EECS 658_________________________PROBLEM SET #4_________________________Fall 1999

**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.

*Prime-Factor FFT for lengths 11 and 5:*
- 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).
- Map the 11-point DFT of {v
_{0}...v_{10}} 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.
- Show that the 5-point prime factor FFT includes the product (v
_{1}+...+v_{4})×(MESS).

Show (MESS)=-5! So 5-point DFTs require only 4 mults! Can you see this directly?

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

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

*Reconstruction of a 2×2 image from samples of its 5×4 2-D DFT:*
- 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.
- Use recursive Lagrange interpolation to reconstruct the image from its 2-D DFT values:

Apply this to find the | Freq. (k_{1},k_{2}) |
(3,2) | (2,3) | (1,0) | (0,1) | (4,2) |

2×2 image satisfying | Real I(k_{1},k_{2}) |
-1.000 | 1.6032 | 7.6180 | 4.0000 | -1.000 |

the following points: | Imag I(k_{1},k_{2}) |
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.
- 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!
- 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.