5.3 Methods for Analysis and Processing of Discrete Biosignals

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185

power density

signal

Spectrogram with wide window

Fig. 5.26: Spectrogram (power density spectrum as a function of window range) of a signal with four

sinusoidal oscillations with 1, 4, 6 and 8 Hz in an interval of 20 s and a sliding Gaussian window with

a width of 4 s.

the window is therefore chosen to be odd so that the window is symmetrical about its

centre iM := (NF + 1)/2. Within the entire period with N samples, the window centre

shall lie at sample i = iF. From this follows for the digital STFT:

fp(i, iF)w(i −iF) = ¹

N

N−1

∑

l=0

STFT^(w)

fp ^{(iF, l)ej2πil/N}

(5.72)

STFT^(w)

fp ^{(iF, l) =}

iF+iM−1

∑

i=iF−iM+1

fp(i)w(i −iF)e^−j2πil/N

(5.73)

where iF = iM/2 to N −1 −iM/2.

In principle, it is now a two-dimensional Fourier transformation; because the loc-

ation of the window iF has now been added as an additional variable. The original

signal fp(i) is obtained if the original spectrum is also used in the reverse transforma-

tion. Since the Fourier transform of a sum is the sum of the Fourier transforms of the

individual summands, the original signal fp(i) could also be obtained by summing

over the Fourier transforms with all windows when the individual windows would

gradually fade out the total spectrum.

Another possibility is that in the frequency domain the spectrum is multiplied by

a window function H(l) that cancels the effect of the temporal window w(i −iF) [33],

ie.

fp(i) = ¹

N

N−1

∑

l=0

STFT^(w)

fp ^{(iF, l)H(l)ej2πil/N .}

(5.74)