EECS 658_________________________PROBLEM SET #3_________________________Fall 1999

ASSIGNED: Sept 30, 1999. READ: FFT (Fast Fourier) handout given out on Tuesday Sept. 28.
DUE DATE: Oct. 07, 1999. THIS WEEK: More fast convolutions and fast Fourier transforms.

1. Show that the minimum #multiplications required for an 11-point cyclic convolution is 20.
Show that the minimum #multiplications required for an 12-point cyclic convolution is 18.
Thus a longer convolution can require fewer multiplications, even for the lower bound!
2. Stubborn, you still want to convolve (real) hn and un by multiplying their DFTs.
Show how to compute the DFTs of both hn and un using only one complex DFT.
1. Compute the cyclic convolution of complex constants (say that 5 times) of order 2N:
2. Show that Winograd breaks down the cyclic convolution into a series of smaller convolutions: Yr(z)+jYi(z)=(Hr(z)+jHi(z))(Ur(z)+jUi(z)) mod(z2K+1), 0 < K < N.
3. Show that Yr(z)+z2K-1Yi(z)=(Hr(z)+z2K-1Hi(z)) (Ur(z)+z2K-1Ui(z)) mod(z2K+1).
4. Show computing Yr(z)+z2K-1Yi(z) and Yr(z)-z2K-1Yi(z) recovers both Yr(z) and Yi(z).
5. What is the interpretation of the computation in (c) in terms of usual complex numbers?

Note that this computes a complex convolution using two real convolutions, doubling #mults,
while simply replacing each real mult with a complex mult would triple or quadruple #mults.
1. Recursive implementation of the Lagrange interpolation formula:
2. Show that any polynomial of degree N can be written in the following form:
X(z)= a+b(z-z1)+c(z-z1)(z-z2)+d(z-z1)(z-z2)(z-z3)+...
3. Show that the constants a,b,c,d... can be computed recursively by setting z=z1, etc.