(New page: = Homework 5, ECE301, Spring 2011, Prof. Boutin = WRITING IN PROGRESS. = Due in dropbox , Monday February 14, 2011 = == Important Notes == *Justify all your ans...) |
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| − | = Due in dropbox , | + | = Due by 6pm in course dropbox , Wednesday March 2, 2011 = |
== Important Notes == | == Important Notes == | ||
| − | |||
*Write your answers clearly and cleaning. | *Write your answers clearly and cleaning. | ||
| − | |||
*Do not permute the order of the problems. | *Do not permute the order of the problems. | ||
| − | * | + | *Include a cover sheet containing the assignment number, course number, semester, instructor, but NOT your name. |
| − | * | + | *Drop in the course drop box here |
| + | *Next week, we will do a double blind peer review. Each student will be assigned one homework to grade. The name of the author of the homework will not be revealed to the grader, and the name of the grader will not be revealer to the author. However, your instructor/TA/grader will know who is writing/grading what thanks to the peer review system, which keeps track of the Purdue career logins of the authors/graders. | ||
== If you have questions == | == If you have questions == | ||
| − | If you have questions or wish to discuss the homework with your peers, you may use the [[ | + | If you have questions or wish to discuss the homework with your peers, you may use the [[Discussion_HW5_ECE301_Spring2011|hw5 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 == | == Question 1 == | ||
| − | Compute the Fourier transform of the continuous-time signal <math>x(t)=e^{-3 |t|}</math>. (Use the definition of the Fourier transform, not a table of pairs and properties.) | + | Compute the Fourier transform of the continuous-time signal <math>x(t)=e^{-3 |t|}</math>. (Use the definition of the Fourier transform, not a table of pairs and properties.) Then check your answer using this [[CT_Fourier_Transform_%28frequency_in_radians_per_time_unit%29| table of Fourier transform pairs and properties]]. (Explain how you checked your answer.) |
== Question 2 == | == Question 2 == | ||
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x(t)=\sin^2 ( \pi t + \frac{\pi}{12}) | x(t)=\sin^2 ( \pi t + \frac{\pi}{12}) | ||
</math> | </math> | ||
| + | |||
| + | (Use the definition of the Fourier transform, not a table of pairs and properties.) Then check your answer using this [[CT_Fourier_Transform_%28frequency_in_radians_per_time_unit%29| table of Fourier transform pairs and properties]]. (Explain how you checked your answer.) | ||
==Question 3 == | ==Question 3 == | ||
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An LTI system has unit impulse response <math class="inline">h(t)= e^{-3t} u(t) </math>. | An LTI system has unit impulse response <math class="inline">h(t)= e^{-3t} u(t) </math>. | ||
| − | a) | + | a) Compute the frequency response <math class="inline">{\mathcal H} (\omega) </math> of this system. |
| − | b) Compute the system's response to the input <math class="inline">x(t)= e^{- | + | b) Compute the system's response to the input <math class="inline">x(t)= e^{-2(t-2)} u(t-2) </math>. |
==Question 6 == | ==Question 6 == | ||
Consider the causal LTI system defined by the differential equation | Consider the causal LTI system defined by the differential equation | ||
| − | <math>\frac{d^ | + | <math>\frac{d^2y(t)}{dt^2}=2 \frac{dy(t)}{dt}+x(t)</math> |
| + | |||
| + | a) What is the frequency response of this system. | ||
| + | b) What is the unit impulse response of this system. | ||
---- | ---- | ||
Revision as of 13:26, 23 February 2011
Contents
Homework 5, ECE301, Spring 2011, Prof. Boutin
WRITING IN PROGRESS.
Due by 6pm in course dropbox , Wednesday March 2, 2011
Important Notes
- Write your answers clearly and cleaning.
- Do not permute the order of the problems.
- Include a cover sheet containing the assignment number, course number, semester, instructor, but NOT your name.
- Drop in the course drop box here
- Next week, we will do a double blind peer review. Each student will be assigned one homework to grade. The name of the author of the homework will not be revealed to the grader, and the name of the grader will not be revealer to the author. However, your instructor/TA/grader will know who is writing/grading what thanks to the peer review system, which keeps track of the Purdue career logins of the authors/graders.
If you have questions
If you have questions or wish to discuss the homework with your peers, you may use the hw5 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.) Then check your answer using this table of Fourier transform pairs and properties. (Explain how you checked your answer.)
Question 2
Compute the Fourier transform of the signal
$ x(t)=\sin^2 ( \pi t + \frac{\pi}{12}) $
(Use the definition of the Fourier transform, not a table of pairs and properties.) Then check your answer using this table of Fourier transform pairs and properties. (Explain how you checked your answer.)
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) Compute the frequency response $ {\mathcal H} (\omega) $ of this system.
b) Compute the system's response to the input $ x(t)= e^{-2(t-2)} u(t-2) $.
Question 6
Consider the causal LTI system defined by the differential equation
$ \frac{d^2y(t)}{dt^2}=2 \frac{dy(t)}{dt}+x(t) $
a) What is the frequency response of this system.
b) What is the unit impulse response of this system.
