## Contents

# Homework 4, ECE301, Spring 2011, Prof. Boutin

WRITING IN PROGRESS.

Note that Question 2(h) is corrected. --Ahmadi 23:03, 11 February 2011 (UTC)

Question 6(c,d,e) has been corrected too. -pm

# Due in class, Monday February 14, 2011

## Important Notes

- Justify all your answers.
- Write your answers clearly and cleaning.
- Write on one side of the paper only.
- Do not permute the order of the problems.
- Make a cover sheet containing your name, course number, semester, instructor, and assignment number.
- Staple your homework.

## If you have questions

If you have questions or wish to discuss the homework with your peers, you may use the hw4 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

The unit impulse response of some LTI systems are given below. Which of these systems are memoryless? Causal? Stable? (Justify your answers mathematically.)

a) $ h[n]= e^{j 2 \pi n} \ $

b) $ h(t) = e^{j 2 \pi t} \ $

c) $ h(t) = e^{j 2 \pi t} u(-t) \ $

d) $ h[n]= e^{j 2 \pi n} \delta[n] \ $

d) $ h[n]= e^{j 2 \pi n} \left( u[n+7] - u[n] \right) \ $

## Question 2

Obtain the Fourier series coefficients of the following signals.

a) $ x(t) = \sin \left( 2 \pi t \right) . \ $

b) $ x(t) = \sin \left( 2 \pi t \right) \cos \left( \frac{\pi}{2} t \right) . \ $

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

d) $ x[n] = j ^n \ $

e) $ x[n]=e^{j \frac{3}{5}\pi \left( n-\frac{1}{2} \right)} \ $

f) $ x(t)= \cos^2 t \ $

g) $ x[n]= 1+e^{j \frac{4\pi n}{7}}- e^{j \frac{2\pi n}{5}} \ $

h) $ x(t)= \sum_{k=-\infty}^\infty f(t+{\color{red}{2}}k) $, where

$ f(t)=\left\{ \begin{array}{ll} t+1, & \text{ for } -1 \leq t <0, \\ 1-t, & \text{ for } 0 \leq t <1, \\ 0, \text{ else}. \end{array} \right. \ $

## Question 3

What is the Fourier series for each of the signals in Question 2?

## Question 4

An LTI system has unit impulse response $ h(t) = e^{ t} \left( u(t-100)-u(t) \right) \ $.

a) Compute the transfer function of this system.

b) Use your answer in a) to compute the system's response to each of the CT signals in Question 3.

## Question 5

An LTI system has unit impulse response $ h[n] = \sum_{k=-7}^8 \delta [n-k] \ $.

a) Compute the transfer function of this system.

b) Use your answer in a) to compute the system's response to each of the DT signals in Question 3.

## Question 6

The signal $ x(t)=\sin (2 \pi 440 t) $ is the input of a system. For each of the outputs listed below, indicate whether the system could be LTI. Justify your answers.

a) $ y(t)=sin (2 \pi 880 t) $.

b) $ y(t)= 0 \ $.

c) $ y(t)= e^{ 2 {\color{red}{j}}\pi 440 t} \ $.

d) $ y(t)= e^{ 2{\color{red}{j}} \pi 880 t} \ $.

e) $ y(t)= e^{ - 2{\color{red}{j}} \pi 440 t} \ $.

## Question 7

The signal $ x[n]=j^n $ is the input of a system. For each of the outputs listed below, indicate whether the system could be LTI. Justify your answers.

a) $ y[n] = 7 j^n \ $.

b) $ y[n] = e^{j \frac{\pi}{2} n } \ $.

c) $ y[n] = e^{j \frac{\pi}{2} } \ $.

d) $ y[n] = e^{j \pi n }\ $.