1. Fast Fourier Transform
CS 498LVK
Hassan Jafri

2. Overview
An FFT is an efficient algorithm to compute the Discrete Fourier Transform (DFT) and it inverse
Complexity of Direct computation of DFT is O(n^2)

3. FFT Algorithms FFT algorithms reduce the complexity to O(n log n)
However, these algorithms are not cache friendly
Radix-2, Radix-4, Radix-8 etc.

4. The Matrix Algorithm
Matrix Fourier Algorithm (4-step algorithm) has better cache locality
Works for composite data lenghth. For input set size n = R x C
Consider input array as RxC matrix

5. The Matrix Algorithm THE ALGORITHM
Apply a (length R) FFT on each column
Multiply each matrix element (index r, c) by the twiddle factor
Apply a (length C) transform on each row
Transpose the Matrix

6. MFA with Slight Variation
n1 simultaneous n2-point multirow FFTs with twiddle factor multiplication
n2 individual n1-point multicolumn FFTs
Transpose

7. The Code subroutine parallel_fft(A, W, U, N)
double complex A(*), W(*), U(*)
if (N .LE. CACHESIZE) then
CALL in_cache_fft(A, W, U, N)
return
end if

8. Step 1 !$OMP PARALLEL !$OMP DO do I=1, N/2 W(I) = A(I) + A(I+N/2)
W(I+N/2) = (A(I)-A(I+N/2)) * U(I)
end do

9. Step 2 !$OMP DO do J=1, 2 call rec_fft(W((J-1)*(N/2)+1),
A(J-1)*(N/2)+1, U(N/2+1), N/2)
end do

10. Step 3 !$OMP DO do I=1, N/2 A(2*I-1)=W(I) A(2*I)= W(I+N/2) end do
!$OMP END PARALLEL
return
end

11. For Reference
Swarztrauber, P.N.: Multiprocessor FFTs. Parallel Computing 5 (1987) 197–210
Cochrane, W.T., Cooley, J.W., Favin, D.L., Helms, H.D., Kaenel, R.A., Lang,W.W., Maling, Jr., G.C., Nelson, D.E., Rader, C.M., Welch, P.D.: What is the fast Fourier transform? IEEE Trans. Audio Electroacoust. 15 (1967) 45–55
Swarztrauber, P.N.: FFT algorithms for vector computers. Parallel Computing 1 (1984) 45–63
Frigo, M., Johnson, S.G.: Fftw. (
Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19 (1965) 297–301.
Daisuke Takahashi, Mitsuhisa Sato, Taisuke Boku: "An OpenMP Implementation of Parallel FFT and Its Performance on IA-64 Processors". WOMPAT 2003:
Wadleigh, K.R.: High performance FFT algorithms for cache-coherent multiprocessors.The International Journal of High Performance Computing Applications 13 (1999) 163–171 Takahashi,
D.: A blocking algorithm for parallel 1-D FFT on shared-memory parallel computers. In: Proc. 6th International Conference on Applied Parallel Computing (PARA 2002). Volume 2367 of Lecture Notes in Computer Science., Springer-Verlag (2002) 380–389