EECS 658_________________________PROBLEM SET #6_________________________Fall 1999
ASSIGNED: Nov. 04, 1999. READ: HANDOUTS: Polynomial Transforms/Use of Polynomial...
DUE DATE: Nov. 11, 1999. THIS WEEK: Number Theoretic Transforms, Fourier reconstruction.
PROVE: If n is prime then group {{1,2...n-1},× mod(n)} has exactly Ø(Ø(n)) generators.
- PROVE THIS USING: the procedure given for constructing generators; and then using:
- Fermat's theorem to show that if q is a generator then any other generator is a power of q.
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.
Fermat Number Theoretic Transforms in the finite field GF(17):
- What is the order of the element "2" in the finite field GF(17)?
- Give the equation for the 8-point DFT in GF(17). What is w? How many mults?
- Give the equation for the 4-point DFT in GF(17). What is w? How many mults?
- 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?
Mersenne Number Theoretic Transforms in the finite field GF(31):
- What is the order of the element "2" in the finite field GF(31)?
- Give the equation for the 5-point prime factor (Rader) FFT in GF(31).
- 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).
- What is the longest possible blocklength of a DFT in GF(31²)? Explain your answer.
- Give the equation for the 8-point DFT in GF(31²). What is w? How many mults?
Matlab computation of Mersenne Number Theoretic Transform of order 14:
- 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}?
- 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.
- Print out the NTT matrix (analogous to the DFT matrix) that implements the NTT.