EECS 658_________________________PROBLEM SET #4_________________________Fall 1999

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ASSIGNED: Oct. 07, 1999. READ: HANDOUTS: Winograd FFT, Phi(N), Cyclotomic polynomials.
DUE DATE: Oct. 14, 1999. THIS WEEK: Good-Thomas and Winograd FFTs and some applications.

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1. Prime-Factor FFT for lengths 11 and 5:
   1. Prove that 2 generates the cyclic group {{1,2...10}, × mod(11)}, with elements reordered.
      This is equivalent to stating that 2 is a primitive element of the finite field GF(11).
   2. Map the 11-point DFT of {v₀...v₁₀} into a 10-point cyclic convolution.
      Specify explicitly the two sequences (one involving the v_i, other involving powers of W) that are being cyclically convolved. Don't bother to give the convolution algorithm.
   3. Show that the 5-point prime factor FFT includes the product (v₁+...+v₄)×(MESS).
      Show (MESS)=-5! So 5-point DFTs require only 4 mults! Can you see this directly?

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2. We want to compute a 25-point DFT. How many mults do each of the following require:
   1. 25-point Winograd FFT. HINT: Reduce to 20-point and two 4-point cyclic convolutions;
   2. 25-point Cooley-Tukey FFT. HINTS: 25=5×5 (big help!); answer is 50% bigger than (a).

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3. Explicit formulae using Euler's totient function Ø(N) (Phi(N)):
   1. Show that the solution to ax=b mod(c) is x=a^(Ø(c)-1)b mod(c). Assume GCD(a,c)=1.
   2. Show that if i is a unit in the ring Z/(n), its mult. inverse is i^-1=i^(Ø(n)-1) mod(n).
   3. Derive formulae using Ø for reindexing input and output in the Good-Thomas FFT

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4. Reconstruction of a 2×2 image from samples of its 5×4 2-D DFT:
   1. Using the Good-Thomas FFT, reformulate the 5×4 2-D DFT of a 2×2 image
      as the 20-point 1-D DFT of a 1-D signal of length 10, as shown below.
   2. Use recursive Lagrange interpolation to reconstruct the image from its 2-D DFT values:
      

      Apply this to find the	Freq. (k1,k2)	(3,2)	(2,3)	(1,0)	(0,1)	(4,2)
      2×2 image satisfying	Real I(k1,k2)	-1.000	1.6032	7.6180	4.0000	-1.000
      the following points:	Imag I(k1,k2)	0.0000	3.7788	1.9021	-5.000	0.0000

      and the complex conjugates of these values at the complex conjugate frequencies.
      To ensure consistency, follow the Matlab format shown below (my actual output).
      Of course, we can translate the image; this only introduces a known linear phase term.
   3. This seem wasteful! We shouldn't need 10 2-D DFT values to reconstruct a 2×2 image!
      But we do! Give two different 2×2 images whose 5×4 2-D DFT agree at 8 points!
   4. Explain why 10 points (not fewer) of the 5×4 2-D DFT guarantee a unique 2×2 image.
      Good-Thomas FFT and Lagrange interpolation are smart enough to ensure uniqueness.
      Although this is a tiny example, the extension to larger problems should be evident.

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