Wavelet transform

Continuous wavelet transform of a signal containing a frequency change; created using simlets - variations of Daubechies wavelets

Wavelet transform ( English Wavelet transform ) - an integral transformation , which is a convolution of the wavelet function with a signal. The wavelet transform translates the signal from a temporal representation into a frequency-temporal one .

A method of converting a function (or signal) into a form that either makes some values of the original signal more studyable, or allows you to compress the original data set. Wavelet signal conversion is a generalization of spectral analysis. The term ( English wavelet ) in translation from English means "small wave". Wavelets are a generic name for mathematical functions of a certain form, which are local in time and frequency and in which all functions are obtained from one basic one, changing it (shifting, stretching).

Content

1 Wavelet Requirements
2 Wavelet transform properties
3 Continuous wavelet transform
4 Discrete Wavelet Transform
5 Graphical representation
6 Application
7 notes
8 References
9 See also
10 Literature

Wavelet Requirements

To implement the wavelet transform, wavelet functions must satisfy the following criteria ^[1] :

1. Wavelet $\psi (t)$ ${\ displaystyle \ psi (t)}$ must have finite energy:

$E=\int \limits _{-\infty }^{\infty }{|\psi (t)|}^{2}\,dt<\infty$ ${\ displaystyle E = \ int \ limits _ {- \ infty} ^ {\ infty} {| \ psi (t) |} ^ {2} \, dt <\ infty}$

2. If ${\hat {\psi }}(f)$ ${\ displaystyle {\ hat {\ psi}} (f)}$ Fourier transform for wavelet $\psi (t)$ ${\ displaystyle \ psi (t)}$ , i.e

${\hat {\psi }}(f)=\int \limits _{-\infty }^{\infty }\psi (t)e^{-i(2\pi f)t}\,dt$ ${\ displaystyle {\ hat {\ psi}} (f) = \ int \ limits _ {- \ infty} ^ {\ infty} \ psi (t) e ^ {- i (2 \ pi f) t} \, dt}$

then the following condition must be fulfilled:

$C_{\psi }=\int \limits _{0}^{\infty }{\frac {{|{\hat {\psi }}(f)|}^{2}}{f}}\,df<\infty$ ${\ displaystyle C _ {\ psi} = \ int \ limits _ {0} ^ {\ infty} {\ frac {{| {\ hat {\ psi}} (f) |} ^ {2}} {f}} \, df <\ infty}$

This condition is called the admissibility condition, and it follows from it that the wavelet at the zero frequency component must satisfy the condition ${\hat {\psi }}(0)=0$ ${\ displaystyle {\ hat {\ psi}} (0) = 0}$ or, otherwise, a wavelet $\psi (t)$ ${\ displaystyle \ psi (t)}$ must have an average of zero.

3. An additional criterion is presented for complex wavelets, namely, for them the Fourier transform must be real at the same time and should decrease for negative frequencies.

4. Localization: the wavelet must be continuous, integrable, have a compact medium and be localized both in time (in space) and in frequency. If a wavelet narrows in space, then its average frequency increases, the wavelet spectrum moves to the region of higher frequencies and expands. This process should be linear - narrowing the wavelet by half should increase its average frequency and the width of the spectrum also by half.

Wavelet Transform Properties

1. Linearity

${\text{WT}}[\alpha s_{1}(t)+\beta s_{2}(t)]=\alpha \,{\text{WT}}[s_{1}(t)]+\beta \,{\text{WT}}[s_{2}(t)]$ ${\ displaystyle {\ text {WT}} [\ alpha s_ {1} (t) + \ beta s_ {2} (t)] = \ alpha \, {\ text {WT}} [s_ {1} (t )] + \ beta \, {\ text {WT}} [s_ {2} (t)]}$

2. Shear invariance

${\text{WT}}[s(t-t_{0})]=C(a,b-t_{0})$ ${\ displaystyle {\ text {WT}} [s (t-t_ {0})] = C (a, b-t_ {0})}$

The time shift of the signal by t ₀ leads to a shift of the wavelet spectrum also by t ₀ .

