Homework 7, ECE301, Spring 2011, Prof. Boutin

Due by 23:59:59 Wednesday March 16, 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 7" 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 hw7 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 discrete-time signal $x[n]=5^{-|n+2|}$ . (Use the definition of the Fourier transform, not a table of pairs and properties.) Then check your answer using this table of DT Fourier transform pairs and properties. (Explain how you checked your answer.)

Question 2

Use the definition of the inverse DT Fourier transform (i.e., the integral formula) to compute the inverse Discrete-time Fourier transform of the signal

${\mathcal X} (\omega) = \sum_{k=-\infty}^\infty \delta (\omega-2\pi k)+ \pi \delta (\omega-\frac{\pi}{2}-2\pi k)+ \pi \delta (\omega+\frac{\pi}{2}-2\pi k)$

Question 3

Consider a discrete-time LTI system with impulse response

$h[n]=\left(\frac{1}{2} \right)^n u[n].$

Use Fourier transforms to determine the response to each of the following input signals

a) $x[n]= \left( \frac{3}{4} \right)^n u[n]$

b) $x[n]= (n+1) \left( \frac{1}{4} \right)^n u[n]$

c) $$ x[n]= (-1)^n $$

(You may use this table of DT Fourier transform pairs and properties to answer this question.)

Question 4

Consider an LTI system whose response to the input signal

$x[n]= \cos \omega_0 n \$

is $\omega_0 \cos \omega_0 n \$ , for any $-\pi \leq \omega_0 \leq \pi$ . Determine the frequency response ${\mathcal X} (\omega)$ and the unit impulse response h[n] of this system. Justify your answers.

(You may use this table of indefinite integrals to answer this question.)

Question 5

Consider the LTI system consisting of the cascade of two LTI systems with frequency responses

${\mathcal H}_1 (\omega)=\frac{2-e^{-j\omega}}{1+\frac{1}{2}e^{-j\omega}}$ and ${\mathcal H}_2 (\omega)=\frac{1}{1-\frac{1}{2}e^{-j\omega}+\frac{1}{4}e^{-2j\omega}}$ , respectively.

a) Obtain a difference equation describing this system (i.e., the cascade). Justify your answer.

b) What is the unit impulse response of this system (i.e., the cascade)? Justify your answer.