EECS 658_________________________PROBLEM SET #3_________________________Fall 1999

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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.

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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!

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2. Stubborn, you still want to convolve (real) h_n and u_n by multiplying their DFTs.
   Show how to compute the DFTs of both h_n and u_n using only one complex DFT.

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3. Compute the cyclic convolution of complex constants (say that 5 times) of order 2^N:
   1. 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(z^2K+1), 0 < K < N.
   2. Show that Yr(z)+z^2K-1Yi(z)=(Hr(z)+z^2K-1Hi(z)) (Ur(z)+z^2K-1Ui(z)) mod(z^2K+1).
   3. Show computing Yr(z)+z^2K-1Yi(z) and Yr(z)-z^2K-1Yi(z) recovers both Yr(z) and Yi(z).
   4. What is the interpretation of the computation in (c) in terms of usual complex numbers?

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   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.

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4. Recursive implementation of the Lagrange interpolation formula:
   1. 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)+...
   2. Show that the constants a,b,c,d... can be computed recursively by setting z=z₁, etc.
   3. Show the Kth recursion requires about 2K mults+adds, for a total of about N² mults+adds.
   4. Show that the standard Lagrange interpolation formula requires about N³ mults+adds.

   5. Apply this to find the unique	INTERPOLATE POINT	1	2	3	4
      cubic polynomial satisfying	POLYNOMIAL VALUE	2	7	18	41

   
   

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5. Multidimensional digit reversal in the Cooley-Tukey n₁n₂...-point FFT:
   1. We use the Cooley-Tukey FFT to break down a large n₁n₂...n_N-point DFT
      into small n₁-point DFTs, n₂-point DFTs...n_N-point DFTs and twiddle mults.
      

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      Show that the Cooley-Tukey FFT maps input v_i to output V_k, where
      input index i=i₁+n₁i₂+n₁n₂i₃+...and output index k=i_N+n_Ni_N-1+n_Nn_N-1i_n-2+...
   2. Set n₁=n₂=...=n_N=2 to derive bit-reversal for 2^N-point DFT.
   3. Show how to compute i₁, i₂, etc. quickly from i (and similarly for k).
      You need not show uniqueness of this mixed integer radix representation.

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