## Contents

# Homework 6, ECE301, Spring 2011, Prof. Boutin

# Due by 6pm Friday March 11, 2011

This homework consists of a double-blind peer-review of **homework 5**. In other words, you will be grading each other's homework! Of course, we are giving you the solution.

Every student who handed in a scan of their homework in the appropriate homework section of the instructor's Assignment Box can participate in this review. To begin the review, go to your own dropbox and click on your "Assignment" tab. There, you should see a file (anonymous) waiting to be reviewed, along with a space to write/upload your review. Open the file and grade each question according to the instructions described below. It is recommended that you print out the file and mark out all mistakes using a red pen; you can then scan and upload the graded homework (in pdf format). When you are done grading, add up all the points and write the total on the front page of the assignment.

Note 1: if you are having problems with the system or the file, please let us know by writing a comment on the hw6 discussion page or by emailing your instructor as soon as possible

Note 2: If the answer to a problem is pretty much unreadable, feel free to give zero points for that problem.

## Overall Presentation: 10 points

All problems solutions should be clearly written, in the correct order. The paper should be free of stains and tears (e.g., not torn out of a spiral book). The scan should be of good enough quality to be easy to read. The file should be in pdf format.

## Question 1: 20 points

- 15 pts for the computation of the Fourier transform of the continuous-time signal. Check every step of the computation and remove point for any mistakes. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down.Make sure to explain why you are taking off points and how many points. If the student used the table of pairs or the table or properties at any point in this question, give 0 points.
- 5 points for checking the answer using the table. If the answer obtained with the table does not match with the answer previously obtained, give zero points. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points.

## Question 2: 20 points

- 15 pts for the computation of the Fourier transform of the signal. Check every step of the computation and remove point for any mistakes. Take points off if properties or pairs from a table were used. (Anything except linearity should incur a large penalty). Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points. If the student used the definition of the Fourier transform (i.e., the integral formula) to obtain the Fourier transform of either the constant function 1 or of the complex exponential functions, give no more than 3 points total for this part.

- 5 points for checking the answer using the table. If the answer obtained with the table does not match with the answer previously obtained, give zero points. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points.

## Question 3: 15 points

Note that there are the two correct ways to answer this problem in the solution: the easy way (based on Parseval's equation) and the hard way (based on inverting $ {\mathcal X} (\omega) $ to get x(t)). If the hard way was chosen, write a note describing the easier solution (but do not take off points). In either case, check every step of the computation and remove points for any mistakes. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points.

## Question 4: 15 points

Note that there are the two correct ways to answer this problem (see the solutions). If the solution chosen is to split the transformation into a time scaling and a time shifting and the time shift value is wrong, take off 5 points. Check every step of the computation and remove points for any mistakes. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points.

## Question 5: 25 points

a) 5 points . Since the question states "compute the frequency response", the answer should include a computation. Give no more than 2 points if only the answer is given.

b) 20 points Compute the system's response to the input $ x(t)= e^{-2(t-2)} u(t-2) $. There are two correct ways to answer this problem: convolving x(t) with h(t), and taking the inverse Fourier transform of the product $ {\mathcal X} (\omega){\mathcal H} (\omega) $. Check every step of the computation and remove points for any mistakes. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points.

## Question 6: 25 points

a) 10 points Give no more than 3 points if the formula given in class was used without justification. Check every step of the computation and remove points for any mistakes. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points.

b) 15 points Check every step of the computation and remove points for any mistakes. Take points off if the explanation is not clear/logical/rigorous. In particular, any statement that is not true should be marked down. Make sure to explain why you are taking off points and how many points.