization of the Continuous Wavelet Transform: The Wavelet Series

This section describes the main idea of wavelet analysis theory, which can also be considered to be the underlying concept of most of the signal analysis techniques. The FT defined by Fourier use basis functions to analyze and reconstruct a function. Every vector in a vector space can be written as a linear combination of the basis vectors in that vector space , i.e., by multiplying the vectors by some constant numbers, and then by taking the summation of the products. The analysis of the signal involves the estimation of these constant numbers (transform coefficients, or Fourier coefficients, wavelet coefficients, etc). The synthesis, or the reconstruction, corresponds to computing the linear combination equation.

All the definitions and theorems related to this subject can be found in Keiser’s book, A Friendly Guide to Wavelets but an introductory level knowledge of how basis functions work is necessary to understand the underlying principles of the wavelet theory. Therefore, this information will be presented in this section.

Basis Vectors

Note: Most of the equations include letters of the Greek alphabet. These letters are written out explicitly in the text with their names, such as tau, psi, phi etc. For capital letters, the first letter of the name has been capitalized, such as, Tau, Psi, Phi etc. Also, subscripts are shown by the underscore character _ , and superscripts are shown by the ^ character. Also note that all letters or letter names written in bold type face represent vectors, Some important points are also written in bold face, but the meaning should be clear from the context.

A basis of a vector space VVV is a set of linearly independent vectors, such that any vector vvv in VVV can be written as a linear combination of these basis vectors. There may be more than one basis for a vector space. However, all of them have the same number of vectors, and this number is known as the dimension of the vector space. For example in two-dimensional space, the basis will have two vectors.

Equation 3.2
Equation 3.2 shows how any vector v can be written as a linear combination of the basis vectors bkb_kbk and the corresponding coefficients νk\nu^kνk .

This concept, given in terms of vectors, can easily be generalized to functions, by replacing the basis vectors bkb_kbk with basis functions ϕk(t)\phi_k(t)ϕk(t), and the vector vvv with a function f(t)f(t)f(t). Equation 3.2 then becomes

Equation 3.2a

The complex exponential (sines and cosines) functions are the basis functions for the FT. Furthermore, they are orthogonal functions, which provide some desirable properties for reconstruction.

Let f(t)f(t)f(t) and g(t)g(t)g(t) be two functions in L2[a,b]L^2 [a,b]L2[a,b]. ( L2[a,b]L^2 [a,b]L2[a,b] denotes the set of square integrable functions in the interval [a,b][a,b][a,b]). The inner product of two functions is defined by Equation 3.3:

Equation 3.3

According to the above definition of the inner product, the CWT can be thought of as the inner product of the test signal with the basis functions ψ(τ,s)(t)\psi_(\tau ,s)(t)ψ(τ,s)(t):

Equation 3.4
where,

Equation 3.5

This definition of the CWT shows that the wavelet analysis is a measure of similarity between the basis functions (wavelets) and the signal itself. Here the similarity is in the sense of similar frequency content. The calculated CWT coefficients refer to the closeness of the signal to the wavelet at the current scale.

This further clarifies the previous discussion on the correlation of the signal with the wavelet at a certain scale. If the signal has a major component of the frequency corresponding to the current scale, then the wavelet (the basis function) at the current scale will be similar or close to the signal at the particular location where this frequency component occurs. Therefore, the CWT coefficient computed at this point in the time-scale plane will be a relatively large number.

Inner Products, Orthogonality, and Orthonormality

Two vectors v , w are said to be orthogonal if their inner product equals zero:

Equation 3.6

Similarly, two functions fff and ggg are said to be orthogonal to each other if their inner product is zero:

Equation 3.7

A set of vectors {v1,v2,....,vn}\{v_1, v_2, ....,v_n\}{v1,v2,....,vn} is said to be orthonormal , if they are pairwise orthogonal to each
other, and all have length ‘‘1’’. This can be expressed as:

Equation 3.8

Similarly, a set of functions ϕk(t),k=1,2,3,...,{\phi_k(t)}, k=1,2,3,...,ϕk(t),k=1,2,3,..., is said to be orthonormal if

Equation 3.9

and

Equation 3.10

or equivalently

Equation 3.11

where, δkl\delta_{kl}δkl is the Kronecker delta function, defined as:

Equation 3.12

As stated above, there may be more than one set of basis functions (or vectors). Among them, the orthonormal basis functions (or vectors) are of particular importance because of the nice properties they provide in finding these analysis coefficients. The orthonormal bases allow computation of these coefficients in a very simple and straightforward way using the orthonormality property.

For orthonormal bases, the coefficients, μk\mu_kμk , can be calculated as

Equation 3.13

and the function f(t) can then be reconstructed by Equation 3.2_a by substituting the mu_k coefficients. This yields
Equation 3.14

Orthonormal bases may not be available for every type of application where a generalized version, biorthogonal bases can be used. The term ‘‘biorthogonal’’ refers to two different bases which are orthogonal to each other, but each do not form an orthogonal set.
In some applications, however, biorthogonal bases also may not be available in which case frames can be used. Frames constitute an important part of wavelet theory, and interested readers are referred to Kaiser’s book mentioned earlier.
Following the same order as in chapter 2 for the STFT, some examples of continuous wavelet transform are presented next. The figures given in the examples were generated by a program written to compute the CWT.

