Contents
Homework 5, ECE301, Spring 2011, Prof. Boutin
WRITING IN PROGRESS.
Due in dropbox , 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
Compute the Fourier transform of the continuous-time signal $ x(t)=e^{-3 |t|} $. (Use the definition of the Fourier transform, not a table of pairs and properties.)
Question 2
Compute the Fourier transform of the signal
$ x(t)=\sin^2 ( \pi t + \frac{\pi}{12}) $
Question 3
Compute the energy of the signal x(t) whose Fourier transform is
$ {\mathcal X} (\omega) = \omega^4 \left( u(\omega+8)-u(\omega-5)\right) $
Do not simply write the answer: write the intermediate steps of your computation.
Question 4
Let x(t) be a continuous time signal with Fourier transform $ {\mathcal X} (\omega) $. Derive an expression for the Fourier transform of y(t)=x(-3t+2) in terms of $ {\mathcal X} (\omega) $. Do not simply write the answer: write the intermediate steps of your derivation.
Question 5
An LTI system has unit impulse response $ h(t)= e^{-3t} u(t) $.
a) Obtain the frequency response $ {\mathcal H} (\omega) $ of this system.
b) Compute the system's response to the input $ x(t)= e^{-3t} u(t-2) $.
Question 6
Consider the causal LTI system defined by the differential equation
$ \frac{d^3y}{dt^3}=\frac{dy}{dt} $
