5.3 Methods for Analysis and Processing of Discrete Biosignals

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159

From the integral with limits from −^Ta

2 ^{to + Ta}

2 ^{, a closed orbital integral has now}

emerged, and the imaginary axis in the p := σ + jω-plane is thereby mapped onto the

unit circle in the z = e^pTa-plane. The z- transformation can be interpreted as a map-

ping between the p- and the z-plane. However, it is ambiguous because the imaginary

axis in the p-plane is mapped several times to the unit circle in the z-plane. In this

case, the integration must be done on a closed path in the mathematically positive

direction in the convergence domain of the z-transform. The relation in Equation 5.28

is valid only for absolutely summable causal time signals.

Example

Exponential sequence: f(n) = e^−knT

Thereby, Equation 5.29

F(z) =

∞

∑

n=0

e^−knTz⁻ⁿ =

∞

∑

n=0

(e^−kTz⁻¹)

n =

1

1 −e^−kTz^{−1 ,}

|z| > e^−kT .

(5.30)

In the range |z| > e^−kT, the ztransform converges. Only for k < 0 is the exponential se-

quence f(n) absolutely summable. Then it also has the discrete-time Fourier-transform

FD(f) = F(z = e^jωTa) =

e^jωTa

e^jωTa −e^{−kT ,}

k > 0 .

(5.31)

5.3 Methods for Analysis and Processing of Discrete Biosignals

As in the continuous-time case, the biosignal can also be studied using its samples

in the discrete-time domain, as long as the latter is in good agreement with the

continuous-time signal. This requires compliance with the sampling theorem, low

sampling distortion, and sufficient accuracy in quantization to avoid excessive round-

ing noise.⁴

5.3.1 Time Domain Signal Analysis and Matching

The analysis of the biosignal in the time domain, in contrast to the frequency domain,

can be clearly illustrated by its sampled values in the course of the curve. In addition

to the determination of distinctive points (e.g. minima, inflection points, maxima) in

the curve progression by means of a curve discussion, the deviations of individual

function values, for example, can also be determined by calculating the statistical

4 Rounding noise arises from the error that occurs in digitization due to truncation or rounding in the

course of converting an exact sample to a sample with finite accuracy.