3. Invariance with respect to scaling

${\text{WT}}{\biggl [}s{\biggl (}{\frac {t}{a_{0}}}{\biggr )}{\biggr ]}={\frac {1}{a_{0}}}C{\biggl (}{\frac {a}{a_{0}}},\,{\frac {b}{a_{0}}}{\biggr )}$ ${\ displaystyle {\ text {WT}} {\ biggl [} s {\ biggl (} {\ frac {t} {a_ {0}}} {\ biggr)} {\ biggr]} = {\ frac {1 } {a_ {0}}} C {\ biggl (} {\ frac {a} {a_ {0}}}, \, {\ frac {b} {a_ {0}}} {\ biggr)}}$

Stretching (compression) of the signal leads to compression (stretching) of the wavelet spectrum of the signal.

4. Differentiation

${\frac {d^{n}}{dt^{n}}}{\text{WT}}[s(t)]={\text{WT}}{\biggl [}{\frac {d^{n}s(t)}{dt^{n}}}{\biggr ]},\quad {\text{WT}}{\biggl [}{\frac {d^{n}s(t)}{dt^{n}}}{\biggr ]}=(-1)^{n}\!\int \limits _{-\infty }^{\infty }s(t){\frac {d^{n}\psi (t)}{dt^{n}}}\,dt$ ${\ displaystyle {\ frac {d ^ {n}} {dt ^ {n}}} {\ text {WT}} [s (t)] = {\ text {WT}} {\ biggl [} {\ frac {d ^ {n} s (t)} {dt ^ {n}}} {\ biggr]}, \ quad {\ text {WT}} {\ biggl [} {\ frac {d ^ {n} s ( t)} {dt ^ {n}}} {\ biggr]} = (- 1) ^ {n} \! \ int \ limits _ {- \ infty} ^ {\ infty} s (t) {\ frac { d ^ {n} \ psi (t)} {dt ^ {n}}}, dt}$

It follows that it makes no difference whether to differentiate a function or an analyzing wavelet. If the analyzing wavelet is given by the formula, then this can be very useful for signal analysis. This property is especially useful if the signal is given by a discrete row.

Continuous Wavelet Transform

The wavelet transform for a continuous signal relative to the wavelet function is defined as follows [1]:

$T(a,b)={\frac {1}{\sqrt {a}}}\int \limits _{-\infty }^{\infty }x(t)\psi ^{*}\left({\frac {t-b}{a}}\right)\,dt$ {\ displaystyle T (a, b) = {\ frac {1} {\ sqrt {a}}} \ int \ limits _ {- \ infty} ^ {\ infty} x (t) \ psi ^ {*} \ left ({\ frac {tb} {a}} \ right) \, dt}

Where ${\psi }^{*}$ ${\ displaystyle {\ psi} ^ {*}}$ means complex pairing for $\psi$ ${\ displaystyle \ psi}$ , parameter $b\in R$ ${\ displaystyle b \ in R}$ corresponds to a time shift, and is called a position parameter, the parameter $a>0$ ${\ displaystyle a> 0}$ sets the scaling and is called the stretch parameter.

$w(a)\equiv {\frac {1}{\sqrt {a}}}$ ${\ displaystyle w (a) \ equiv {\ frac {1} {\ sqrt {a}}}}$ - weight function.

We can define a normalized function as follows

${\psi }_{a,b}={\frac {1}{\sqrt {a}}}{\psi }\left({\frac {t-b}{a}}\right)$ ${\ displaystyle {\ psi} _ {a, b} = {\ frac {1} {\ sqrt {a}}} {\ psi} \ left ({\ frac {tb} {a}} \ right)}$

which means time shift by b and time scaling by a . Then the wavelet transform formula will change to

$T(a,b)=\int \limits _{-\infty }^{\infty }x(t)\,\psi _{a,b}^{*}\,dt$ ${\ displaystyle T (a, b) = \ int \ limits _ {- \ infty} ^ {\ infty} x (t) \, \ psi _ {a, b} ^ {*} \, dt}$

