2.3 Definition and Classification of Signals

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Covariance and Correlation

Another important quantity in statistics is the covariance of two signals. If X and Y

are two real, integrable random variables whose product is also integrable, i.e., the

expected values E(X), E(Y) and E(XY) exist, then the covariance of X and Y is defined

as follows:

Cov(X, Y) = E ((X −E(X)) ⋅(Y −E(Y))) .

(2.42)

The value of the covariance makes tendential statements about the values of two ran-

dom variables X and Y: Positive covariances stand for a monotonic relationship (com-

mon tendency of X and Y), negative ones for an inverse monotonic relationship (op-

posite tendency of X and Y) of the random variables. The random variables X and Y

have no monotonic relationship if the covariance is zero.

The covariance thus indicates the direction of a monotonic relationship between

two random variables, but the strength of the relationship cannot be read from it.

Comparability with other signal pairs is achieved, for example, by normalising the

covariance with the standard deviation. As shown in subsection 2.3.6, this leads to

the correlation coefficient of the two random variables X and Y.

Signal Components of Biosignals

Normally, real biosignals U(t), like the ECG raw signal in Figure 2.9, basically contain

a deterministic signal part s(t) – mostly the biosignal of the physiological process to

be measured – and a stochastic signal part consisting of noise and artefacts r(t)+a(t).

The measured biosignal is thus:

U(t) = s(t) + r(t) + a(t) .

(2.43)

The relevant information of the biosignals is generally contained in the deterministic

part s(t), while the stochastic signal part often only contains noise and artefacts. How-

ever, there are also numerous applications that deal with the random deviations in the

signals, i.e. the stochastic component, during evaluation. For example, in repeatedly

measured motion sequences, the deviations from the mean value curve of the motion

can provide important information about the accuracy of the motion sequence. This

issue is discussed in detail in section 6.2.

Stationarity of Stochastic Signals

The class of stochastic signals can be further divided into stationary and stationary

stochastic signals. In terms of content, stationarity means that certain stochastic prop-

erties of the process Xt, t = 0, 1, 2, . . . are time- or location-invariant.. Basically, time

or location invariance means that a certain stochastic property is independent of a

shift in time or location, i.e. it does not change, for example, if the time origin is shif-

ted by some value τ.

Strictly speaking, stationary signals are divided into strong and weak stationar-

ity. A stochastic signal is covariance stationary or weakly stationary if the first two