Biosignal Processing Fundamentals and Recent Applications with MATLAB (Stefan Bernhard, Andreas Brensing etc.)

156

5 Methods for Discrete Processing and Analysis of Biosignals

The periodic continuation of a time limited signal results in a discrete

spectrum with Dirac pulses whose weighting values, except for the factor

1/Tp, correspond to the values at the individual frequency points of the

continuous spectrum F(f) of the original non-periodic signal f(t).

In principle, the only requirement for the type of time signal is that it must be that it

should be limited in time and integrable. It can therefore also be a square-wave pulse

train, as it arises when sampling with square-wave pulses according to Figure 5.4. For

this purpose, we investigate the discrete-time Fourier transform according to Equa-

tion 5.9

FD(f) =

∞

∑

k=−∞

f(k)e⁻^j²^π^fkT^a

(5.15)

of a sampled signal f(k) and determine the spectrum periodic with sampling frequency

fa = 1/Ta for one period at N frequency points, i.e. at frequencies l ⋅fa/N = l/(NTa)

with l = 0, 1, . . . , N −1, i.e.

FD (

NTa

) =

∞

∑

k=−∞

f(k)e⁻^j²^π^kl^/^N,

l = 0, 1, . . . , N −1 .

(5.16)

The infinite sum in Equation 5.16 is now divided into an infinite sum of these periods,

each having N values (cf. [62]), ie:

FD (

NTa

) = ⋅⋅⋅+

−1

∑

k=−N

f(k)e⁻^j²^π^kl^/^N+

N−1

∑

k=0

f(k)e⁻^j²^π^kl^/^N+

2N−1

∑

k=N

f(k)e⁻^j²^π^kl^/^N+ ⋅⋅⋅

∞

∑

m=−∞

N(m+1)−1

∑

k=mN

f(k)e⁻^j²^π^kl^/^N.

(5.17)

In the case of a double sum as in the last equation, the order may be reversed. If we

further replace the running variable k by k := i + mN, it follows from Equation 5.17:

FD (

NTa

) =

N−1

∑

i=0

∞

∑

m=−∞

f(i + mN)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

fp(i)

e⁻^j²^π^il^/^N.

(5.18)

A comparison of the last Equation 5.18 with Equation 5.14 shows that the under-

clamped part of the periodically repeating signal f(t) is given at the sampling instants

ti = i ⋅Ta. If the signal f(t) is limited so that it disappears after the period NTa, the

original signal f(t) can be determined from the periodically repeating signal fp(t) by

truncation after N samples.

However, the values of the spectrum FD(l/NTa) of the discrete-time Fourier-

transform of the sampled signal f(t) can also be described by the coefficients of a com-

plex Fourier series of the periodic signal fp(t). With the discrete Fourier-transformation