The original signal can be restored by the inverse transformation formula

$x(t)={\frac {1}{C_{\psi }}}\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }T(a,b)\,{\psi }_{a,b}(t)\,da\,db$ ${\ displaystyle x (t) = {\ frac {1} {C _ {\ psi}}} \ int \ limits _ {- \ infty} ^ {\ infty} \ int \ limits _ {- \ infty} ^ {\ infty} T (a, b) \, {\ psi} _ {a, b} (t) \, da \, db}$

Discrete Wavelet Transform

In the discrete case, the parameters of scaling a and shift b are represented by discrete values:

$a=a_{0}^{m},\quad b=nb_{0}$ ${\ displaystyle a = a_ {0} ^ {m}, \ quad b = nb_ {0}}$

Then the analyzing wavelet has the following form:

$\psi _{m,n}=a_{0}^{-m/2}\psi \left({\frac {t-nb_{0}}{a_{0}^{m}}}\right)$ ${\ displaystyle \ psi _ {m, n} = a_ {0} ^ {- m / 2} \ psi \ left ({\ frac {t-nb_ {0}} {a_ {0} ^ {m}}} \ right)}$

where m and n are integers.

In this case, for a continuous signal, the discrete wavelet transform and its inverse transform are written as follows:

$T_{m,n}=\int \limits _{-\infty }^{\infty }x(t)\,\psi _{m,n}^{*}(t)\,dt$ ${\ displaystyle T_ {m, n} = \ int \ limits _ {- \ infty} ^ {\ infty} x (t) \, \ psi _ {m, n} ^ {*} (t) \, dt}$

Quantities $T_{m,n}$ ${\ displaystyle T_ {m, n}}$ also known as wavelet coefficients.

$x(t)=K_{\psi }\sum \limits _{m=-\infty }^{\infty }\sum \limits _{n=-\infty }^{\infty }T_{m,n}\psi _{m,n}(t)$ ${\ displaystyle x (t) = K _ {\ psi} \ sum \ limits _ {m = - \ infty} ^ {\ infty} \ sum \ limits _ {n = - \ infty} ^ {\ infty} T_ {m , n} \ psi _ {m, n} (t)}$

Where $K_{\psi }$ ${\ displaystyle K _ {\ psi}}$ - constant normalization.

Graphical View

Temporal and spectral representations of a WAVE wavelet

Temporal and spectral representations of the Morlet wavelet

Application

Wavelet transform is widely used for signal analysis. In addition, it finds great application in the field of data compression. In a discrete wavelet transform, the most significant information in the signal is contained at high amplitudes, and less useful at low amplitudes. Data compression can be obtained by dropping low amplitudes. Wavelet transform allows you to get a high compression ratio in combination with good quality of the restored signal. The wavelet transform was chosen for JPEG2000 and ICER image compression standards. However, at low compressions, the wavelet transform is inferior in quality compared to the window Fourier transform , which underlies the JPEG standard.

The choice of a specific type and type of wavelets in many respects depends on the analyzed signals and analysis tasks. To obtain optimal conversion algorithms, certain criteria have been developed, but they still cannot be considered final, since they are internal to the conversion algorithms themselves and, as a rule, do not take into account external criteria related to the signals and goals of their transformations. It follows that in the practical use of wavelets it is necessary to pay sufficient attention to checking their operability and effectiveness for the set goals in comparison with the known methods of processing and analysis.

Notes

Advantages:

Wavelet transforms have all the advantages of Fourier transforms.
Wavelet bases can be well localized both in frequency and in time. When isolating well-localized different-scale processes in signals, one can consider only those scale levels of decomposition that are of interest.
Basic wavelets can be implemented with functions of various smoothness.

Disadvantages:

One drawback can be distinguished, this is the relative complexity of the conversion.

Links

↑ Addison PS The Illustrated Wavelet Transform Handbook. - IOP, 2002.

Literature

[1] Addison PS The Illustrated Wavelet Transform Handbook. - IOP, 2002.