Fourier transform of PSD

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The discussion centers on the relationship between the Power Spectral Density (PSD) and the Fourier Transform (FT) of the Autocorrelation function, emphasizing that PSD is derived from the FT. Participants explore potential applications of applying the FT to PSD in electrical engineering (EE), noting that it can help understand harmonic content and evaluate power spectra. The conversation touches on the mathematical intricacies of performing sequential Fourier transforms and their implications, particularly in analyzing spatial frequencies in 2D images. Additionally, concepts like Parseval's Theorem are mentioned as relevant to EE applications. Overall, the exploration of Fourier transforms in relation to PSD reveals both theoretical interest and practical implications in signal processing.
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So PSD is the Fourier transform of the Autocorrelation function.
Is there any application of the Fourier transform on PSD in EE?
Or it's like in Newtonian dynamics a second derivative wrt time is as far as we can get (more than that it's called a Jerk...).
 
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mad mathematician said:
Is there any application of the Fourier transform on PSD in EE?
If you think it will help you to understand the harmonic content, then there is an application.

Each application of the FT, folds the signal between the time and the frequency domains. Inverse transforms return things in time sequence, while two sequential forward transforms, does not seem to make much sense, but can be useful if you are computing the spatial frequencies of 2D images.

Most of the useful transforms evaluate the power spectrum, maybe take the log, then fold the frequency domain back into the time domain using the IFT. Searching for echos is one application.

There is plenty of Fourier space to explore, and a whole inverted vocabulary.
https://en.wikipedia.org/wiki/Cepstrum
 
mad mathematician said:
So PSD is the Fourier transform of the Autocorrelation function.
Is there any application of the Fourier transform on PSD in EE?
Or it's like in Newtonian dynamics a second derivative wrt time is as far as we can get (more than that it's called a Jerk...).
You might take a look at Parseval's Theorem and it's application in electrical engineering.
 
mad mathematician said:
So PSD is the Fourier transform of the Autocorrelation function.
Is there any application of the Fourier transform on PSD in EE?
The Fourier Transform transforms a function of time into a function of frequencies. How can you then transform that function of frequencies into another function of frequencies using a Fourier Transform?
 
berkeman said:
The Fourier Transform transforms a function of time into a function of frequencies. How can you then transform that function of frequencies into another function of frequencies using a Fourier Transform?
Well, mathematically what I had in mind is as follows:
Given the autocorrelation function ##R(\tau)##, then the PSD is ##S(f)=\int R(t)\exp(-ift)dt##, another fourier transform: ##W(\omega)=\int S(f)\exp(-if\omega)df=\int\int R(t)\exp(-if(\omega+t))dfdt##; one can change variables in the integration as follows: ##u=ft ,v=f\omega##, and then to calcualte the Jacobian determinant w.r.t u and v.

I don't know of any applications to this "bizzare" operation, just something that I thought about.
In the introduction to Harmonic Analysis course that I took more than 10 years ago there was some theorem
of Thorin which seems relevant to my idea of repeating the Fourier transform.

So I guess it's part nostalgia part concerning me nowadays in my EE studies.
 
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