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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?

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