5.3 Methods for Analysis and Processing of Discrete Biosignals

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189

QRS-complex

sym4 wavelet

discrete time n

voltage U / mV

Fig. 5.29: QRS complex compared with a wavelet of the symlet-family.

3.

The further decomposition of the low-frequency signal xA2 is done as in the 2nd

stage. In further stages the procedure is the same and so many stages are chosen

until a sufficiently large accuracy in the reconstruction is achieved. Thus the

whole spectrum X(l) of the original signal x(i) can be described by adding the

spectra of the last low pass XAN and all high passes XDi, i = 1, . . . , N, i.e.

X(l) = XAN(l) +

N

∑

i=1

XDi(l) ,

N : number of steps .

(5.77)

4.

To reconstruct the original signal, start from the low-pass signal of the last stage,

reverse the downsampling by adding a new sample between the old samples by

interpolation, and apply this to the input of the last low-pass. Proceed in the same

way with the higher-frequency signal component. The output signals of the low-

pass and high-pass are now added, and one gets again the signal XA2 of the pen-

ultimate stage (reverse order as in Figure 5.28). With the following calculation of

the low-pass signal xA2, the procedure is repeated for all previous stages up to the

first stage, which then completes the reconstruction.

When processing the spectra, they do not necessarily have to be divided with ideal

high or low passes. The transfer functions of the high and low pass can also overlap

somewhat at one stage and have a smooth transition between passband and stopband,

which makes them easier to realise with the help of digital filters. In Matlab, wavelets

of the families Daubechies, Coif|lets, Symlets, Fejer-Korovkin Filters, Discrete Meyer,

Biorthogonal and Reverse Biorthogonal can also be used. Some wavelets can also be

quite similar to the signal under investigation, and thus fewer stages are needed in

the decomposition. For example, the symlet Sym4 resembles a QRS complex in an ECG

(see Figure 5.29).