## Contents

# Homework 10, ECE301, Spring 2011, Prof. Boutin

# Due by 6pm Wednesday April 13, 2011

To hand in your homework, first go to your instructor's dropbox. Then click on the button titled "mboutin Assignments". You will find an assignment titled "ECE301 Homework 10" at the bottom of the page. Click the "submit new" button of this assignment to submit your homework. After you submit your homework, you should be able to see your submission by clicking the button titled "view Submission". Note that the dropbox and peer review software have been developed by Purdue students. If you do not like this software the way it is, just join the Rhea development team and fix it!

## Important Notes

- Write your answers clearly and cleaning.
- Do not permute the order of the problems.
- Include a cover sheet containing the assignment number, course number, semester, instructor, but NOT your name.
- Drop in your instructor's drop box following the above instructions.
- Next week, we will do a double blind peer review of this homework. Each student will be assigned one homework to grade. The name of the author of the homework will not be revealed to the grader, and the name of the grader will not be revealed to the author. However, your instructor/TA/grader will know who is writing/grading what thanks to the peer review system, which keeps track of the Purdue career login of the authors/graders.

## If you have questions

If you have questions or wish to discuss the homework with your peers, you may use the hw10 discussion page. All students are encouraged to help each other on this page. Your TA and instructor will read this page regularly and attempt to answer your questions as soon as possible.

## Question 1

Let x(t) be a signal with Nyquist rate equal to $ 2,000 \pi $. Consider the signals

$ y_1(t)= e^{j (\omega_c t+\theta_c) } x(t) $

and

$ y_2(t)= \sin(\omega_c t ) x(t). \ $

a) What conditions should be put on $ \omega_c $ to insure that x(t) can be recovered from $ y_1(t) $?

b) Assuming the condition you stated in a) are met, how can one recover x(t) from $ y_1(t) $?

c) What conditions should be put on $ \omega_c $ to insure that x(t) can be recovered from $ y_2(t) $?

d) Assuming the condition you stated in c) are met, how can one recover x(t) from $ y_2(t) $?

## Question 2

A signal x(t) is multiplied by a rectangular pulse train c(t), where

$ c(t)=\sum_{k=-\infty}^\infty u(t+\frac{1}{4}10^{-3}+k10^{-3} )-u(t-\frac{1}{4}10^{-3}+k 10^{-3}). $

a) What conditions should be put on the signal x(t) to insure that it can be recovered from x(t)c(t)?

b) Assuming the conditions you stated in a) are met, how can one recover x(t) from x(t)c(t)?

## Question 3

A speech signal x(t) is band-limited to $ |\omega|<\omega_M $. For privacy reasons, the speech signal is "scrambled" into a new signal y(t) such that

$ {\mathcal Y}(\omega) = \left\{ \begin{array}{ll} {\mathcal X} (\omega -\omega_m), & \omega> 0 \\ {\mathcal X} (\omega + \omega_m), & \omega<0 \\ \end{array} \right. $

a) Sketch the spectrum (i.e., the Fourier transform) of y(t).

b) Draw the block diagram of a system that would produce such a scrambled signal (i.e. a "scrambler").

c) Draw the block diagram of a system that would descramble the signal (i.e., a "descrambler").

## Question 4

Compute the Laplace transform of the following signals. (Do not forget to indicate the ROC.)

a) $ e^{-5t} u(t+3)\ $

b) $ e^{-5t} u(-t+5) \ $

c) $ e^{-5t} \left( u(t)-u(t-3) \right) \ $

d) $ e^{-2 |t| } \ $

## Question 5

Let x(t) be a signal whose Laplace transform has exactly two poles located at s=-1 and s=-3, respectively. If the Fourier transform of $ g(t)=e^{2t}x(t) $ converges, could x(t) be

a) left-sided?

b) right-sided?

c) two-sided?

## Question 5

We have shown in class that the Laplace transform of $ e^{-at} u(t) $ is

$ \frac{1}{s+a}, \text{ with ROC } {\mathcal Re}(s)>{\mathcal Re}(-a). $

Use this fact to compute the inverse Laplace transform of

$ X(s)=\frac{2(s+2)}{s^2+7s+12}, \text{ with ROC }{\mathcal Re}(s)> -3 . $