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Let $X_t$ be a zero-mean, stationary random process.

Let $X_f$ be the Fourier transform of $X_t$; $X_f$ is also a random process, but as a function of $f$.

Let us denote the power spectral density of $X_t$ by $S_X(f)$, which is the Fourier transform of the process's auto-correlation function, $R_X(\tau)=E[X_{t+\tau}X_t]$.

Claim: $S_X(f)$ can be approximated by the variance of $X_fe^{2\pi ift}$.

Is the claim correct, and if so, under what circumstances does the approximation work best?

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  • $\begingroup$ $X_t$ and $X_{t+\tau}$ must be independent. $\endgroup$ – Seyhmus Güngören Jul 2 '15 at 1:18
  • $\begingroup$ @SeyhmusGüngören thanks. Would you know how I can show the claim to be true under the condition you cited? If you have references that shows this that would be helpful too. $\endgroup$ – TSJ Jul 2 '15 at 3:48
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If they are independent then $R_X(\tau)$ will be zero everywhere except at $\tau=0$, i.e. $R_X(\tau)=\sigma_X^2\delta(\tau)$, because $X_t$ is zero mean.

Now, if you take the fourier transform of $R_X(\tau)=\sigma_X^2\delta(\tau)$ then you get simply $S_X(f)=\sigma_X^2$ for all $f$. There is no phase shift in this case because $\tau_0=0$, otherwise for $R_X(\tau)=\sigma_X^2\delta(\tau-\tau_0)$ we would have $S_X(f)=\sigma_X^2 e^{-j2\pi f\tau_0}$.

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  • $\begingroup$ Seyhmus Güngören, thanks - I will think about this and let you know. $\endgroup$ – TSJ Jul 7 '15 at 10:01

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