EECS 658_________________________PROBLEM SET #6_________________________Fall 1999

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ASSIGNED: Nov. 04, 1999. READ: HANDOUTS: Polynomial Transforms/Use of Polynomial...
DUE DATE: Nov. 11, 1999. THIS WEEK: Number Theoretic Transforms, Fourier reconstruction.

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1. PROVE: If n is prime then group {{1,2...n-1},× mod(n)} has exactly Ø(Ø(n)) generators.
   1. PROVE THIS USING: the procedure given for constructing generators; and then using:
   2. Fermat's theorem to show that if q is a generator then any other generator is a power of q.

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2. Image reconstruction from Fourier samples on a polar raster:
   • In SAR (Synthetic Aperture Radar) and MRI (Magnetic Resonance Imaging), we need to
     reconstruct a discrete image from samples of its Fourier transform along radial slices.
   • Apply the algorithm of Problem #4 of Problem Set #4 to reconstruct a 3×3 slanted image
     from samples of its DTFT along radial slices with slopes: 0; ±7/6; infinity (see below).
   • The image and values of its 6×7 2-D DFT as computed by Matlab are as shown below.
   • The extension to larger problems should again be evident. Note slanting the image helps.

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3. Fermat Number Theoretic Transforms in the finite field GF(17):
   1. What is the order of the element "2" in the finite field GF(17)?
   2. Give the equation for the 8-point DFT in GF(17). What is w? How many mults?
   3. Give the equation for the 4-point DFT in GF(17). What is w? How many mults?
   4. 11 is a square root of 2 in GF(17). Is 11 a primitive element in GF(17)?
      Give the equation for the 16-point DFT in GF(17). What is w? How many mults?

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4. Mersenne Number Theoretic Transforms in the finite field GF(31):
   1. What is the order of the element "2" in the finite field GF(31)?
   2. Give the equation for the 5-point prime factor (Rader) FFT in GF(31).
   3. There is no square root of -1 in GF(31), i.e., the polynomial x²+1 is prime over GF(31).
      What is the order of the element "1+x" in the extension field (of GF(31)) GF(31²)?
      GF(31²)={{ax+b}, +,× mod(x²+1)} over (coefficients in) the field GF(31) (0 < a,b < 31).
   4. What is the longest possible blocklength of a DFT in GF(31²)? Explain your answer.
   5. Give the equation for the 8-point DFT in GF(31²). What is w? How many mults?

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5. Matlab computation of Mersenne Number Theoretic Transform of order 14:
   1. Write a Matlab program that uses a 14-point NTT in GF(127) to compute the linear
      convolution y of h={1,4,1,5,6,2,6} and u={7,1,8,2,5,1,8}. What are w and 14^-1?
   2. Write a Matlab program that uses the same NTT to deconvolve h from y to get u.
      To "divide" Y by H: Use U=YH^-1 computed using Problem #3 of Problem Set #4.
   3. Print out the NTT matrix (analogous to the DFT matrix) that implements the NTT.

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