5.3 Methods for Analysis and Processing of Discrete Biosignals

|

167

5.3.1.2 The Auto- and Cross-Correlation

The correlation generally describes the similarity of the random variables X1 and X2

and is proportional to the energy of their difference according to [45]

E[(X1 −X2)²] = E[X²

1^{] −2 E[X1X2]}

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

RX1X2

+E[X²

2^{] .}

(5.34)

The operator E forms the expected value of a random variable. Therefore, E[X1²] and

E[X2²] are the expected values of the squared random variables X1 and X2, respect-

ively, and E[X1X2] is the expected value of the product of the random variables. This

is also denoted by RX1X2 and is defined as their associated correlation.

The random variables are often signals that depend on time. If they are different,

e.g. two different measurements, RX1X2is called cross-correlation. If it is one and the

same signal, i.e. X1 = X2 = X, only at different measuring points, RXX is the corres-

ponding auto-correlation.

In the case of random variables, the expected value is also the mean value, which

is formed either over a large group (group mean index, e.g. mean value for several

identical dice) or over time (time mean index, e.g. mean value for one dice when dice

are rolled several times).

In biosignal processing, different time averages of measured values are usually

examined, which can arise, for example, from measurements with electrodes at differ-

ent points on the body, such as in ECG and EEG, or from the measurement of electrical

fields in computer or magnetic resonance tomography. For discrete-time signals, one

then obtains for the correlation RX1X2 in general:

RX1X2 =

∞

∑

i=−∞

∞

∑

j=−∞

Xi1Xj2PXi1Xi2(Xi1, Xj2) .

(5.35)

Here PXi1Xj2(Xi1, Xj2) gives the probability density function for the values Xi1 and Xj2

to occur simultaneously. If this function is the same for all combinations Xi1, Xi2, it

is simply given by the inverse of the number of possibilities – e.g., for a die with 6

possible rolls {1 to 6}, the probability of a given number being rolled is ¹

6^{. Then we get:}

RX1X2 = lim

L→∞

1

(2L + 1)²

L

∑

i=−L

L

∑

j=−N

Xi1Xj2 .

(5.36)

The probability density function PXi1Xj2(Xi1, Xj2)cantherebybeexpressedby1/(2L + 1)².

In auto-correlation, time dependence is usually investigated, and one obtains

with the definitions Xi := X(t = ti) and Xj = X(t = ti + τj) when summing up all signal

values with the same difference τj of the measurement time points in an interval with

2N + 1 values:

RXX(ti, τj) = lim

N→∞

1

2N + 1

N

∑

l=−N

X(ti + tl)X(ti + tl + τj) .

(5.37)