Homework 11, ECE438, Fall 2013, Prof. Boutin

Harcopy of your solution due in class, Friday November 22, 2013


Presentation Guidelines

  • Write only on one side of the paper.
  • Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
  • Staple the pages together.
  • Include a cover page.
  • Do not let your dog play with your homework.

Question 1

Compute the CSFT of the following signals:

a) $ f(x,y)=\frac{ e^{j 2\pi x} \sin(\pi y)}{y} $

b) $ f(x,y)=rect(x-x_0) $

c) $ f(x,y)=cos(\pi x) $

Make sure to specify what property you are using at every step.

Question 2

Consider the following FIR filter:

$ h[m,n]: \begin{array}{cccc} & m=-1 & m=0 & m=1 \\ n=1&-\frac{1}{8} & \frac{1}{2} & -\frac{1}{8} \\ n=0&-\frac{1}{4} & 1 & -\frac{1}{4} \\ n=-1&-\frac{1}{8} & \frac{1}{2} & -\frac{1}{8} \end{array} $

a) Write a difference equation that can be used to implement this filter.

b) Is this filter separable? Answer yes/no and justify your answer.

c) Compute the CSFT H(u,v) of this filter. Sketch the plot of H(u,0). Sketch the plot of H(0,v).

d) What is the output image when this filter is applied to the following image (using symmetric boundary conditions)?

$ g[m,n]: \begin{array}{ccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0\\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{array} $


Discussion

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