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In my slides about signal processing, there is one that mentions the same thing as the beginning of this and this answer, namely that the Fourier transform of a signal, squared is the power spectrum density of the signal.

In this talk, it is mentioned that coherence is calculated by dividing the squared cross-spectrum by the product of the two auto-spectra.

However, the formula in my slides divides the squared cross-spectrum by the product of the formula of our previously seen PSD and another PSD.

So, is autospectrum the same as PSD? I can find a lot of information about PSD, but not on autospectrum.

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  • $\begingroup$ I don't know yet how to put in formulas nicely. I think it'll become clearer what I mean. I'll try to edit those in tomorrow. $\endgroup$ – Mien Jun 24 '12 at 22:04
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    $\begingroup$ Neither answer that you link to says "the Fourier transform of a signal, squared is the power spectrum density of the signal." as you assert. The power spectral density is the Fourier transform of $R_x(\tau)$, the autocorrelation function of a signal $x(t)$ which is not the same thing as Fourier transform of the signal squared, i.e. $\mathcal F[x^2(t)]$$, **nor** the Fourier transform squared, i.e. $[X(f)]^2$. The power spectral density is equal to $|X(f)|^2$, but not equal to $[X(f)]^2$ in general. $\endgroup$ – Dilip Sarwate Jun 25 '12 at 0:40
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This may be just a matter of semantics. Indeed it is true that "The Fourier Transform of the auto correlation x[n] is identical to the magnitude squared of the Fourier Transform of x[n]". This is just a mathematical identity.

You could call that the power spectrum density (PSD) but in most practical applications any actual PSD computation will involve some sort of framing and windowing. The choice of framing and windowing will impact the result (and it's a complicated trade off), so there isn't really one clear and unambiguous definition of PSD.

You can still use the mathematical identity but it needs to be properly adjust with respect to framing, windowing and circular vs. linear convolution.

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