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5 Methods for Discrete Processing and Analysis of Biosignals

5.2 Discrete Transformations of Signal Processing

5.2.1 The Discrete-Time Fourier Transform

A discrete sequence could arise, for example, when sampling a signal with square

pulses according to Equation 5.3, if the square width goes ∆T →0. If we now calculate

the spectrum of this rectangular sequence not according to Equation 5.6, but add up

the spectra of these rectangular pulses weighted with the samples and delayed with

kTa, we obtain from Equation 5.3 with the help of the similarity theorem Equation 5.5:

F∆T(f) = F (f∆T(t)) = A

∞

∑

k=−∞

f(kTa) ⋅F (rect ( ^{t −kTa}

∆T

))

= A

∞

∑

k=−∞

f(kTa) ⋅e^−j2πfkTaF (rect ( ^t

∆T ⁾⁾

= A ⋅∆T si(πf∆T) ⋅

∞

∑

k=−∞

f(kTa) ⋅e^−j2πfkTa .

The factor e^−jωkTa results from the delay of the square pulses by kTa, which can be

interpreted as passing through a delay element that has this transfer function. The

sum

FD(f) :=

∞

∑

k=−∞

f(kTa) ⋅e^−j2πfkTa

(5.8)

is the discrete-time Fourier-transformation of the samples f(kTa), where one normally

omits the factor Ta in the function f(kTa) (cf. e.g. [52]). One now obtains altogether for

the outward and backward transformation:

f(k) = Ta

Ta/2

∫

−Ta/2

FD(f)e^j2πfkTadf, FD(f) =

∞

∑

k=−∞

f(k)e^−j2πfkTa ,

(5.9)

with the relation

F∆T(f) = A ⋅∆T si(πf∆T) ⋅FD(f) .

(5.10)

If the square pulse gF(t) = A ⋅rect(t/∆T) when sampled according to Figure 5.4 goes to

infinite amplitude height, i.e. A →∞and vanishing width, i.e. i.e. ∆T →0, but with

A⋅∆T = 1 into a dirac pulse δ(t), it follows from Equation 5.10 with lim∆T→0 si(πf∆T) =

1:

F∆T(f) = FD(f) ,

when sampling with δ(t) pulses.

(5.11)

The discrete-time Fourier-transform therefore describes in the frequency domain the

spectrum of a signal sampled with Dirac pulses.