## Contents

# Homework 9

# Due in class, Wednesday April 6, 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 hw9 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

Determine whether the following signals are band-limited or not. If they are band-limited, specify what is the Nyquist rate for these signals.

a) $ x(t)= u(t+7)-u(t-7)\ $

b) $ x(t)= u(t)-u(t-14)\ $

b) $ x(t)= \frac{\sin (3 \pi t)}{t} \ $

c) $ x(t)= \frac{\sin (3 \pi (t+5))}{t+5} \ $

## Question 2

The signal $ x(t)= e^{j \pi t }\frac{\sin (\pi t)}{t} $ is sampled with a sampling period $ T $. For each of the values of T below, sketch the Fourier transform of the sampling and indicate whether or not one can recover $ x(t) $ from the sampling. If you answered yes, explain how.

a) $ T= \frac{1}{4}\ $

b) $ T= \frac{2}{3}\ $

c) $ T= 2 \ $

## Question 3

Let x(t) be a continuous-time signal with $ \left| {\mathcal X} (\omega)\right| =0 $ for $ \left| \omega \right| > \omega_m $. Can one recover the signal x(t) from the signal $ y(t)=x(t) p(t-3) $, where

$ p(t)= \sum_{k=-\infty}^\infty \delta (t- \frac{2\pi}{\omega_s} {\color{red}k}) ? $

If so, explain how. If not, explain why not.

## Question 4

The input x(t) and output y(t) of a continuous-time system are related by the equation

$ y(t)= \frac{d x(t)}{dt}. $

Assuming that the input x(t) is band-limited with Nyquist rate $ \omega_0 $, describe a discrete-time LTI system that would process the sampling $ x_d[n]=x(n \frac{ \pi }{\omega_0}) $ in such a way that the processed samples $ y_d[n] $ would satisfy $ y_d[n]=y(n\frac{ \pi }{\omega_0}) $.