## Contents

# Homework 3, ECE301, Spring 2011, Prof. Boutin

# Due in class, Monday February 7 , 2011

- Oops! Monday is the 7th, not the 8. -pm

## 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 hw3 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

Which of the following systems are invertible? Memoryless? Causal? Stable? Linear? Time-invariant? (Justify your answers mathematically.)

a) $ y[n]=x[n]x[n-1] \ $

b) $ y(t)= x\left( \sin(t) \right) \ $

c) $ y[n]= \sum_{k=n-10}^{n+10} x[k] \ $

d) $ y(t)= t^2 x(t+1) \ $

## Question 2

Determine the unit impulse response of each of the four systems described in Question 1.

## Question 3

Show that the CT unit impulse satisfies the equation

$ \delta (2 t ) = \frac{1}{2} \delta (t) $

## Question 4

True of False? (Justify your answer)

The cascade of two time-invariant systems is itself time-invariant.

## Question 5

The unit impulse response of an LTI system is

$ h[n]= \delta[n+1] +2 \delta[n-1]. $

Compute the system's response to the input

$ x[n] = u[n-1]- u[n-7]. $

## Question 6

The unit impulse response of an LTI system is

$ h[n]= u[n-3]. $

Compute the system's response to the input

$ x[n] = \left( \frac{1}{3} \right) ^{-n} u[-n-1]. $

## Question 7

The unit impulse response of an LTI system is

$ h(t)= u(t+5)-u(t-7). $

Compute the system's response to the input

$ x(t) = u(- t) $

## Question 8

The unit impulse response of an LTI system is

$ h(t)= e^{t} u(-t+5) $

Compute the system's response to the input

$ x(t) = u(-t-8) $