Theoretical approach to wavelets

The standard Fourier transform also gives a representation of the frequency con- tent of the signal, but information concerning time-localization of the interesting frequency is lost. The short-time Fourier transform, allows getting time-frequency

localization, by windowing the signal in small frames, and then computing its Fourier Transform.

However, the time-frequency localization properties are inherent in the wavelet transformation, so that the dierence between the windowed Fourier transform and the wavelet transform remains in the transformation function: while Fourier trans- form always consists of the same envelope, translated to the proper time location and "lled in" with higher frequency oscillations from the data, the wavelet transform has time-width adapted to their frequency: high frequency transformation functions are very narrow, while low frequency functions are much broader. The result is the wavelet transform is better able to capture both low frequency and very brief high frequency phenomena.

The Continuous Wavelet Transform (CWT) is the basis of the wavelet anal- ysis [She96], and it is written as:

T_s,τwav = |s|−1/2 Z dtf (t)ψ t − τ s  (5.1)

This equation produces the decomposition of the function f(t) into the dierent ψt,τ wavelets, all of them generated from the mother wavelet (Eq.5.2) by adjusting

s, τ, the scaling and translation parameters.

ψs,τ(t) = 1 √ sψ  t − τ s  (5.2)

5.1.1

Basis of multiresolution analysis

Considering a function f(t) and labeling the resolution level by j the scale below which all uctuations on that resolution are ignored is 1/2j_{. The function that}

approximates f(t) is fj(t). At the next resolution level j + 1, the details, denoted

by dj(t), are included in function fj+1(t) = fj(t) + dj(t). This procedure can be

repeated several times. The function f(t) can be viewed as f (t) = fj +

k=∞

X

k=j

dk (5.3)

Similarly, the space of square integrable functions L2

(R) can be viewed as com- positions of subspaces Wk and subspace Vj. Wkcontains details dk(t). The subspace

The requirements of multiresolution analysis are [Dau92]:

1. Subspace V , must be contained in all subspaces on higher resolutions

. . . ⊂ V0 ⊂ V1 ⊂ . . . ⊂L2(R) (5.4)

2. All square integrable functions must be included at the nest resolution level (5.5) and only zero function on the coarsest level

∪jVj =L2(R) (5.5)

∩jVj = {0} (5.6)

3. All the spaces {Vj} are scaled versions of the central space V0. If f(t) is in

space Vj and it contains no details on scales smaller than 1/2j, then function

f (2t) contains no details on scales smaller than 1/2j+1 _{and it is from space}

Vj+1.

f (t) ∈ Vj ⇔ f (2t) ∈ Vj+1 (5.7)

4. If f(t) ∈ V0, so do its translates by integer k, {f(t − k)}.

f (t) ∈ V0 ⇔ f (t − k) ∈ V0 (5.8)

Once these properties are satised by a ladder of spaces Vj there exists a function

ψ(t) ∈ V0 so that {ψ(t − k)} constitute an orthonormal basis for V0.

5.1.2

Discretization of the wavelets

Since the CWT has innite wavelets representations, depending on the values of s and τ, it is not practical using it as an analysis tool. Moreover, depending on the selected values of the translation and scaling parameters, the mother wavelet could constitute an orthogonal basis [Dau92], facilitating the posterior reconstruction.

The Discrete Wavelet Transform (DWT) is dened by the discretization of s and τ: T_m,nwav = |a0|−m/2 Z dtf (t)ψ t − nb0a m 0 am 0  = |a0|−m/2 Z dtf (t)ψ a−m₀ t − nb0
, (5.9) where a0 > 1, b0 > 0, and m and n range over Z. The selection of the values of a0and

b0 is important for the correct representation of the information contained in f(t),

for L2

(R). In particular, if a0 = 2, b0 = 1, there exist ψ with good time-frequency

localization properties, such that

ψm,n(t) = 2−m/2ψ (2mt − n) (5.10)

constitute an orthonormal basis for L2

(R) as long as ψm,n are orthonormal.

The rst known function which satises the orthonormal requirements was the Haar function [Haa10]:

ψ(x) =          1 0 ≤ x < 1₂ −1 1 2 ≤ x < 1 0 otherwise (5.11) but in the recent years, several functions have proven its properties as orthonor- mal basis, such as the Meyer wavelet, the Daubechies family of wavelets or the Mexican Hat wavelet (Fig. 5.1).

Figure 5.1: Several mother wavelet functions, its progressive dilation constitutes an or- thonormal basis. a) Haar wavelet b) Meyer wavelet c) Daubechies1 wavelet d) Mexican Hat Wavelet

5.1.3

The problem of reconstruction

When the discretization of the wavelets is used on a transformed continuous signal the result is understood as a series of wavelet coecients, but it appears a couple of important questions:

Do the discrete wavelets coecient completely characterize the original func- tion? Can the original function be reconstructed from the discrete wavelets coecients?

It has been proven that is possible getting a stable reconstruction as long as the wavelets coecients satisfy the following condition [Dau92]:

A kf k2 ≤X

j,k

| hf, ψn,mi |2 ≤ B kf k2 (5.12)

being kfk2 _{the energy of the original function, A > 0, B < ∞ and A, B independent}

from f. If ψm,,(t); m, n ∈ Z satises this condition, also can be understood as an

stability requirement, it is referred as a frame. The connection between frames and numerically stable reconstruction from discretized wavelets has been proved by several authors [You80; Gro85].

[Dau92] and [She96] have also proven that if the mother wavelet, joint with the dilation and scaling parameters, constitutes an orthonormal basis, the reconstruction of an arbitrarily signal f(t) can be accomplished by summing the orthogonal wavelet basis function, weighted by the wavelet transform coecients:

f (t) =X

m,n

T_m,nwav(t)ψm,n(t) (5.13)

In conclusion, as long as these conditions are achieved, the signal is fully charac- terized by the discretized wavelet levels and what is more, the original signal can be reconstructed.

5.1.4

The equivalence between wavelets and filters

The principal property of the wavelets transformation is the capability of analyzing a time-variant signal on dierent frequency scales. The details shown on each wavelet level, will depend on the scaling parameter s , or its discrete equivalent m (See Eq. 5.1 and Eq. 5.9), ltering the interesting frequencies on each level.

[Dau92], the progressive dilations of wavelet can be understood as a band-pass lter bank (5.2).

Figure 5.2: Filter bank produced by the progressive dilation of Mexican Hat wavelet, starting from 40 ms and taking ten representation scales.

Adjusting correctly the dilation parameter, the whole spectrum of the signal is covered by the spectra of dilated wavelets. Once again, if a0 is set to 2 and m ∈ Z,

it ensures the full representation of the signal's spectrum and good time-frequency localization properties.