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Setup:

A Gaussian white noise process $x(t)$ is defined such that

  1. The autocovariance $R(\tau) = \left< x(t) x(t-\tau) \right> = \sigma^2\delta(\tau)$ (uncorrelated in time)

  2. The ensemble distribution at any time $t$, $P[x(t)]$, is $\mathcal{N}[0, \sigma]$.

We can write the Fourier transform in terms of amplitude and phase as

$$\mathcal{F}[x(t)] = \hat{x}(\omega) = \int x(t) e^{i 2\pi \omega t} dt = a(\omega)e^{i\theta(\omega)}$$

A simple integral show s that the power spectral density

$$S(\omega) = \left< \left| a(\omega) \right|^2 \right> = \sigma^2 = \int R(\tau) e^{i 2\pi \omega \tau} d\tau$$

Question:

It is commonly stated that the ensemble statistics of $a(\omega)$ and $\theta(\omega)$ follow

  1. $P[a] = \mathcal{N}[0, \sigma]$
  2. $P[\theta] = \mathcal{U}[-\pi, \pi]$

In other words, now $a$ and $\theta$ are random processes in the frequency domain with the given distributions.

How does one derive this? What can we say about the correlation structure?

abalter
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    Can you provide a reference for these two claims? Also, the Fourier transform of a white noise process is not well-defined. The Fourier integral does not converge for every realization of $x(t)$. – Stelios Feb 01 '18 at 07:12
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    I could provide 100 references. I'll list a few. This is basically the definition of white noise--uncorrelated gaussian random variables. The FFT of any given realization is going to be different. This is why the amplitudes and phases have a statistical distribution. https://en.wikipedia.org/wiki/White_noise. https://www.encyclopediaofmath.org/index.php/White_noise. https://stats.stackexchange.com/questions/7070/what-is-a-white-noise-process. https://stats.stackexchange.com/questions/309719/white-noise-in-statistics. http://www3.ul.ie/gleesonj/Papers/SDEs/Applied_SDEs_notes.pdf. – abalter Feb 01 '18 at 07:44
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    https://www.gaussianwaves.com/2013/11/simulation-and-analysis-of-white-noise-in-matlab/. https://www.researchgate.net/post/How_do_I_generate_time_series_data_from_given_PSD_of_random_vibration_input. http://people.duke.edu/~hpgavin/cee541/PowerSpectra.pdf. – abalter Feb 01 '18 at 07:45
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    Or simply, https://math.stackexchange.com/a/2630833/145777 – abalter Feb 01 '18 at 07:49
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    I could not identify the claims on the fourier transform properties of your question in these references. These references seem to describe white Gaussian noise properties (which are, of course, well known) but not the properties of the Fourier transform of white noise. – Stelios Feb 01 '18 at 08:42
  • @Stelios The PSD of a signal is an estimate. The FFT of noise gives a noisy spectrum. An ensemble of FFTs will give you statistical properties. In fact, the PSD is defined in theoretical work as the ensemble average of amplitudes squared. See this or p. 186 of this. – abalter Feb 01 '18 at 18:06
  • The answer frequently comes up when someone is trying to simulate a signal based on a given PSD. They find out that the analytical form of the PSD is on only the mean value of an ensemble of PSDs. 1) Look at Mina Abdallah's answer to this question. 2) this 3) The empirically generated plots starting on page 22 of this. – abalter Feb 01 '18 at 18:06
  • In your question, you write the standard formula for the fourier transform (FT), the wikipedia page you referred me to states that this transform does not always exist and gives a different "FT" formula (is this what you are considering?), and in your comments you mention FFT, which is concept that makes sense only for discrete time (random) sequences/processes (whereas your question and references you provided imply a process indexed by a continuous variable). Please make sure you understand the difference among these concepts and revise your question accordingly. – Stelios Feb 01 '18 at 19:48
  • I understand the difference between all of these. I reality, there is no such thing as a continuous signal. The only way to approximate the Fourier transform is by performing an FFT on data. By doing so, as in the empirical example, one can easily show that an ensemble of GWN signals produces a statistical ensemble of FFTs. The question is, how to theoretically predict that result. I'm totally clear on the relationship between a Fourier transform on a hypothetical continuous signal, and the empirical fact of a signal always being discrete and using the FFT. – abalter Feb 01 '18 at 21:32

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