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.



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