Hossein Sameti Department of Computer Engineering Sharif University of Technology

 Many real-life systems can be modeled by LTI systems  use convolution for computing the output  use DFT to compute convolution  Fast Fourier Transform (FFT) is a method for calculating Discrete Fourier Transform (DFT)  Only faster!  Definition of DFT:  How many computations? 2 N pt. DFT of x(n) Q: For each k:How many adds and how many mults? A: (N-1) complex adds and N complex mults. How many k values do we have? N Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

3 Direct computation: FFT: Ideal case: NDirectFFT 10^3O(10^6)O(10^3*log10^3)=O(10^4) 10^6O(10^12)O(10^6*log10^6)=O(2*10^7) Example: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

4 FFT Decimation in time Decimation in frequency Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

5 The main idea: use the divide and conquer method It works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

6 N: power of 2 n: evenn: odd n: evenn=2r r:0  N/2-1 n:0  N-2 n: oddn=2r+1 r:0  N/2-1 n:1  N-1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

7 Suppose: What are G(k) and H(k)?

8 In G(k) and H(k), k varies between 0 and N/2-1. However, in X(k), k varies between 0 and N-1. Solution: use the relationship between DFS and DFT. We thus need to replicate G(k) and H(k) “once”, to get X(k). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

9 g(r) h(r) pt. DFT + (twiddle factor) After replication Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

10 g(r) h(r) pt. DFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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12 N/2 pt. DFT block Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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14 r(0) r(1) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

15 r(0) r(1) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

16 Flow graph of a the 2-pt. DFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

17 How many stages do we have? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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20 2 mults+ 2 adds 1 mult+ 2 adds Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

21 In-place computation (only N storage locations are needed) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

22 How many stages do we have Each stage has N inputs and N outputs. Each butterfly has 2 inputs and 2 outputs. Each stage has butterflies. Each butterfly needs 1 mult and 2 adds. Total number of operations: adds mults Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

23 Output indexing is in order. input indexing is shuffled. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

24 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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27 The main idea: use the divide and conquer method (this time in the frequency domain) Divide the computation into two parts: even indices of k and odd indices of k. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

28 1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

29 N/2 pt. DFT of g(n) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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33 N/2 pt. DFT of h(n) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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39 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

 Change x with X (i.e., input nodes with output nodes)  Change X with x (i.e., output nodes with input nodes)  Reverse the order of the flow graphs.  The same system function is achieved. 40 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

41 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

42 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

 How can we deal with twiddle factors?  Should we store them in a table (i.e, use a lookup table) or should we calculate them?  What happens if N is not a factor of 2?  It can be shown that if N=RQ, then an N pt. DFT can be expressed in terms of R Q-pt. DFT or Q R pt. DFTs (Cooley-Tukey algorithm). 43 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology