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5 Methods for Discrete Processing and Analysis of Biosignals

In many cases, the auto-correlation depends only on the difference τj of the measure-

ment time points and not on the absolute time ti at which the measurement took place.

Equation 5.37 can therefore be simplified as follows:

—RXX(τj) = lim

N→∞

1

2N + 1

N

∑

k=−N

X(tk)X(tk + τj)

(5.38)

or even simpler:

—RXX(j) = lim

N→∞

1

2N + 1

N

∑

k=−N

X(k)X(k + j) .

(5.39)

Mean Influences

When analysing biosignals, it is often important to examine the correlations in the

changes of a measurand rather than the measurand itself. For example, the pulse rate

of a heart is influenced by respiration. This influence could then be determined by cal-

culating the auto-correlation. This then oscillates around an individual mean value.

However, the interesting values can only be recognised in the oscillating values and

can often be difficult to determine if a large constant value also comes into play in the

correlation. Therefore, the auto-correlation of deviations from the mean is also used,

which is called auto-covariance ^—CXX(j), and is defined as follows:

—CXX(j) = lim

N→∞

1

2N + 1

N

∑

k=−N

{X(k) −E[X]}{X(k + j) −E[X]} ,

(5.40)

with

E[X] :=

1

2N + 1

N

∑

k=−N

X(k) .

(5.41)

The relationship between auto-correlation and auto-covariance is obtained by decom-

posing the random signal X into a mean-free random signal ^̃X(j) and its mean accord-

ing to X(μ) = ^̃X(μ) + E[X]. Then one obtains for the auto-correlation ^—R̃X̃X(m) of the

mean-free random signal or the auto-covariance^—CXX(j):

—CXX(j) = R̃X̃X(j) = E[̃X(μ)̃X(μ + j)]

= E[(X(μ) −E[X(μ)])(X(μ + j) −E[X(μ + j)])]

= E[X(μ)(X(μ + j)] −E[X(μ)]E[X(μ + j)]

−E[X(μ)]E[X(μ + j)] + E[X(μ)]E[X(μ + j)]

= E[X(μ)(X(μ + j)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

RXX(j)

−E[X(μ)]E[X(μ + j)] .

(5.42)