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[[Category:problem solving]] | [[Category:problem solving]] | ||
[[Category:random variables]] | [[Category:random variables]] | ||
| + | [[Category:probability]] | ||
| + | |||
| + | <center> | ||
| + | <font size= 4> | ||
| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
| + | </font size> | ||
| + | |||
| + | <font size= 4> | ||
| + | Communication, Networking, Signal and Image Processing (CS) | ||
| + | |||
| + | Question 1: Probability and Random Processes | ||
| + | </font size> | ||
| + | |||
| + | January 2001 | ||
| + | </center> | ||
| + | ---- | ||
| + | ---- | ||
| + | =Part 4= | ||
| + | Let <math class="inline">\mathbf{X}_{t}</math> be a band-limited white noise strictly stationary random process with bandwidth 10 KHz. It is also known that <math class="inline">\mathbf{X}_{t}</math> is uniformly distributed between <math class="inline">\pm5</math> volts. Find: | ||
| + | |||
| + | '''(a) (10 pts)''' | ||
| + | |||
| + | Let <math class="inline">\mathbf{Y}_{t}=\left(\mathbf{X}_{t}\right)^{2}</math> . Find the mean square value of <math class="inline">\mathbf{Y}_{t}</math> . | ||
| + | |||
| + | '''(b) (10 pts)''' | ||
| + | |||
| + | Let <math class="inline">\mathbf{X}_{t}</math> be the input to a linear shift-invariant system with transfer function: | ||
| + | <br> | ||
| + | <math class="inline">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}</math> | ||
| + | <br> | ||
| + | |||
| + | Find the mean and variance of the output. | ||
| − | |||
| − | |||
---- | ---- | ||
==Share and discuss your solutions below.== | ==Share and discuss your solutions below.== | ||
---- | ---- | ||
| − | =Solution 1 | + | =Solution 1 = |
Write it here | Write it here | ||
---- | ---- | ||
Latest revision as of 09:37, 13 September 2013
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.
Solution 1
Write it here
Solution 2
Write it here.
