Assuming that this process passes through an ideal lowpass

Question

X (t) denotes a zero-mean WSS Gaussian random process with autocorrelation function

1. What is the power in this process?

2. Determine the power spectral density, SX (f), for this process.

3. What is the bandwidth of this process?

4. Assuming that this process passes through an ideal lowpass filter with a bandwidth of 5 kHz and the output is denoted by Y(t), determine Sy(f), the power spectral density of Y (t), and the total power in the output process.

5. Determine the PDF (probability density function) of random variables X(0), X (10^-4 ), and X (1.5 × 10^-4).

6. Show that random variables X (0) and X(10^-4 ) are independent but X (0) and X(1.5 × 10^-4 ) are dependent.