Does a Fourier Transform of a white noise exist? If so, what is its general form?

It is possible to compute the fourier transform of a Power Spectral Density?

My problem in detail. I have to compute the response of a dynamic system given the PSD of an input ($ \Phi_{vv} $).

I thought to solve the problem in three step:

  1. Build a shape filter (system which, given a white noise input ($ n $), gives a colored noise as output)

  2. Express both the shape filter and the system dynamic with a fourier transform

  3. If i can manage to express also the input (the white noise) in the fourier domain I should have solved the problem

The Fourier Transform of stationary white noise exists: in fact "white" here refers to the characteristics of the Fourier Transform, in that the power spectral density will be constant over the entire spectrum, and also implies that each sample in the time domain is independent of all other samples.

The distribution of the magnitude of the samples in time is also usually defined, such as being "Gaussian" or "Uniform" depending on what distribution is actually used. For white noise samples the phase is also distributed uniformly $\in [0, 2\pi)$. Due to the central limit theorem, the Fourier Transform of any distribution in time that is white will tend toward Gaussian in frequency. The resulting Fourier Transform will be Gaussian distributed in magnitude, uniformly distributed in phase, and each sample in the frequency domain is independent of all other samples.

The Power Spectral Density is computed from the Fourier Transform as the complex conjugate product, and not the other way around. The reason is once computed all phase information is lost so there is a many to one relationship in the computation.


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