ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2001

Part 4

Let $ \mathbf{X}_{t} $ be a band-limited white noise strictly stationary random process with bandwidth 10 KHz. It is also known that $ \mathbf{X}_{t} $ is uniformly distributed between $ \pm5 $ volts. Find:

(a) (10 pts)

Let $ \mathbf{Y}_{t}=\left(\mathbf{X}_{t}\right)^{2} $ . Find the mean square value of $ \mathbf{Y}_{t} $ .

(b) (10 pts)

Let $ \mathbf{X}_{t} $ be the input to a linear shift-invariant system with transfer function:
$ H\left(f\right)=\begin{cases} \begin{array}{lll} 1 \text{ for }\left|f\right|\leq5\text{ KHz}\\ 0.5 \text{ for }5\text{ KHz}\leq\left|f\right|\leq50\text{ KHz}\\ 0 \text{ elsewhere. } \end{array}\end{cases} $

Find the mean and variance of the output.

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