5.3 Methods for Analysis and Processing of Discrete Biosignals

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181

Between the auto-correlation and the power density⁹ (or energy density) there is a

close relationship which can be described as follows:

The convolution sum (Equation 5.49) can be generally interpreted, as in the calcu-

lation of the output signal by convolution of an input signal with the impulse response

of a digital system (cf. Figure 5.16), such that one signal, e.g. x1, is the input signal and

the other, x2, is the impulse response of a linear digital system. In the frequency do-

main, this means that to determine the output spectrum X2, the input spectrum is

multiplied by the transfer function G, or in abbreviated notation:

x(̃n) := x1(̃n) ∗x2(̃n) ∘−−∙X1(m) ⋅X2(m) ,

(5.56)

with

x1(̃n) ∗x2(̃n):=

∞

∑

̃ν=−∞

x1(̃ν)x2(̃n −̃ν) .

(5.57)

Using the substitution n := −̃ν and ν := ̃n we get:

−∞

∑

n=∞

x1(−n)x2(ν + n) =

∞

∑

n=−∞

x1(−n)x2(ν + n) .

(5.58)

If x1(−n) := x(n) and x2(n + ν) := x(n + ν) still hold, it follows.

x(̃n) =

∞

∑

n=−∞

x1(−n)x2(ν + n) =

∞

∑

n=−∞

x(n)x(ν + n) = ^—RXX(ν) .

(5.59)

Result

The mean auto-correlation function RXX(ν) can therefore be interpreted as a convolu-

tion of the function x1(n) = x(−n) with the function x2(n) = x(n).

Insertion into Equation 5.56 further gives the relation between the time and fre-

quency domains:

—RXX(ν) ∘−−∙X1(m) ⋅X2(m) = F(x(−ν)) ⋅F(x(ν)) .

(5.60)

Because of

x(n) ∘−−∙X(m)

and

x(−n) ∘−−∙X^∗(m)

(similarity theorem)

then follows from this as well:

RXX(ν) ∘−−∙X^∗(m)X(m) = |X(m)|² = SXX(m)

(5.61)

with SXX(m) as the spectral power or energy density.