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5 Methods for Discrete Processing and Analysis of Biosignals

Fig. 5.25: Pulse-width in time and frequency domain defined over equal areas of a rectangular pulse.

By applying the Fourier-transformation (Equation 5.63), for the time t = 0 and the

frequency f = 0, we obtain

I(0) =

∞

∫

−∞

i(t)e^−j0tdt =

∞

∫

−∞

i(t) dt ,

(5.68)

i(0) =

∞

∫

−∞

I(f)e^j2πf0df =

∞

∫

−∞

I(f) df ,

(5.69)

and you get the result

I(0) = T ⋅i(0) ;

i(0) = B ⋅I(0) .

(5.70)

Substituting finally yields the sought relationship between the pulse widths in the

time and frequency domain

B ⋅T = 1 .

(5.71)

Thus the bandwidth B of a pulse is equal to the reciprocal of its temporal pulse width T

and vice versa. A narrow window in the time domain therefore produces a wide win-

dow in the frequency domain. So if one wants to know exactly when, for example, a

heart sound occurs, this can be done by continuously shifting a time window by vary-

ing t0, but the frequency resolution is very small. However, if one wishes to examine

the frequency of a heart sound with its frequency components in detail, it is not pos-

sible to determine exactly when it occurred.

5.3.3.1 The Short-Time Fourier Transform (STFT)

In the short time-Fourier-transformation (STFT), a uniform time function w(t−t0) with

the same width is multiplied by the signal f(t) to be investigated and the centre of the

window function t0 is continuously shifted until the entire signal waveform has been

windowed out and investigated. Independent of t0, the width of the window function

remains constant. In the discrete-time case, the STFT can be calculated analogously

to the discrete Fourier-transformation according to Equation 5.23, but here it is not

necessary to sum up over the entire period length with N samples, but only over the

window width with an odd number of NF samples. The number of samples NF within