5.3 Methods for Analysis and Processing of Discrete Biosignals

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shift j

fT(j)

Fig. 5.9: Preparation to increase the sampling frequency by three times by inserting two values

between each of the old samples, but setting them all equal to zero.

2.

If the sampling frequency is to be increased, this would in principle mean re-

constructing the analogue signal and then sampling it again at an increased fre-

quency. An interpolation lowpass according to Figure 5.1 and Figure 5.7 and an

additional analogue/digital converter would therefore be necessary. But there is

also different:

First, depending on the requirements M −1, additional additional values are in-

serted between each two samples corresponding to the new sampling frequency,

but all of them are set equal to zero (see Figure 5.9), i.e.

̃fTa(j) = fTa(j/M)

for j = i ⋅M

= 0

other.

The sampling frequency is thus increased by M times and the sampling inter-

val T ­u = Ta/M is decreased by M times. In the frequency domain, according to

the discrete-time Fourier-transformation to Equation 5.9, one then obtains for the

spectrum of this signal:

̃FTa(f) =

∞

∑

j=−∞

̃fTa(j)e^−j2πfjT ­ub =

∞

∑

i=−∞

̃fTa(i ⋅M)e^{−j2πfiMT/M}

=

∞

∑

i=−∞

fTa(i)e^−j2πfiTa = FTa(f).

Attention: The spectrum does not change after inserting M zeros between every

two samples!

However, since the sampling frequency has now been increased by M times, the

new half sampling frequency also changes and thus there are also additional spec-

tral components in the signal that must be suppressed by a low-pass filter (see

Figure 5.10).