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5 Methods for Discrete Processing and Analysis of Biosignals

1st level

2nd level

3rd level

LP

LP

LP

Fig. 5.28: Principle of a digital wavelet-transformation (DWT) up to level 3 using subband cod-

ing [46].

the wavelet according to Morlet or the Mexican-Hat have very large overlaps and thus

a lot of redundancy [33]. Less expensive wavelets can be obtained by applying sub-

band coding, which has long been used in message transmission in the form of the

quadrature-mirror-filter (QMF) [38].

In 1988, Stephane Mallat and Yves Meyer developed a method based on a di-

gital high-pass and low-pass filter pair, in which the algorithm is the same as in the

subband coding known in digital signal processing with quadrature-mirror-filter [46].

They showed that continued digital filtering of the low-pass component produces sets

of coefficients corresponding to a wavelet-decomposition. The wavelet itself is the im-

pulse response of a high-pass, the scaling function that of a low-pass. The principle

of this procedure presented in Figure 5.28 is:

1.

The frequency range of the signal x(i) to be investigated by wavelets is divided

in the middle in the 1st stage, whereby the low-frequency part is generated by a

low-pass filter and the higher-frequency part by a high-pass filter. However, these

parts of the signal only have half the bandwidth each and can therefore also be

reconstructed with half the samples. This is achieved by discarding every second

sample (downsampling). Altogether, the partial signals xA1 and xD1 are now cre-

ated. The low-frequency signal xA1 is also referred to in the literature as a scaling

function, the higher-frequency xD1 as a wavelet.

2.

The frequency range of the low-frequency signal xA1 is now divided further in

the middle in the 2nd stage, divided into a low-frequency and a higher-frequency

range by the same low-pass and high-pass filters –but this time with half the

cut-off frequency –and then further processed with a quarter of the sampling fre-

quency due to the bandwidth being half smaller (after downsampling again with

a factor of 2). The signals xA2 and xD2 are now produced.