Fourier transform of PSD

[mad mathematician]

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...).

 

[Baluncore]

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

 

[osilmag]

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.

 

[berkeman]

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?

 

[mad mathematician]

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.