Fast wavelet transform

The fast wavelet transformation is an efficient method for computing a discrete wavelet transform. It can be compared with the application of Fast Fourier Transform for calculating the coefficients of a Fourier series.

Construction

A given continuous signal is first converted by the orthogonal projection on a sub-space of an orthogonal multi-scale analysis to a time-discrete sequence of coefficients. The larger, the more accurate then the resulting approximation. In many cases it is sufficient to

Generated. Together, these form an analysis filter bank, which has operations in it are explained below.

After steps of the recursion the conse-quences

The goal of this transformation is that the are "thin " busy and therefore can compress well.

The filter and sufficiently frequency-selective, the output signal was band- limited, and has been obtained the WCS - sampling according to the first sequence of coefficients, the first low-pass result contains all the signal components up to half the Nyquist frequency, the band pass result, the overlying both times with a bandwidth corresponding sampling.

Analysis and Synthesis

The herringbone decomposition in the multi-scale analysis corresponds to a composite of the low-pass and band-pass discrete-time filter bank, it is a discrete-time signal is divided into a high band and a low band ( convolution of episodes). Then both signals are clocked down (English downsampling ) to

With this is the time reversed sequence

Referred to. The clock speed of a sequence means that a new sequence is formed from the elements with even index,

All together these operations results in a term by term calculation procedure of the analysis filter bank

From orthogonality implies that the output signal can be recovered, first, the low-pass and band-pass portions and in the sampling rate can be extrapolated, it is referred to as up-sampling, is folded with the scaling and Waveletmasken and then added

Or coefficient-wise

The transition from to is called analysis, the inverse synthesis. It will be appreciated that the transform of a finite signal now has about the same number of samples as the signal itself, that contains as much information.

Extensions

It is not necessary that the consequences of the analysis filter bank with those in the synthesis filter bank identical as above, but is not then guarantee that the combination of the two filter banks reconstruct the output signal. Is this still the case, one speaks of reconstruction of complete (English perfect reconstruction ) or biorthogonality of the wavelet bases.

• Numerical Mathematics
• Wavelet
• Discrete transformation