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I was reading here in equation (12) that the power spectral density (PSD) for a signal $f(t)$ and its corresponding Fourier Transform $f(\omega)$ is defined as

$$<{f(\omega)f^*(\omega)}>=S_f(\omega)\delta(\omega-\omega')$$

I just know the usual definition of the PSD as the Fourier transformation of the Autocorrelation function:

$$S(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dt ~r(t)e^{-i\omega t}~~~(*)$$

where the Autocorrelation function is given by:

$$r(t)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T} d\tau~ f(\tau)f^*(\tau+t)$$

My question: equation $(*)$ is equation (13) in the paper which I linked at the beginning (one has to do an inverse FT and I think they forgot the $e^{i\omega t}$!?). But is this equation equivalent to equation (12) in the paper? How is the $<...>$ in equation (12) of the paper defined?

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The average power carried by a wave (e.g. an electromagnetic wave) is proportional to its squared amplitude. In this sense it could be considered more natural to define the power spectral density, i.e. the power density as a function of the frequency, as the function $|f(\omega)|^2 = f(\omega)f^\ast(\omega)$. Note that this could just be considered etymology, the nature of $f$ may be such that it doesn't have anything to do with power or energy.

That this is equivalent to the definition that you know is a consequence of the Wiener-Khinchin theorem, which essentially says that the autocorrelation of a function $f$ can be obtained as the Fourier transform of $|f(\omega)|^2$, the absolute square of the Fourier transform of $f$ itself.

This is what relates your definition of the PSD to that from the article. The fine details of the notation change from text to text. I am not sure for example why they introduce an $\omega'$ here (note that you are missing a prime on the LHS). The triangular brackets usually indicate some kind of average.

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I have not read the paper you linked to, only looked at eq (12). If you include the exponential in $<...>$ then the term $\delta (\omega-\omega')$ will be multiplied by $exp[\mathfrak {j} (\omega-\omega')t]$ but that is $1$ when $\omega=\omega'$ the only location where the $\delta \ne 0$, so it does not matter in the product. The $<>$ almost certainly denotes an average in the frequency domain. For normal stochastic processes, ones that can be approximated with a finite periodogram (derived from the finite interval $[-T,T]$ Fourier Transform of the random time function, and let $T \to \infty$ ) the frequency components are uncorrelated, and this is expressed in eq 12.

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