Parallel Fast Fourier Transform Ryan Liu

Introduction The Discrete Fourier Transform could be applied in science and engineering. Examples: ◦ Voice recognition ◦ Image processing Discrete Fourier Transform (DFT): O(n 2 ) Fast Fourier Transform (FFT): O(n log n) 2

Fourier Analysis Fourier analysis: Represent continuous functions by potentially infinite series of sine and cosine functions. Fourier Series: A function can be expressed as the sum of a series of sines and cosines : 3

Fourier Transform Fourier Series can be generalized to derive the Fourier Transform Forward Fourier Transform: Inverse Fourier Transform: Note: 4

Fourier Transform Fourier Transform maps a time series into the series of frequencies that composed the time series. Inverse Fourier Transform maps the series of frequencies back into the corresponding time series. The two functions are inverses of each other. 5

Discrete Fourier Transform Maps a sequence over time to another sequence over frequency. The Discrete Fourier Transform (DFT): k = 0, … N -1 n = 0, … N -1 Let k represent the discrete time signal, and F n represent discrete frequency transform function. 6

Speech example of DFT “Angora cats are furrier…” 7 Signal Frequency and amplitude

DFT Computation n elements vector x. DFT matrix vector product F n x ◦ f i,j = w n ij for, j < n, and w n is the primitive nth root of unity. 8

DFT Example1 DFT of vector (2,3) ◦ The primitive square root of unity for w 2 is -1 ◦ The inverse of DFT 9

DFT Example2 DFT of vector (1,2,4,3) The primitive 4 th root of unity for w 4 is i 10

DFT Example2 Inverse DFT 11

Fast Fourier Transform DFT requires O(n 2 ) time to process for n samples: So, using DFT is not a best way in practice. Fast Fourier Transform: ◦ Produces exactly the same result as the DFT. ◦ Time complexity O(n log(n)). ◦ divide-and-conquer strategy. 12

FFT Recursively breaks down a DFT of any composite size N = N 1 N 2 into many smaller DFTs of sizes N1 and N2, along with O(N) multiplication. It’s to divide the transform into two pieces of size N/2 at each step. 13

Parallel FFT Algorithm: recursive 14 FFT(1,2,4,3) FFT(1,4) FFT(1) FFT(2,3) FFT(4)FFT(3)FFT(2)

Parallel FFT 15 Tracking the flow of data values.

Phases of Parallel FFT Phase 1: Processes permute the input sequence. Phase 2: ◦ First log n – log p iterations of FFT ◦ No message passing is required Phase 3: ◦ Final log p iterations ◦ In each iteration every process swaps values with partner. 16

Time Complexity Analysis FFT time complexity O(n log n). Parallel FFT ◦ Each process controls n/p elements ◦ The overall communication time complexity is O( (n/p) log p) ◦ Computational time complexity of parallel FFT is O(n log n/p) 17

Reference Quinn,M.J (2004). Parallel programming in C with MPI and OpenMP Chu, E., & George. A., (2000). Serial and Parallel Fast Fourier Transform Algorithms Bi,G & Zeng, Y. (2003). Transforms and Fast Algorithm for Signal Analysis and Representation. The Fast Fourier Transform. (n.d.). Retrieved from Chu, E., & George, A., (1999). Inside the FFT black box: serial and parallel fast Fourier transform algorithms. Boca Raton, Fla.: CRC Press,