# Proving a cyclostationary processes signal

Suppose a random signal $$x(t)=\sum\limits_{n=-\infty}^\infty Z_n \delta(t-n\tau)$$, where $$z_n = Z$$ and $$Z$$ is a random variable with equal probability to be $$+-1$$, is passing through a low pass filter $$h(t) = sinc(\frac t \tau)$$ and the output is $$y(t)$$. and I want to prove the output signal is cyclostationary process signal.

I know the $$\mu_y(t) = \int\limits_{-\infty}^\infty E[X(z)]h(t-z) dz = \sum\limits_{n=-\infty}^\infty \int\limits_{-\infty}^{\infty} E[Z_n] \delta(z-n\tau)h(t-z) dz$$

and the autocorrelation of $$Y(t)$$ is $$R_Y(t, t+x)=E[Y(t_1)Y(t_2)]=\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty E[X(z)X(u)] h(t-z) h(t+x-u) dz du = \sum\limits_{n=-\infty}^\infty \sum\limits_{m=-\infty}^\infty \int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty E[Z_n Z_m] \delta (z-n \tau) h(t-z) \delta (u-m \tau) h(t+x-u) dz du$$

And this is what I need to prove

Cyclostationary Process

A random process $$X(t)$$ with mean $$\mu(t)$$ and auto correlation function $$R_x(t+\tau, t)$$ is called cyclostationary. if both the mean and the auto correlation are periodic in t with some period $$T_0$$, i.e. if

• $$\mu_x (t + T_0) = \mu(t)$$

• $$R_x (t + \tau + T_0, t + T_0) = R_x (t + \tau , t)$$

And I think $$\mu_y(t)$$ is 0 because $$E[Z_n]$$ is 0 (right?), But how can I prove the auto-correlation are periodic?

Last thing, I want to thanks all you guys for your comments and intuitive answer that help me so much on my study and get my head straight. Thanks.