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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.

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  • $\begingroup$ 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, 2020 at 18:35
  • $\begingroup$ I mean the pdf in the frequency domain. $\endgroup$
    – Am Ki
    Jul 3, 2020 at 18:47

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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.

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  • $\begingroup$ Assume it is iid at each time steps. $\endgroup$
    – Am Ki
    Jul 3, 2020 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, 2020 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, 2020 at 19:10
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In my opinion as frequency domain is a exponential transformaion random variabl so the distribution is also exponential or a close distribution such as lognormal or may be Rayleigh.

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  • $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
    – ZaellixA
    Jun 14, 2023 at 15:00
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The Fourier transform (FT) of the probability density function (PDF) of a random variable is called characteristic function of that random variable, defined as

$$\phi_X(\omega)=\int_{-\infty}^{\infty}f(x)e^{j\omega x} dx.$$

where $f(x)$ is the PDF of the random variable $X$. If $j\omega$ is replaced by $s$, the resulting integral is called the moment (generating) function.

In particular, if $X$ is a random variable of normal distribution $N(\mu,\sigma)$, then its characteristic function is

$$\phi_X(\omega)=e^{j\mu\omega-\frac{1}{2}\sigma^2\omega^2}.$$

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A normal Gaussian transforms into another Gaussian with the assumption of a linear transform.

In general a transformation does not change the underlying statistics. The transformation preserves the information and allows you to analyze the information in another domain.

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  • $\begingroup$ Assuming linear transform. $\endgroup$
    – Mark
    Jun 16, 2023 at 5:17

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