A random force F(t) has average value given by

〈F(t)〉 = 0 (33.85)

and its autocorrelation function is given by

$〈F(t)F(t^{\prime })〉 = Aδ(t−t^{\prime }),$ (33.86)

where $δ(t−t^{\prime })$ is a Dirac delta function. ${}^{9}$ Find the power spectrum.

${}^{9}$ See Appendix C.10.

Question

A random force F(t) has average value given by

〈F(t)〉 = 0 (33.85)

and its autocorrelation function is given by

[latex]〈F(t)F(t^{\prime })〉 = Aδ(t−t^{\prime }),[/latex] (33.86)

where [latex]δ(t−t^{\prime })[/latex] is a Dirac delta function.[latex]{}^{9}[/latex] Find the power spectrum.

[latex]{}^{9}[/latex]See Appendix C.10.

Accepted Answer

By the Wiener–Khinchin theorem, the power spectrum is simply the Fourier transform of the autocorrelation function, and hence

〈|F(ω)|²〉 = A (33.87)

is a ﬂat power spectrum.

Question 33.6: A random force F(t) has average value given by 〈F(t)〉 = 0 ......