# Probability distribution of phase of fourier transform of white noise, colored noise (1/f noise in particular)--- Definitions and standards

What is the probability distribution of phase of fourier transform of white noise and colored noise (1/f noise in particular) ? Is there a standardised definition?

Often, the colored noise is derived from filtered white noise (whose magnitude pdf can be guassian or uniform or one of the other distributions). While this matches the definition for the magnitude spectrum, the phase spectrum is usually not considered. Are there sources which describe the pdf of phase of fourier transform in detail?

For AWGN, there is a related post where the pdf of phase of fourier transform is described as a uniform distribution (What is the phase and magnitude response of white noise?).The attached file lists it in a table , but the details as to how the result was arrived at was not discussed.

• Pretty much all noise signals I've ever dealt with have a uniform phase distribution. That's completely independent of the spectrum of the noise. Sep 7, 2023 at 18:05
• In particular: $$p_\phi(\theta) = \frac{1}{2 \pi}\operatorname{rect}\left(\frac{\theta}{2 \pi} \right)$$ where $$\operatorname{rect}(u) \triangleq \begin{cases} 1 \qquad & \text{ if } |u| < \tfrac12 \\ \tfrac12 \qquad & \text{ if } |u| = \tfrac12 \\ 0 \qquad & \text{ if } |u| > \tfrac12 \\ \end{cases}$$ Sep 7, 2023 at 19:19