## Contents

# Homework 2

# Due in class, Wednesday January 26, 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 hw2 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 energy $ E_\infty $ and the power $ P_\infty $ of the following signals.

a) $ x(t)=e^{-t}u(t) \ $

b) $ x(t)=e^{jt}u(t) \ $

c) $ x[n]=\frac{1}{3} u[n] \ $

## Question 2

Find the fundamental period of the following signals.

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

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

c) $ x[n]= \cos^2 n \ $

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

e) $ x(t)= \sum_{k=-\infty}^\infty \frac{1}{1+(t-7k)^2} \ $

## Question 3

Prove that, for any DT signal x[n], we have

$ \sum_{n=-\infty}^\infty x^2[n] = \sum_{n=-\infty}^\infty x_e^2[n]+ \sum_{n=-\infty}^\infty x_o^2[n], $

where $ x_e[n] \ $ and $ x_o[n] \ $ are the even and odd parts of the signal, respectively.

## Question 4

Use

- time shifting
- time scaling
- multiplication of a signal by the unit step function
- signal addition

to write the "smoke on the water" tune from the last homework assignment as a function z(t) of a single note lasting one second. (HInt: recall Prof. Mimi's Opus #1 presented in the lecture.)

## Question 5

Consider the following systems

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 1} & \\ & & \end{array}\right] \rightarrow y(t)=x(3t+7) $

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow y(t)=x(5t-1) $

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow y(t)=x(-t) $

What is the output of the following cascade?

$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 1} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 2} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{system 3} & \\ & & \end{array}\right] \rightarrow y(t) $