I have a complex signal in the time domain normally distributed. What will be its distribution in the frequency domain?

I assumed since the frequency domain is a linear transformation the distribution will not change.

  • $\begingroup$ You got a problem here... When you say frequency domain distribution be careful. Do you mean frequency domain power distribution of the (WSS) random process? Or do you mean the probability density function of the associated random process (or random variable) which is obtained when the power sectrum is also viewed as a random process... These are two different things. Please clarify. $\endgroup$
    – Fat32
    Jul 3 '20 at 18:35
  • $\begingroup$ I mean the pdf in the frequency domain. $\endgroup$
    – Am Ki
    Jul 3 '20 at 18:47

I assume you mean that at each time $t$, the signal is a normally-distributed random variable. This tells you the probability that the signal will take a value in any given range, but it does not tell you whether/how the signal values at different times are related to one another. Typically, for a random signal one defines not only the probability distribution for the signal at each time, but also the autocorrelation function $E \{ x(t) x(u) \}$ or some other measure of how signal values at different times are related or independent. This is required in order to determine how the signal is distributed in the frequency domain.

  • $\begingroup$ Assume it is iid at each time steps. $\endgroup$
    – Am Ki
    Jul 3 '20 at 18:51
  • $\begingroup$ Then the autocorrelation function is $R_x(\tau) = K \delta(\tau)$, and the PSD is constant, $S_x(\Omega) = K$. But I infer from the previous comments that you are not looking for the PSD. $\endgroup$
    – Steve J.
    Jul 3 '20 at 18:55
  • $\begingroup$ @Steve If you consider in the time domain at each time t, the signal is normally distributed, then why is there any concern with how the signal varies across frequencies? Wouldn't it be the same consideration--- the distribution of the frequency waveform at each frequency f. Also, in the time domain we clarify if the system is ergodic or not to determine if we can use the statisistics of what occurs at each t to also be across all time. I don't know what the frequency domain equivalent of ergodicity is. $\endgroup$ Jul 3 '20 at 19:10

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