1. Chapter 9. Computation of Discrete Fourier Transform 9.1 Introduction 9.2 Decimation-in-Time Factorization 9.3 Decimation-in-Frequency Factorization 9.4 Application of FFT 9.5 Fast Computation of DCT 9.6 Matrix Approach 9.7 Prime Factor Algorithm BGL/SNU

2. 1. Introduction BGL/SNU

5. - Example of fast computation BGL/SNU

7. 2. Decimation-in-Time Factorization BGL/SNU

11. Decimation-in-time FFT flow graphs BGL/SNU

12. Decimation-in-time FFT flow graphs BGL/SNU

13. Decimation-in-time FFT flow graphs BGL/SNU

14. (Sande- Tuckey) FFT 3. Decimation-in-frequency Factorization BGL/SNU

15. g(n) h(n) g[0] g[1] g[2] g[3] h[0] h[1] h[2] h[3] X[0] X[2] X[4] X[6] X[1] X[3] X[5] X[7] x[0] x[1] x[2] x[3] x[4] x[5] x[6] x[7]

16. Final flow graph X[0] X[4] X[2] X[6] X[1] X[5] X[3] X[7] x[0] x[1] x[2] x[3] x[4] x[5] x[6] x[7] BGL/SNU

17. -Remarks - # Stages - # butterflies - # computations - inplace computations - output data ordering : bit-reversed -Question The flow graph for D-I-F is obtained by reversing. The direction of the flow graph for D-I-T. Why? -Omit Sections 9.5-9.7 BGL/SNU

18. (1) Spectrum Analysis - is the spectrum of x[n], n=0,1,…,N-1 - Inverse transform can be done through the same mechanism i) Take the complex conjugate of X[k] ii) Pass it through the FFT process, But with one shift right(/2) operation at each stage iii) Finally, take the complex conjugate of the result 4. Applications of FFT BGL/SNU

19. (2) Convolution ( Filtering ) - Operation reduction : h[n] x[n] y[n] N 2N N x[n] 0 N-1 n h[n] 0 N-1 n y[n 0 2N-2 n #computation(multi)? 1+2+…+N+N-1+…+1+0 =N 2

20. -Utilize FFT of 2N-point ~ h[n] 0 N-1 2N n x[n] 0 N-1 2N n y[n] 0 2N-2 2N n R 2N [n] 1 0 2N-1 n ~ ~ 2N-pt DFTs BGL/SNU

21. 2N-pt FFT 2N-pt FFT 2N-pt IFFT x[n] h[n] X[k] H[k] Y[k] y[n] 2N # operation (multi) - operation reduction : BGL/SNU

22. (3) Correlation /Power Spectrum 2N-point DFTs # Operation : - Power spectrum P[k] = X[k] X * [k] BGL/SNU

23. $ Comparison of # computation 5121024 1 10 6 10 2 10 3 10 4 10 5 10 0.45k 1k 2k 5k 35k 16k 7.25k 3.3k 16k 62.5k 250k 1M Direct Computation FFT-based Convolution Correlation FFT N BGL/SNU

24. 5. Fast Computation of DCT BGL/SNU

25. - Example: Lee’s Algorithm (1984, IEEE Trans, ASSP, Dec) 1D x[ 0] x[1] x[2] x[3] x[4] x[5] x[6] x[7] X [ 0] X[4] X[2] X[6] X[1] X[5] X[3] X[7] BGL/SNU

26. - Example: 2D DCT Algorithm (1991, N.I.Cho and S.U.Lee) BGL/SNU Separable Transform NxN 2D DCT = N 1-D DCT into row direction followed by N 1-D DCT into column direction. Totally 2N 1-D DCT (each N-point) are required.

27. Fast Algorithm reduces the number of 1-D DCTs into N. By using the trigonometric properties, 2D DCT is decomposed into 1-D DCTs.

28. Signal flow graph of 2-D DCT 8x8 DCT 4x4 DCT

29. 6. Matrix Approach · Decimation-in-time BGL/SNU

30.    

31. · Decimation-in-frequency   BGL/SNU

32. · General expression for N=2 case

33. · Extension to general N (Cooley/Tuckey) BGL/SNU

34. · # computations (complex) BGL/SNU

35. 7. Prime Factor Algorithm (Thomas/Good) (1) Basics from Number Theory  Euler’s Phi function  Euler’s Theorem 6mod125,2)(,6,5)(.mod1then,1),(If )( )(   N aNNaeg NaNa     Chinese Remainder Theorem (CRT) N

36.  Second Integer Representation (SIR) BGL/SNU

38. (2) Prime Factor Algorithm Set Then BGL/SNU

39. Therefore Note that the only difference is in the “twiddle factor” BGL/SNU

40. (3) Comparison Example 12-Point DFT (N=12, p=3, q=4)C/T : Cooley/Tuckey T/G : Thomas/Good · Transform · Index Mappings

41. · Diagram 4pt DFT 0 3 6 9 0 3 6 9 (0,0) (1,0) (2,0) (3,0) 4pt DFT 4 7 10 1 1 4 7 (0,1) (1,1) (2,1) (3,1) 4pt DFT 8 11 2 5 2 5 8 (0,2) (1,2) (2,2) (3,2) 3pt DFT ),( 01 kkX (0,0) (0,1) (0,2) 3pt DFT (1,0) (1,1) (1,2) 3pt DFT (2,0) (2,1) (2,2) 3pt DFT (3,0) (3,1) (3,2) 00 44 88 19 51 95 26 610 2 33 77 11 T/GC/T T/G BGL/SNU 001 ),(nkx (0,0) (0,1) 102 ),(kkx (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) (2,2) (2,3) (2,0) (2,1)

42. - Radix-2 algorithms: algorithms in textbook : - Radix-4 algorithms : Radix-4 algorithm BGL/JWL/SNU

43. - Radix-4 butterfly BGL/JWL/SNU

44. - Radix-4 butterfly -j j 1 j -j BGL/JWL/SNU