5.3 Methods for Analysis and Processing of Discrete Biosignals

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169

shift j

normalized

Fig. 5.13: Matlab-Example of a normalized auto-covariance CXX(j) of a Gaussian distributed mean-

free random number sequence with 5000 values and a variance of one.

Furthermore, since the random signal is ergodic⁵ and thus the expected value is time-

independent, it follows because of

E[X(μ)] = E[X(μ + j)] := E[X]:

—CXX(j) = RXX(j) −(E[X])² .

(5.43)

As a result, the mean-free random signal^—CXX(j) is simply obtained by subtracting the

root mean square (E[X])² from the auto-correlation function.

Redundancy-Free Biosignals

A redundancy-free signal exists if X(μ + j) for j

̸= 0 is independent of the preceding

measured values X(μ):⁶

RXX(j) = E[X(μ)X(μ + j)] = E[X(μ)] ⋅E[X(μ + j)] = (E[X])² ,

j

̸= 0 .

It follows for the auto-covariance^—CXX(j) according to Equation 5.43:

—CXX(j) = .

{

{

{

E[X²] −(E[X])² = σ²

X ^,

at j = 0

.0 ,

other

= σ²

X ^{⋅δ(j), δ(j) :}

discrete unit pulse.

(5.44)

Thus, it follows that for redundancy-free and mean-free signals, the auto-correlation

consists only of a discrete unit momentum weighted by the variance σ²

X ^{of the ran-}

dom variable X. An example of such a covariance function is shown in Figure 5.13.

5 A random signal is called ergodic if the mean values are equal over the multitude and over time.

6 For independent or redundancy-free random signals, the expected value of the product of two ran-

dom signals, e.g. A and B, is equal to the product of their expected values, i.e. E[A ⋅B] = E[A] ⋅E[B].