## Contents

# Homework 5, ECE301, Spring 2011, Prof. Boutin

# Due by 23:59:59 Wednesday March 2, 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 5" 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 logins 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 hw5 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

Compute the Fourier transform of the continuous-time signal $ x(t)=e^{-3 |t|} $. (Use the definition of the Fourier transform, not a table of pairs and properties.) Then check your answer using this table of Fourier transform pairs and properties. (Explain how you checked your answer.)

## Question 2

Compute the Fourier transform of the signal

$ x(t)=\sin^2 ( \pi t + \frac{\pi}{12}) $

(Use the definition of the Fourier transform, not a table of pairs and properties.) Then check your answer using this table of Fourier transform pairs and properties. (Explain how you checked your answer.)

## Question 3

Compute the energy of the signal x(t) whose Fourier transform is

$ {\mathcal X} (\omega) = | \omega | ^3 \left( u(\omega+8)-u(\omega-5)\right) $

Do not simply write the answer: write the intermediate steps of your computation.

## Question 4

Let x(t) be a continuous time signal with Fourier transform $ {\mathcal X} (\omega) $. Derive an expression for the Fourier transform of y(t)=x(-3t+2) in terms of $ {\mathcal X} (\omega) $. Do not simply write the answer: write the intermediate steps of your derivation.

## Question 5

An LTI system has unit impulse response $ h(t)= e^{-3t} u(t) $.

a) Compute the frequency response $ {\mathcal H} (\omega) $ of this system.

b) Compute the system's response to the input $ x(t)= e^{-2(t-2)} u(t-2) $.

## Question 6

Consider the causal LTI system defined by the differential equation

$ \frac{d^2y(t)}{dt^2}=3 \frac{dy(t)}{dt}-2y(t)+x(t) $

a) What is the frequency response of this system. (Justify your answer)

b) What is the unit impulse response of this system. (Justify your answer)