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Consequently,

$(**)-\chi (t)=\beta {\frac {d}{dt}}(A(t)\theta (t)).$

becomes

$(**)-\chi (t)=\beta {\operatorname {d} A(t) \over \operatorname {d} t}\theta (t).$

For stationary processes, the Wiener-Khinchin theorem states that the power spectrum equals twice the Fourier transform of the auto-correlation function

$S_{x}(\omega )=2{\tilde {A}}(\omega ).$

The last step is to Fourier transform equation (**) and to take the imaginary part. For this it is useful to recall that the Fourier transform of a real symmetric function is real, while the Fourier transform of a real antisymmetric function is purely imaginary. We can split ${\operatorname {d} A(t) \over \operatorname {d} t}\theta (t)$ into a symmetric and an anti-symmetric part

$2{\operatorname {d} A(t) \over \operatorname {d} t}\theta (t)\ ={\operatorname {d} A(t) \over \operatorname {d} t}+{\operatorname {d} A(t) \over \operatorname {d} t}{\rm {sign}}(t).$

Now the fluctuation-dissipation theorem follows. Add a few random words