| Physics 198 | May 2, 1997 |
Problem 8. Given a complete set of values for 2 identical periodic functions, compute their FFTs and check the validity of the linearity property of the discrete Fourier transform..
For 0 > k or for k > 7 , x(k) = x(k mod 8) ; i.e., the function has a period of 8.
A second function is also defined as y(k) = x(k) for all k.
Definition of Fourier transform:
where T is the sampling interval (either spatial or temporal) here set equal to unity.
Calculating
is pretty much a brute force enterprise, using the relation
and watching the sign of the cosine and sine functions in their different quadrants.
n = 0 :
n = 1 :
n = 2 :
Similarly :
Now y(kT) is defined as identical to x(kT), so .
So Z = X + Y = 2X.
Now want to take the transform of z = x + y. This is obviously the same as z(k) = 2x(k) and so z can be defined as :
Starting to take the transform of z(k) quickly shows that all values are the same as the corresponding transform values of x(k) except for a common factor of 2, which can be taken outside. Then the resulting transform is given by
Thus the transform of z = x + y gives Z which is identical to Z = X + Y. This agrees with the linearity property, but of course does not prove it because this is a special case where x = y.
Problems 9 and 10 are not finished yet. I have all the others, and will put them up now rather than wait for these two. They will appear soon (e.g., later on Saturday)
Problem 9. Demonstrate the symmetry problem
for x(k).
Problem 10. Compute the discrete Fourier transform of x(k-3)
Send Mail or Comments: matthysd@vms.csd.mu.edu
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