Assignment 5: Continuous Time Fourier Transform

• If your final grade is between 90-100, the full credit of this assignment adds 1 point to your final grade and the maximum is 100.
• If your final grade is between 80-90, the full credit of this assignment adds 2 points to your fianl grade and the maximum is 90.
• If your final grade is between 70-80, the full credit of this assignment adds 4 points to your fianl grade and the maximum is 80.
• If your final grade is between 60-70, the full credit of this assignment adds 8 points to your fianl grade and the maximum is 70.
• If your final grade is below 60, the full credit of this assignment adds 16 points to your fianl grade and the maximum is 60.

Problems

1. The step size of data set is too large (&Delta x=1). We can resolve this problem by interpolate those L+1 data points by natural cubic spline. (see lecture note 7, page 15). (20%)

   The interface of your subroutine would be like

   function [coeff] = ncp(L, x)

   where L is a scalar indicating the number of data set, x is an array of length L. The return argument coeff is an L× 4 matrix, in which the ith row is the coefficients of the cubic polynomial, from degree 0 to degree 3, in the ith iterval.

2. Compare your result in 1 with the spline function in Matlab/Octave. (20%)

3. Write a function to evaluate queried function points of the cubic spline computed in 1. (20%)

   The interface is

   function [y] = queryncp(coeff, x)

   where coeff is the matrix computed in 1, and x is the data points to be queried, which is a sorted arry of arbitrary length (all elements are in [0,L]).
   The return argument y is an array of the same length as x, whose elements is the corresponding function values of x.

4. Compare your results in 3 with the function ppval. (20%)

5. Now we can use functions defined in 1 and 3, and the composite Simpson's rule to compute the integration,
   .
   (20%)

   The interface is like

   function intval = ctft(L, fx, m, t)

   where

   • L is the number of data points.
   • fx is an array of length L that contains function values of 0,1,...,L-1. (Use the first element of fx as the function value at L.)
   • m is the number of subintervals inside each interval. For example, if m=4, then the evaluation points of Composite Simpson's rule become
     0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, ..., L-1, L-0.75, L-0.5, L-0.25, L.

   • t is the parameter in CTFT.
   • intval is the computed result of integration.

Submission

Updated Jan 13, 2009