192

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5 Methods for Discrete Processing and Analysis of Biosignals

Fig. 5.31: Input and output-Signals of a discrete-time system.

Fig. 5.32: Building elements of a discrete-time linear system.

In general, the input numbers are the encoded samples of an analogue signal. After

processing by the mathematical algorithm, the output numbers are then converted

back to analogue values and interpolated between sampling times (see Figure 5.36).

As explained in section 5.1, the frequency spectrum of the sampled analogue signal

is periodic with the sampling frequency (cf. e.g. Figure 5.5). However, this also means

that the transfer function of the digital filter, which represents the ratio between the

output and input spectrum, must also be periodic. Figure 5.37 shows as an example

the periodic spectra of an ideal low and high pass. Next, the spectrum of the high pass

can be easily generated from the spectrum of the low pass by shifting the spectrum of

the low pass to the left or right by an odd multiple of fa/2, ie:

Ghp(f) = Gtp(f + [2k + 1] fa/2) ,

k = 0, ±1, ±2, . . .

(5.80)

Since according to the frequency shift theorem in the Fourier-transformation a fre-

quency shift with the frequency fV in the spectral domain causes a multiplication by

the factor e^{j2πfV t} in the time domain, a frequency shift of fa/2 =

1

2Ta ^{(i.e. i.e. k = 0 in}