Wavelet transform

With wavelet transform (. WT, English wavelet transform) a certain family of linear time-frequency transformations in mathematics and the engineering sciences ( primary: telecommunications, computer science ) refers. WT is comprised of the wavelet analysis, which indicates the transition of the time representation in the spectral or Waveletdarstellung, and the wavelet synthesis, which means the inverse transformation of the Wavelettransformierten in the time display.

The term refers to the wavelet basis function used for the transformation, with the signal to be analyzed or image - in general, an N- dimensional function - " compared " is.

The roots of Waveletschule are in France, where the original French term was coined ondelette, but its English counterpart wavelet has later set as the label. Translated into German as much as wavelet means small wave or wavelets and expresses the fact that in contrast to the Fourier transform used temporally localized waves or functions as a basis, whereby the above- mentioned time and frequency resolution is possible. As all linear time-frequency transformation is also subject to the wavelet transform to the uncertainty of communication engineering, ie an event can not be localized simultaneously with arbitrary precision in time and frequency. There is always a compromise of good temporal resolution or good resolution in the frequency domain.

The wavelet transform is divided primarily into two camps, namely the continuous wavelet transform, which has its main application in mathematics and data analysis, and the discrete wavelet transform, which is more likely to find in the engineering sciences and their application in is the area of ​​data reduction, data compression, and signal processing.

• 2.1 Properties of Wavelets
• 2.2 wavelet synthesis
• 2.3 reproducing kernel

• 3.1 Fast wavelet transform
• 3.2 wavelet packet transform and best basis algorithms

Operation

The wavelet transform can be regarded as an improvement of the short-time Fourier transform ( STFT ).

Weaken the short -time Fourier transform

In the STFT, the signal to be examined with a window function - such as the Gaussian bell-shaped curve as the Gabor transform - compared. For each point of the STFT window to the considered point in time and the frequency to be considered ( modulation in the time domain ) will be moved. The absolute time duration and bandwidth of the window ( " width" in the time and frequency domain) - and thus the resolution - this does not change.

The resolutions in time and frequency domains are only dependent on the shape of the window. Due to the time-frequency blur is the resolution in the time domain is inversely proportional to the resolution in the frequency domain. So it can not be achieved simultaneously in the time domain and in the frequency domain, the best possible resolution.

Now includes a signal frequency components at both high and low frequencies, one would obtain at low frequencies a good ( absolute ) frequency resolution, since a small absolute change in frequency falls here significant. At a high frequency, a good time resolution is important, since a complete oscillation here takes less time and the instantaneous frequency can therefore change faster.

If you have for example a signal with frequency components at 1 Hz and 1 kHz and would like to resolve the frequency to 10 percent, at 1 Hz, a frequency resolution of 0.1 Hz necessary. At 1 kHz, this corresponds to a resolution of 0.01 percent - such a good resolution is not necessary here.

On the other hand, performs the signal at 1 kHz ten complete oscillations in 10 ms. In order to resolve frequency changes during this period, a time resolution better than 10 ms is needed. At 1 Hz, this time period corresponds to one hundredth of a vibration. Such a good time resolution is therefore not necessary here.

So you wish for at low frequencies a good frequency resolution at the expense of poor time resolution at high frequencies and good time resolution in poorer frequency resolution. The Short -Time Fourier Transform does not make this.

Summary of Operation

As the STFT, the signal to be examined is compared with a window function. However, instead of moving the window, and modulating ( shift in the frequency domain ) ( as in the STFT ), the window is shifted and scaled. By scaling is as also by the modulation, a frequency shift, however, is at the same time with an increase in frequency, the duration ( " width" in the time domain ) of the window is reduced. This results in higher frequencies, a better temporal resolution. At low frequencies, the frequency resolution is better, but the time resolution is worse.

Continuous wavelet transform

The continuous wavelet transform (CWT, Eng. Continuous wavelet transform) is given by

This can be explained by the introduction of the wavelet family

Compact as the scalar product

Interpret, which immediately follows the linearity of the WT.

Properties of wavelets

The most important property of wavelets is its Admissibilität

From which it follows that the Fourier transform of the wavelet at the point 0 disappears

Which means that the wavelet must have a bandpass character. Furthermore, it follows that the first moment of wavelets, ie its mean vanishes:

Wavelet synthesis

The original function x (t) can be recovered up to an additive constant back from the reconstruction formula Wavelettransformierten

With

Reproducing kernel

When reproducing kernel (English Reproducing kernel ), the wavelet transform of the wavelet is called self. Thus referred

The core of the wavelets.

Reproducing the attribute is the core, because the wavelet transform of the convolution reproduced with the core, that is, the wavelet transform is invariant under the convolution with the core. This convolution is given by

This is no ordinary folding, since it is not commutative; However, it is associative.

A more important role is replaced, therefore, the reproducing core that it indicates the minimum correlation between two points ( A, B ) and ( A ', B ') in the Waveletraum. This can be shown by looking at the autocorrelation of white noise in Waveletraum. If we denote by a Gaussian white noise with variance 1, then its autocorrelation is given by. The correlation in Waveletraum is then ( without executing the statement)

So just given by the reproducing kernel.

Discrete Wavelet Transform

• The discrete wavelet transform or DWT is a wavelet transform, which is performed in a time - and frequency- discrete.
• It has been shown that in spite of the reduction information to a discrete portion of at fully maintained.
• A DWT can be very efficient as a series of discrete-time filters implement the continuous wavelet transform is practically calculated in this way.

Fast wavelet transformation

• The fast wavelet transform ( wavelet transform almost Sheet FWT ) is an algorithm that implements the discrete wavelet transform using the theory of the multi-scale analysis. Thereby forming the inner product of the signal with each wavelet is replaced by successively dividing the signal into frequency bands. This increases the complexity of the wavelet transform (see, fast Fourier transform ) is reduced to.

Wavelet packet transform and best basis algorithms

The wavelet packet transformation is an extension of the fast wavelet transform ( FWT ), not only by the low-pass channel, but also the band-pass channel can be further split by means of the wavelet filter bank. This can serve as example to get from a conventional 2-channel DWT the Daubechies wavelets an M -channel DWT, where M is a power of 2; the exponent is called the depth of the package tree. This method is used in the broadband data transfer as an alternative for the fast Fourier transform.

Is transformed into a recursion of FWT a white noise as an input signal, the result due to the orthogonal nature of the DWT is again a white noise, the energy ( = Sum of squares of samples) is evenly distributed to low-and band -pass channel. Assuming the highest possible deviation from this behavior, ie complete as possible a concentration of signal energy on one of the two channels, as a decision criterion whether the input channel is to be split, and is given to this process for the split channels continues, the result is a variant of a best - basis method.

Important applications

• Image compression and video compression: wavelet compression
• Solution of differential equations
• Signal processing

History

• First wavelet ( Haar wavelet ) of Alfréd Hair (1909 )
• Since the 1950s: Jean Morlet and Alex Grossman
• Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Victor Wickerhauser