# Converting a non-stationary random process into a WSS process by adding a random phase

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:

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

1. PDF of the delay/the variable is:

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?
• when you make everything in a text bold and/or italics, then that ceases to highlight anything; hence, I "normalized" your formatting. – Marcus Müller Dec 9 '18 at 16:22
• 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"! – Marcus Müller Dec 9 '18 at 16:25

$$x(t)=\sum_ka_kp(t-kT)\tag{1}$$
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
$$\tilde{x}(t)=\sum_ka_kp(t-kT+\theta)\tag{2}$$
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 $$(2)$$.