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Homework 11, ECE438, Fall 2015, Prof. Boutin
Harcopy of your solution due in class, Monday November 30, 2015
Question 1
Consider the following filter:
$ h[m,n]: \begin{array}{cccc} & m=-1 & m=0 & m=1 \\ n=1&\frac{1}{16} & \frac{2}{16} & \frac{1}{16} \\ n=0&\frac{2}{16} & \frac{4}{16} & \frac{2}{16} \\ n=-1&\frac{1}{16} & \frac{2}{16} & \frac{1}{16} \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 DSFT H(u,v) of this filter. Sketch the plot of H(u,0). Sketch the plot of H(0,v). What are the characteristics of this filter (low-pass, band-pass, or high-pass)?
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 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{array} $
Question 2
Consider the following filter:
$ h[m,n]: \begin{array}{cccc} & m=-1 & m=0 & m=1 \\ n=1&-\frac{1}{9} & -\frac{1}{9} & -\frac{1}{9} \\ n=0&-\frac{1}{9} & \frac{8}{9} & -\frac{1}{9} \\ n=-1&-\frac{1}{9} &- \frac{1}{9} & -\frac{1}{9} \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 DSFT H(u,v) of this filter. Sketch the plot of H(u,0). Sketch the plot of H(0,v). What are the characteristics of this filter (low-pass, band-pass, or high-pass)?
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 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{array} $
Question 3
Consider the following 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 DSFT H(u,v) of this filter. Sketch the plot of H(u,0). Sketch the plot of H(0,v). What are the characteristics of this filter (low-pass, band-pass, or high-pass)?
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 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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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