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5 Methods for Discrete Processing and Analysis of Biosignals

The Fourier-transform of this window function h(i) := F{H(l)} is exactly the inverse of

the window function w(i) and must satisfy the condition

∞

∑

i=−∞

h(i)w(i) = 1

(5.75)

Examples of spectrograms (power density spectra as a function of the position of a

time window) obtained using STFT with two Gaussian windows of different widths are

shown in Figure 5.26 and Figure 5.27. They were calculated with the function tfrsp()

from the toolbox Time Frequency for Matlab and Octave. This toolbox is free and can

be downloaded from http://tftb.nongnu.org/. The Gaussian window function can be

calculated in the discrete-time domain by

w(n) := e⁻¹

2^{( n−iM}

σ⋅iM ⁾

2

,

für n = 0, . . . , NF −1

(5.76)

are described [32]. σ is a measure of the width of this window.

Below the spectrogram, the signal sampled at 20 Hz is shown. To the left of the

spectrogram the complete spectrum (without window function) is shown, where the

spectral lines at the four frequencies (1, 4, 6 and 8 Hz) with the wide Gaussian win-

dow of 4 s match well with the horizontal lines of the spectrogram. The frequency can

therefore be determined well. However, the width of these lines does not reflect the

locations of the power density fluctuations within the duration of a single sinusoidal

oscillation, which can be explained by the uncertainty condition according to Equa-

tion 5.70. A good frequency resolution means a poor time resolution and vice versa.

With the narrow Gaussian window with a width 0.5 s, the frequency of the four sine

oscillations can no longer be determined exactly from the spectrogram. However, it

can be determined exactly when the minima and maxima of the sinusoidal oscillation

with 1 Hz occur with the bar at the bottom left, because the power density is greatest

there. Due to the poorer frequency resolution, one now obtains a better time resolution

than with the wide window in Figure 5.26.

5.3.3.2 The Discrete Wavelet Transform

A disadvantage of the short-time-Fourier-transform is that the window width in the

time domain always has a constant value. If the frequency in a time domain is to be

determined with a certain accuracy, enough samples would have to lie within the time

window. For low-frequency signals, a sufficiently wide window is needed so that this

oscillation can be sampled sufficiently often per period (uncertainty principle of signal

processing). A higher-frequency oscillation does not need this wide time window and

could also be analysed by a shorter one.

As in the continuous-time case, the discrete-time wavelet transformation (DWT)

provides a remedy, whereby the time window depends on the frequency to be invest-

igated. The higher this frequency, the narrower the time window can be and vice versa

(see chapter 2).