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5 Methods for Discrete Processing and Analysis of Biosignals

Fig. 5.5: Spectrum of the signal FTa(f) sampled with Dirac pulses obtained by periodically repeating

the spectrum of the original signal F(f).

results for the resulting spectrum F∆T(t) of the rectangular sequence f∆T(t) after

sampling:²

F∆T(f) = FTa(f) ⋅F {A ⋅rect ( ^t

∆T ^)}

= [ ^A

Ta

∞

∑

ν=−∞

F (f −^t

Ta

)] ⋅∆T si(πf∆T)

= ^{A ⋅∆T}

Ta

si(πf∆T) ⋅

∞

∑

ν=−∞

F (f −^t

Ta

) .

(5.6)

The result for the spectrum of the signal FTa(f) sampled with a Dirac-pulse train is a

periodic repetition of the source signal spectrum F(f) with sampling frequency fa =

1/Ta (cf. Figure 5.5).

However, this only works if the periodic spectrum repetitions do not overlap, as

shown in Figure 5.5. In the case of overlapping spectra, a restoration of the original

analogue signal by a simple low-pass filtering is no longer possible (cf. e.g. Figure 5.6).

For a recovery it is namely necessary that the upper cut-off frequency of the analogue

signal is smaller than half the sampling frequency:

fg < ^fa

2 ^,

Shannon sampling theorem.

(5.7)

The spectrum of the analogue signal, and thus the signal itself, can be reconstructed

from the spectrum of the sampled signal using a low-pass filter, which filters out only

the spectrum component around the frequency zero point. The associated interpola-

tion function between the samples is thus generated by the impulse response of the

low-pass filter (see Figure 5.7).

This type can also be interpreted as ideal sampling with Dirac pulses. However, ac-

cording to Equation 5.6, the signal sampled not ideally but with square wave functions

2 si(x) := ^sin(x)

x