Before we close this section, I would like to include two mother wavelets commonly used in wavelet analysis. The Mexican Hat wavelet is defined as the second derivative of the Gaussian function:
Equation 3.15

which is
Equation 3.16

The Morlet wavelet is defined as

Equation 3.16a

where aaa is a modulation parameter, and σ\sigmaσ is the scaling parameter that affects the width of the window.

EXAMPLES

All of the examples that are given below correspond to real-life non-stationary signals. These signals are drawn from a database signals that includes event related potentials of normal people, and patients with Alzheimer’s disease. Since these are not test signals like simple sinusoids, it is not as easy to interpret them. They are shown here only to give an idea of how real-life CWTs look like.

The following signal shown in Figure 3.11 belongs to a normal person.
Figure 3.11

and the following is its CWT. The numbers on the axes are of no importance to us. those numbers simply show that the CWT was computed at 350 translation and 60 scale locations on the translation-scale plane. The important point to note here is the fact that the computation is not a true continuous WT, as it is apparent from the computation at finite number of locations. This is only a discretized version of the CWT, which is explained later on this page. Note, however, that this is NOT discrete wavelet transform (DWT) which is the topic of Part IV of this tutorial.

Figure 3.12

and the Figure 3.13 plots the same transform from a different angle for better visualization.
Figure 3.13

Figure 3.14 plots an event related potential of a patient diagnosed with Alzheimer’s disease
Figure 3.14

and Figure 3.15 illustrates its CWT:
Figure 3.15

and here is another view from a different angle

Figure 3.16

THE WAVELET SYNTHESIS

The continuous wavelet transform is a reversible transform, provided that Equation 3.18 is satisfied. Fortunately, this is a very non-restrictive requirement. The continuous wavelet transform is reversible if Equation 3.18 is satisfied, even though the basis functions are in general may not be orthonormal. The reconstruction is possible by using the following reconstruction formula:
Equation 3.17 Inverse Wavelet Transform

where KaTeX parse error: Undefined control sequence: \C at position 1: \̲C̲_\psi is a constant that depends on the wavelet used. The success of the reconstruction depends on
this constant called, the admissibility constant , to satisfy the following admissibility condition :

Equation 3.18 Admissibility Condition

where ψ^(ξ)\hat{\psi}(\xi)ψ^(ξ) is the FT of ψ(t)\psi(t)ψ(t). Equation 3.18 implies that psi^hat(0) = 0, which is
Equation 3.19

As stated above, Equation 3.19 is not a very restrictive requirement since many wavelet functions can be found whose integral is zero. For Equation 3.19 to be satisfied, the wavelet must be oscillatory.

Discretization of the Continuous Wavelet Transform: The Wavelet Series

In today’s world, computers are used to do most computations (well,…ok… almost all computations). It is apparent that neither the FT, nor the STFT, nor the CWT can be practically computed by using analytical equations, integrals, etc. It is therefore necessary to discretize the transforms.

As in the FT and STFT, the most intuitive way of doing this is simply sampling the time-frequency (scale) plane. Again intuitively, sampling the plane with a uniform sampling rate sounds like the most natural choice. However, in the case of WT, the scale change can be used to reduce the sampling rate.

At higher scales (lower frequencies), the sampling rate can be decreased, according to Nyquist’s rule. In other words, if the time-scale plane needs to be sampled with a sampling rate of N1N_1N1 at scale s1s_1s1 , the same plane can be sampled with a sampling rate of N2N_2N2 , at scale s2s_2s2 , where, s1<s2s_1 < s_2s1<s2 (corresponding to frequencies f1>f2f1>f2f1>f2 ) and N2<N1N_2 < N_1N2<N1 . The actual relationship between N_1 and N_2 is
Equation 3.20

Equation 3.21

In other words, at lower frequencies the sampling rate can be decreased which will save a considerable amount of computation time.
It should be noted at this time, however, that the discretization can be done in any way without any restriction as far as the analysis of the signal is concerned. If synthesis is not required, even the Nyquist criteria does not need to be satisfied. The restrictions on the discretization and the sampling rate become important if, and only if, the signal reconstruction is desired. Nyquist’s sampling rate is the minimum sampling rate that allows the original continuous time signal to be reconstructed from its discrete samples.

The basis vectors that are mentioned earlier are of particular importance for this reason. As mentioned earlier, the wavelet ψ(tau,s)\psi(tau,s)ψ(tau,s) satisfying Equation 3.18, allows reconstruction of the signal by Equation 3.17. However, this is true for the continuous transform. The question is: can we still reconstruct the signal if we discretize the time and scale parameters? The answer is ‘‘yes’’, under certain conditions (as they always say in commercials: certain restrictions apply !!!).

