Here is an example where this method has been implemented.

We were trying to calculate the spectrum of a transmitted signal(Random signal/weighted pulse)

  1. The auto correlation function of the pulse which is not WSS because it is dependent on t:

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  1. We associated a random variable(which is the delay) with an uniform PDF with the signal:

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  1. PDF of the delay/the variable is:

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I was unable to grasp the concept behind it, probably my basics are a bit rusty.

  • By simply just associating a random variable (with an uniform PDF), how can we just make any random process a wide sense stationary process?
  • Whats the concept behind it?
  • $\begingroup$ when you make everything in a text bold and/or italics, then that ceases to highlight anything; hence, I "normalized" your formatting. $\endgroup$ – Marcus Müller Dec 9 '18 at 16:22
  • $\begingroup$ But since we know nothing about the $P_T$, I don't think we can help you much. Define it! give context! Define what you mean with "associate"! $\endgroup$ – Marcus Müller Dec 9 '18 at 16:25

It is not true that you can convert any non-stationary random process into a WSS random process by just adding a random phase. What is true is that you can make a (wide-sense) cyclostationary process WSS by adding a random phase. A common example is the PAM process mentioned in your question:


The discrete values $a_k$ in $(1)$ are assumed to be random. The process $x(t)$ can be shown to be cyclostationary. The modified process


where $\theta$ is a random variable uniformly distributed on $[0,T]$ is WSS. This random phase reflects our uncertainty about the phase of the signal, i.e., about the origin of the time axis. In this sense, it is physically motivated, and not just a nice "trick" enabling us to compute the power spectrum of $(2)$.


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