182

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5 Methods for Discrete Processing and Analysis of Biosignals

Fig. 5.24: Power-density calculation of heart-rate-variability according to the example given in Fig-

ure 5.14 using the magnitude square of the fast Fourier-transformation.

Further result

The Fourier transform of the mean auto-correlation function RXX(ν) is the spectral

power or energy density SXX(m), i.e. analogous to Equation 5.23 applies for discrete

signals in a range with N values (e.g. periodic signal or clipping by windowing):

RXX(ν) = ¹

N

N−1

∑

m=0

SXX(m)e^j2πνm/N ,

SXX(m) =

N−1

∑

ν=0

RXX(ν)e^−j2πνm/N .

(5.62)

This means that from the measurement of a statistically distributed interference sig-

nal, its power spectral density can be determined by means of auto-correlation or

auto-covariance. From this, the magnitude spectrum |X(m)| can be determined due

to SXX(m) = |X(m)|², but not the phase course. For example, Figure 5.24 shows the

power density of the changes in the pulse rate shown in Figure 5.14. One can clearly

see a high DC component (spectral component at f = 0 Hz) resulting from the fact that

the pulse rate varies around a certain mean value, about 84 beats per minute, and

this variation is mainly due to the fact that the pulse rate varies up to approx. 0.3 Hz

is present. To calculate this, one can apply the fast Fourier-transform to the heart rate

variability in Matlab using the function fft() and then determine the power spec-

tral density according to Equation 5.61 by squaring the magnitude spectrum. Or one

can use the function xcorr() to first determine the auto-correlation and then use the

fast Fourier-transformation and thereby calculate the power spectral density without

squaring. A third possibility is to apply the function periodogram() to the heart vari-

ability, which is the simplest, because the heart rate variability can be passed to it

directly as an argument, and one does not need to calculate either an auto-correlation

or the square of a Fourier-transformation.

9 The power density gives the power frequency for a signal with finite power (e.g. sinusoidal oscilla-

tion), the energy density gives the power frequency for a signal with finite energy (e.g. impulse).