SIMG-782 Homework 3

Problem 1: Plot the Hough Transform parametric graph in rectangular form y=ax+b for the following points. Determine which sets of three or more points fall on a straight line. Your answer has to come out of the Hough Transform analysis unaided by a human plotting the data and seeing the lines. {(0,3), (0,-3), (1,5), (1,-2), (-1,1), (2,-1), (2,7), (-1,-4)}

Problem 2 : Plot the Hough transform parametric graph for the polar form of the relationship, r=x cos q+y sin q for the set of points given in Problem 1. Use the analysis to determine the sets of three or more points that fall on a straight line. (Note: you are not asked here to find the (a,b) parameters of the straight lines. However, that would be a simple thing to do once you have the points.)

Problem 3: Compute the DFT of the following finite-length sequences considered to be of length N. Use pencil and paper calculations to obtain your results. You will need them to check the program you will write in Problem 4. Then plot the DFT coefficients in the complex plane. Label each point with its index.

  1. N=8, x(n)=d(n), where d(n) is the impulse sequence defined as d(0)=1 and d(n)=0 for n=1,2,…N.
  2. N=8, x(n)=d(n-1). Note that the "1" is now shifted to position n=1.
  3. N=8, x(n)=d(n-3).
  4. N=8, x(n)=d(n-2)
  5. N=4, x(n)=d(n-1)

Problem 4: Write an IDL program that will compute the DFT of a sequence x(n) of length N and plot the DFT coefficients as points in the complex plane. You may use the routine FFT. Use the program to calculate the results of Problem 3.

Problem 5: Let x(n)=a^n, n=0,1,…N-1, where |a|<1. What function x(t) could have been sampled to create x(n)? How much information about x(t) is missing? That is, how many different x(t) functions can you come up with that would provide exactly the same data? What additional conditions do you advise to make the selection of x(t) unique?

Problem 6: Apply the program you wrote in Problem 4 to the data in Problem 5 for a=0.5. It is up to you to choose a suitable value for N. Provide some analysis that supports your choice of N. It turns out that the computer to process the data costs 1000*N dollars but errors in reconstructing x(t) cost an average of 2000*P dollars in the system that will use this algorithm, where P is the average percent error between the actual value and the reconstructed value. Your promotion depends on the soundness of your presentation to the VP of engineering.