Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 46

We know from section 1.5.4 that the random signal cannot be Fourier transformed. As a consequence we apply the correlation methods from above to the output of a system excited by random signals. We start with the autocorrelation of the system output g from excitation by random input f

(1.184)

The impulse response can by taken out from the expected value operation, because it does not depend on the time. Hence we get

(1.185)

When we assume a retarded time argument of τ′=τ+τ1−τ2 the expected value can be interpreted as the autocorrelation of f(t). With this assumption we get:

(1.186)

The term in the rectangular brackets can be seen as the convolution with argument (τ+τ1).

(1.187)

By replacing τ₁ by −u this reads as

(1.188)

using that Rff(τ) is symmetric in τ. This equation can be converted into frequency domain using the fact that time reversal in time corresponds to complex conjugate spectra:

(1.189)

So we know now the autospectrum of the system excited by random response. Next we investigate the cross correlation between input and output.

(1.190)

(1.191)

This expression can be simplified by assuming the expected value to be the auto-convolution with argument τ−τ1 to:

(1.192)

Converting this into the frequency domain gives:

(1.193)

This is a very important result: every transfer function (also using deterministic signals) can be determined by the ratio of the cross spectrum to auto spectrum.

(1.194)

In principle this can also be done by using the Fourier transform (1.173), using time-limited excitation and dividing output and input FT, but the cross spectral variant is much more robust against measurement noise.