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- 1. DIF-FFT Presented by : Aleem Alsanbani Saleem Almaqashi
- 2. Fast Fourier Transform FFT - A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and inverse of DFT. - There are many FFT algorithms which involves a wide range of mathematics,. A Discrete Fourier transform decomposes a sequence of values into components of different frequencies. - This operation is very useful in many fields but computing it directly from the definition is often too slow to be practical .
- 3. Cont .. • FFT are special algorithms for speedier implementation of DFT. • FFT requires a smaller number of arithmetic operations such as multiplications and additions than DFT. • FFT also requires lesser computational time than DFT .
- 4. Fast Fourier Transform Algorithms • Direct computation of the DFT is less efficient because it does not exploit the properties of symmetry and periodicity of the phase factor WN = e-j2π/N . • These properties are: - Symmetry property. - Periodicity property. • As we already know that all computationally efficient algorithms for DFT are collectively known as FFT Algorithms and these algorithms exploit the above two properties of phase factor, WN.
- 5. FFT Algorithms Classification Based On Decimation • Another classification of FFT algorithms based on Decimation of s(n) :r S(K). Decimation means decomposition into decimal parts. On the basis of decimation process, FFT algorithms are of two types: • 1. Decimation-in-Time FFT algorithms. • 2. Decimation-in-Frequency FFT algorithms.
- 6. Cont.. • Decimation-in-Time Algorithms: sequence s(n) will be broken up into odd numbered and even numbered subsequences. • This algorithm was first proposed by Cooley and Tukey in 1965. • Decimation-in-Frequency Algorithms. the sequence s(n) will be. Broken up into two equal halves. • This algorithm was first proposed by Gentlemen and Sande in 1966. • Computation reduction factor of FFT algorithms . • Number of computations required for direct DFT / Number of computations required for FFT algorithm • = N2 / N / 2 log2 (N)
- 7. decimation-in-frequency FFT algorithm • In decimation-in-frequency FFT algorithm, the output DFT sequence S(K) is broken into smaller and smaller subsequences. For the derivation of this algorithm, the number of points or samples in a given sequence should be N = 2r where r > 0. For this purpose, we can first-divide the input sequence into the first-half and the second-half of the points. • Flow graph of complete decimation-in-frequency (DIF) decomposition of an N-point DFT computation (N = 8).
- 8. Steps for Computation of Decimation in Frequency FFT Algorithm • Given below are the important steps for the computation of DIF FFT algorithms. • 1. Data shuffling is not required but whole sequence is divided in two parts: first half and second half. From these we calculate g(n) and h(n) as follows: • g(n) = s(n) +s(n+N/2 ) • and h(n) = s(n)-s(n+N/2 ) • where n = 0, 1, ..., N/2 -1 Finally data shuffling is performed. It is also performed by Bit reversal.
- 9. Number Of Complex Multiplications Required In DIF- FFT Algorithm • Number of complex multiplications required in decimation-m-. FFT algorithm are the same as that required in decimation-in- time FFT algorithm. • Number of complex multiplication required in these DFT algorithms are N/2 log2iV, where N= 2r, r>0 and N is the total number of points (or samples) in a discrete-time sequence. Thus the total computations (number of multiplication and addition operations) are the same in both FFT algorithms. • Now we will compare the computational complexity for the direct computation of the DFT and FFT algorithm. This comparison is given in Table
- 10. Number Of Complex Multiplications Required In DIF- FFT Algorithm No. of points Complex Complex Speed (or samples" multiplication multiplication improvement in a sequence s s Factor -A/B s(n(, N in direct in FFT computation algorithms of N/2 log2 N = B DFT NN =A= 4- 22 16 4 =4.0 8 -23 64 12 =5.3 16 - 24 256 32 =8.0
- 11. First stage of the decimation-in-frequency FFT algorithm..
- 13. Alternate DIT FFT structures • DIT structure with input natural, output bit-reversed (OSB 9.14):
- 14. Alternate DIT FFT structures • DIT structure with input bit-reversed, output natural
- 15. Radix-2 Decimation-In-Frequency Solved Example Part1 • Example Find the DFT of the following discrete-time sequence . • s(n) = {1, -1, -1, -1, 1, 1, 1, -1} using Radix-2 decimation-in-frequency FFT algorithm. • Solution. The Twiddle factor or phase rotation factor WN= involved in the FFT calculation are found out as follows for N= 8.
- 16. Example Part1
- 17. Example Part1
- 18. Radix-2 decimation-in-frequency Solved Example Part2 • Fig.Flow graph of Radix-2 decimation-in-frequency (DIF) FFT algorithm N = 8. In Radix-2 decimation-in-frequency (DIF) FFT algorithm, original sequence s(n) is decomposed into two subsequences as first half and second half of a sequence. There is no need of reordering (shuffling) the original sequence as in Radix-2 decimation-in-time (DIT) FFT algorithm. But resultant discrete frequency sequence is shuffled (reordered) into natural order because these are obtained in unnatural order. Flow graph of Radix-2 decimation-in-frequency (DIF) FFT algorithm for N= 8 is shown in Fig. Determination of DFT using Radix-2 DIF FFT algorithm requires three stages because the number of points in a given sequence is 8, i.e., = 2r — N — 8, where r is number of stages required = 3.
- 19. Solv. Stage I : A0 = s(0) + s(4) = 1 + 1 = 2 A1 = s(l) + s(5) = -1 + 1 =0 A2 = s(2) + s(6) = -1 + 1 = 0 A3 = s(3) + s(7) = -1 - 1 = -2 A4 = [s(0)+(-1) s(4)] W80 = [1 + (-1) (1)] x 1 =0 A5 = [s(1) + (-1) s(5)]W81 = [-1 + (-1)(1)]((1-j) /√2= - √2(l - j) A6 = [s(2) + (-1) s(6)]W82 = [-1 + (-1) x 1] (- j) =2j A7 = [s(3) + (-1) s(7)]W83 = [-1 + (-1)(-1)]{(-(1-j) /√2} = 0
- 20. Solv…. S(K) = {S(0), S(l), S(2), S(3), S(4), S(5), S(6), S(6), S(7)} Or S(X) = {0-√2+(2 + √2 )j, 2 -2j √2+(-2 + √2)j,4, √2+ (2 - √2 )j,2 + 2j,- √2 -(2 + √2)7}
- 21. Conclusions • Radix 22, 24… Structure uses less adders and multipliers but still has good efficiency processing DIF DFT • Common Factor Algorithm and Butterfly Structure enable this architecture to reuse its modules numerous times
- 22. References • [1],Shousheng He and Torkelson, M. “A new approach to pipeline FFT processor,” Proceedings of IPPS '96, 15-19, pp766 –770. April 1996 • [2] Alan V.Oppenheim, Ronald W. Schafer, “ Discrete-time signal processing “ 2nd edition • [3] Zhangde Wang “INDEX MAPPING FOR ONE TO MULTI DIMENSIONS “Electronics Letters Publication Volume: 25, pp: 781-782 Jun 1989 • [4] He, S. & Torkelson, M., A systolic array implementation of common factor algorithm to compute DFT, Proc. Int. Symp. on Parallel Architectures, Algorithms and Networks, Kanazawa, Japan, pp. 374-381, 1994. • [5]IJung-YeolOH and Myoung-Seob LIM , ‘Fast Fourier Transform Algorithm for Low-Power and Area-Efficient Implementation’EICETRANS.COMMUN.,VOL.E89–B, APRI • [6]BURRUS, c. s.: 'Index mappings for multidimensional formulation of the DFT and convolution',IEEE Trans., 1977, ASSP-25, (6), pp. 239-242
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