(New page: = Homework 3 = = Due in class, Monday February 4, 2011 = == Important Notes == *Justify all your answers. *Write your answers clearly and cleaning. *Write on one side of the paper only...) |
|||
| (5 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | = Homework 3 | + | = Homework 3, [[ECE301]], Spring 2011, [[user:mboutin|Prof. Boutin]] = |
| − | + | ||
| − | + | ||
| + | = Due in class, Monday February <span style="color:red"> 7 </span>, 2011 = | ||
| + | ;Oops! Monday is the 7th, not the 8. -pm | ||
== Important Notes == | == Important Notes == | ||
*Justify all your answers. | *Justify all your answers. | ||
| Line 15: | Line 15: | ||
== Question 1 == | == Question 1 == | ||
| − | Which of the following systems are invertible? Memoryless? Causal? Stable? Linear? Time-invariant? (Justify your answers mathematically.) | + | Which of the following systems are invertible? Memoryless? Causal? Stable? Linear? Time-invariant? (Justify your answers mathematically.) |
a) <math class="inline"> y[n]=x[n]x[n-1] \ </math> | a) <math class="inline"> y[n]=x[n]x[n-1] \ </math> | ||
| Line 21: | Line 21: | ||
b) <math class="inline">y(t)= x\left( \sin(t) \right) \ </math> | b) <math class="inline">y(t)= x\left( \sin(t) \right) \ </math> | ||
| − | c) <math class="inline">y[n]= \sum_{n-10}^{n+10} x[k] \ </math> | + | c) <math class="inline">y[n]= \sum_{k=n-10}^{n+10} x[k] \ </math> |
d) <math class="inline">y(t)= t^2 x(t+1) \ </math> | d) <math class="inline">y(t)= t^2 x(t+1) \ </math> | ||
| − | == Question 2 == | + | == 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 | Show that the CT unit impulse satisfies the equation | ||
<math> \delta (2 t ) = \frac{1}{2} \delta (t) </math> | <math> \delta (2 t ) = \frac{1}{2} \delta (t) </math> | ||
| − | + | :Note: you may find [[Homework_3_ECE438F09|this page]] useful. It was written by a student in [[ECE438]].-pm | |
| − | == Question | + | == Question 4== |
True of False? (Justify your answer) | True of False? (Justify your answer) | ||
The cascade of two time-invariant systems is itself time-invariant. | The cascade of two time-invariant systems is itself time-invariant. | ||
| − | |||
==Question 5== | ==Question 5== | ||
| + | The unit impulse response of an LTI system is | ||
| + | |||
| + | <math>h[n]= \delta[n+1] +2 \delta[n-1]. </math> | ||
| + | |||
| + | Compute the system's response to the input | ||
| + | |||
| + | <math>x[n] = u[n-1]- u[n-7]. </math> | ||
==Question 6== | ==Question 6== | ||
| + | The unit impulse response of an LTI system is | ||
| + | |||
| + | <math>h[n]= u[n-3].</math> | ||
| + | |||
| + | Compute the system's response to the input | ||
| + | |||
| + | <math>x[n] = \left( \frac{1}{3} \right) ^{-n} u[-n-1].</math> | ||
| + | |||
| + | ==Question 7== | ||
| + | The unit impulse response of an LTI system is | ||
| + | |||
| + | <math>h(t)= u(t+5)-u(t-7). </math> | ||
| + | |||
| + | Compute the system's response to the input | ||
| + | |||
| + | <math> x(t) = u(- t) </math> | ||
| + | |||
| + | ==Question 8== | ||
| + | The unit impulse response of an LTI system is | ||
| + | |||
| + | <math>h(t)= e^{t} u(-t+5) </math> | ||
| + | Compute the system's response to the input | ||
| + | <math> x(t) = u(-t-8) </math> | ||
---- | ---- | ||
Latest revision as of 11:19, 7 February 2011
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) $