The scale parameter s is discretized first on a logarithmic grid. The time parameter is then discretized with respect to the scale parameter , i.e., a different sampling rate is used for every scale. In other words, the sampling is done on the dyadic sampling grid shown in Figure 3.17 :

Figure 3.17

Think of the area covered by the axes as the entire time-scale plane. The CWT assigns a value to the continuum of points on this plane. Therefore, there are an infinite number of CWT coefficients. First consider the discretization of the scale axis. Among that infinite number of points, only a finite number are taken, using a logarithmic rule. The base of the logarithm depends on the user. The most common value is 2 because of its convenience. If 2 is chosen, only the scales 2, 4, 8, 16, 32, 64,…etc. are computed. If the value was 3, the scales 3, 9, 27, 81, 243,…etc. would have been computed. The time axis is then discretized according to the discretization of the scale axis. Since the discrete scale changes by factors of 2 , the sampling rate is reduced for the time axis by a factor of 2 at every scale.

Note that at the lowest scale (s=2), only 32 points of the time axis are sampled (for the particular case given in Figure 3.17). At the next scale value, s=4, the sampling rate of time axis is reduced by a factor of 2 since the scale is increased by a factor of 2, and therefore, only 16 samples are taken. At the next step, s=8 and 8 samples are taken in time, and so on.

Although it is called the time-scale plane, it is more accurate to call it the translation-scale plane, because ‘‘time’’ in the transform domain actually corresponds to the shifting of the wavelet in time. For the wavelet series, the actual time is still continuous.

Similar to the relationship between continuous Fourier transform, Fourier series and the discrete Fourier transform, there is a continuous wavelet transform, a semi-discrete wavelet transform (also known as wavelet series) and a discrete wavelet transform.

Expressing the above discretization procedure in mathematical terms, the scale discretization is s=s0js = s_0^js=s0j, and translation discretization is τ=ks0jτ0\tau = ks_0^j\tau_0τ=ks0jτ0 where s0>1s_0>1s0>1 and τ0>0\tau_0>0τ0>0 . Note, how the translation discretization is dependent on scale discretization with s0s_0s0.

The continuous wavelet function
Equation 3.22

Equation 3.23

by inserting s=s0js = s_0^js=s0j, and τ=k.s0j.τ0\tau = k.s_0^j.\tau_0τ=k.s0j.τ0.

If {ψ(j,k)}\{\psi_(j,k)\}{ψ(j,k)} constitutes an orthonormal basis, the wavelet series transform becomes
Equation 3.24

or
Equation 3.25

A wavelet series requires that ψ(j,k){\psi_(j,k)}ψ(j,k) are either orthonormal, biorthogonal, or frame. If ψ(j,k){\psi_(j,k)}ψ(j,k) are not orthonormal, Equation 3.24 becomes

Equation 3.26

where ψj,k∗(t)^\hat{ \psi_{j,k}^*(t)}ψj,k∗(t)^ , is either the dual biorthogonal basis or dual frame (Note that * denotes the conjugate). If {ψ(j,k)}\{\psi_(j,k) \}{ψ(j,k)} are orthonormal or biorthogonal, the transform will be non-redundant, where as if they form a frame, the transform will be redundant. On the other hand, it is much easier to find frames than it is to find orthonormal or biorthogonal bases.

The following analogy may clear this concept. Consider the whole process as looking at a particular object. The human eyes first determine the coarse view which depends on the distance of the eyes to the object. This corresponds to adjusting the scale parameter s0−js_0^{-j}s0−j.

When looking at a very close object, with great detail, jjj is negative and large (low scale, high frequency, analyses the detail in the signal). Moving the head (or eyes) very slowly and with very small increments (of angle, of distance, depending on the object that is being viewed), corresponds to small values of τ=k.s0j.τ0\tau = k.s_0^j.\tau_0τ=k.s0j.τ0 .

Note that when jjj is negative and large, it corresponds to small changes in time, τ\tauτ , (high sampling rate) and large changes in s0−js_0^{-j}s0−j (low scale, high frequencies, where the sampling rate is high). The scale parameter can be thought of as magnification too.

How low can the sampling rate be and still allow reconstruction of the signal? This is the main question to be answered to optimize the procedure. The most convenient value (in terms of programming) is found to be "2’’ for s_0 and “1” for τ\tauτ. Obviously, when the sampling rate is forced to be as low as possible, the number of available orthonormal wavelets is also reduced.

The continuous wavelet transform examples that were given in this chapter were actually the wavelet series of the given signals. The parameters were chosen depending on the signal. Since the reconstruction was not needed, the sampling rates were sometimes far below the critical value where s0s_0s0 varied from 2 to 10, and τ0\tau_0τ0 varied from 2 to 8, for different examples.

This concludes Part III of this tutorial. I hope you now have a basic understanding of what the wavelet transform is all about. There is one thing left to be discussed however. Even though the discretized wavelet transform can be computed on a computer, this computation may take anywhere from a couple seconds to couple hours depending on your signal size and the resolution you want. An amazingly fast algorithm is actually available to compute the wavelet transform of a signal. The discrete wavelet transform (DWT) is introduced in the final chapter of this tutorial, in Part IV.

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THE WAVELET THEORY: A MATHEMATICAL APPROACH