Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

infinity Endscripts StartFraction 1 Over upper T EndFraction integral Subscript negative upper T slash 2 Superscript upper T slash 2 Baseline f left-parenthesis t right-parenthesis g left-parenthesis t plus tau right-parenthesis d t"/> (1.157)

1.5.4 Fourier Analysis of Random Signals

The Fourier transform of a random signal would lead to infinite results because it is not approaching zero. Fourier series cannot be applied too, because there is no periodicity in the signal. A smart solution is to use the correlation function for the Fourier transform and not the random signal itself. The correlation is a decaying function that is suitable for infinite integration due to the 1/T factor in (1.157). We start with the pair of Fourier transforms of the autocorrelation:

StartLayout 1st Row 1st Column upper S Subscript f f Baseline left-parenthesis omega right-parenthesis 2nd Column equals integral Subscript negative normal infinity Superscript normal infinity Baseline upper R Subscript f f Baseline left-parenthesis tau right-parenthesis e Superscript minus j omega tau Baseline d tau EndLayout (1.158a)

StartLayout 1st Row 1st Column upper R Subscript f f Baseline left-parenthesis tau right-parenthesis 2nd Column equals StartFraction 1 Over 2 pi EndFraction integral Subscript negative normal infinity Superscript normal infinity Baseline upper S Subscript f f Baseline left-parenthesis omega right-parenthesis e Superscript j omega tau Baseline d omega EndLayout (1.158b)

Sff(ω) is called the auto spectral density auto spectral density of the signal f(t). It is a measure of how the signal energy is distributed over the frequency range. This becomes quite obvious if we look at τ = 0 and use (1.153):

upper R Subscript f f Baseline left-parenthesis 0 right-parenthesis equals StartFraction 1 Over 2 pi EndFraction integral Subscript negative normal infinity Superscript normal infinity Baseline upper S Subscript f f Baseline left-parenthesis omega right-parenthesis d omega equals upper E left-bracket f squared left-parenthesis t right-parenthesis right-bracket (1.159)

The autocorrelation is a real symmetric function, thus the auto spectrum as Fourier transform of this is a symmetric real valued function in frequency.

We define the cross spectral density function as the Fourier transform of the cross correlation function from Equation (1.151) and the back transformation:

StartLayout 1st Row 1st Column bold-italic upper S Subscript f g Baseline left-parenthesis omega right-parenthesis 2nd Column equals integral Subscript negative normal infinity Superscript normal infinity Baseline upper R Subscript f g Baseline e Superscript minus j omega tau Baseline d tau EndLayout (1.160)

StartLayout 1st Row 1st Column upper R Subscript f g Baseline left-parenthesis omega right-parenthesis 2nd Column equals StartFraction 1 Over 2 pi EndFraction integral Subscript negative normal infinity Superscript normal infinity Baseline bold-italic upper S Subscript f g Baseline e Superscript j omega tau Baseline d tau period EndLayout (1.161)

Changing the integration constant τ=−τ′ and using symmetry (1.156) reveals

bold-italic upper S Subscript f g Baseline equals bold-italic upper S Subscript g f (1.162)

Mathematically, the expected value of an ergodic processes is derived by the investigation of infinite time periods. However, this can neither be realized in tests nor numerical simulations. We are restricted to specific time slots, so let us assume the Fourier transform of two random signals f(t) and g(t).

(1.163)

It can be shown by a straightforward but lengthy proof by Bendat and Piersol (1980) that the cross spectral density is a complex function given by

(1.164)

Here, we take the signals of infinite duration from an ensemble of recordings. The same can be shown for the autospectrum.

(1.165)

So these important relationships show, that if we have an infinite time period T and we average over a whole ensemble of such records we can determine the auto and cross spectra from this. In practical test and simulation situations we will see that for a given number of time signals of finite time T we can estimate the power- and cross spectral density. For convenience we abbreviate Equation (1.164)

bold-italic upper S Subscript f g Baseline left-parenthesis omega right-parenthesis equals upper E left-bracket bold-italic upper F Superscript asterisk Baseline left-parenthesis omega right-parenthesis bold-italic upper G left-parenthesis omega right-parenthesis right-bracket (1.166)

Note that the cross correlation Rfg(τ) of time signals f(t) and g(t) can be reconstructed from the inverse Fourier transform.

1.5.5 Estimation of Power and Cross Spectra

As shown in section 1.5.4 and Equation (1.164) the expected value of the Fourier transform of an ensemble of signals of infinite duration is given by:

bold-italic upper S Subscript f g Baseline left-parenthesis omega right-parenthesis equals limit Underscript upper T right-arrow normal infinity Endscripts upper E left-bracket StartFraction 1 Over upper T EndFraction upper F Subscript k Superscript asterisk Baseline left-parenthesis omega right-parenthesis upper G Subscript k Baseline left-parenthesis omega right-parenthesis right-bracket

It is helpful to convert this into an expression for a finite number of signals of finite duration. Usually, there is one signal of time length T available that is separated into M partitions of length T_w.

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