6. Fast Fourier Transform

Transcription

1 x[] X[] x[] x[] x[6] X[] X[] X[3] x[] x[5] x[3] x[7] 3 X[] X[5] X[6] X[7]

2 A Historical Perspective The Cooley and Tukey Fast Fourier Transform (FFT) algorithm is a turning point to the computation of DFT Before that, DFT was never practical except running by some very expensive computers FFT gives nearly a factor of improvement in the computation speed over the traditional approach in most computing environments

3 Principal discoveries of efficient methods of computing the DFT are listed as follows: Researcher(s) Date Feature Cooley & Tukey 965 Any n composite integer I.J. Good 95 Any integer with relatively prime factors L.H. Thomas 9 Any integer with relatively prime factors Danielson 9 n K. Stumpff 939 n k, 3 n k C. Runge 93 n k J.D. Everett 6 A. Smith 6,, 6, 3 F. Carlini 3 C.F. Gauss 5 Any composite integer

4 Basic principle of FFT Divide and Conquer To break down a big problem to a number of smaller problems and tackle them individually eed to satisfy the following criterion cost( subproblems) + cost( overhead) < cost( original problem)

5 cost( subproblems) + cost( overhead) < cost( original problem) Big Problem Smaller Problem Smaller Problems 5 Overhead Smaller Problem Apply Divide and Conquer iteratively

6 Operation of FFT Length- DFT Length-/ DFT Length-/ DFTs Length-/ DFT 6 Overhead For each round, the size of the problem is divided by

7 Mathematical Derivation Recall the DFT of x[n], for n,,, X[ k] where n nk x[ n] nk e jπnk / for k,,, Separate the summation into two groups nk X [ k] x[ n] + x[ n] even n odd n nk 7 for k,,,

8 Example If DFT of x[n] is. k. k. k + x[] + x[] x[3] X[ k] x[] + e separate it into two groups with odd and even n 3. k X[ k] {. k. k + } + {. k 3. k x[] x[] x[] + x[3] } nk x[ n] + even n odd n x[ n] nk

9 9 Signal Processing Fundamentals Part I / ) ( / ] [ ] [ ] [ ] [ ] [ r k r r rk n odd nk even n nk r x r x n x n x k X for k,,, + + / / ] [ ] [ r rk k r rk r x r x

10 Since X[ k] rk / x[r] / + r e G[ k] + jπ rk / k rk / rk H[ k] k e jπrk /( / r x[r / ) + ] for k,,, rk / Length-/ DFT of data with even n Overhead Length-/ DFT of data with odd n

11 Example: x[] x[] x[] x[6] / point DFT G[] G[] G[] G[3] 3 X[] X[] X[] X[3] x[] x[3] x[5] x[7] / point DFT H[] H[] H[] H[3] X[] X[5] X[6] X[7]

12 How much it saves? Assume x[n] has data and each data is a complex number X[ k] n nk x[ n] for k,,, For each k, it needs complex multiplications and complex additions In overall, it needs complex multiplications and () complex additions

13 FFT converts an -point DFT to /-points DFTs k X[ k] G[ k] + H[ k] for k,,, (/) complex multiplications /(/) complex additions complex multiplications 3

14 DFT -stage FFT 6x/+ x/+ 6x/56+ x/ x/+ x/+ 3,x/99+ 5x/5+ 6,96x/,3+,x/,+ 6,3x/6,56+,3x/, ,536x/65,+ 33,x/3, ,x/6,63+ 3,5x/3,7+,,576x/,7,55+ 55,3x/5,+

15 Obviously < cost( subproblems) cost( original problem) + cost( overhead) e can re-apply the same approach to the decomposed / problems 5

16 Example: Apply to G[k] and H[k] x[] x[] x[] x[6] / point DFT / point DFT 3 3 X[] X[] X[] X[3] 6 x[] x[5] x[3] x[7] / point DFT / point DFT X[] X[5] X[6] X[7]

17 The complexity is further reduced The above decomposition is recursively performed until a length- DFT is computed A length- DFT can be implemented as a butterfly as follows: X[ k] n X[] X[] x[ n] x[] x[] nk + + x[] x[] x[] x[] X[] / e j π X[] 7

18 x[] x[] x[] x[6] 3 3 X[] X[] X[] X[3] x[] x[5] x[3] x[7] X[] X[5] X[6] X[7]

19 The complexity can be further reduced. For every butterfly, it can be converted as follows: r r ( r / ) + r r Since ( r+ / ) r / r jπ e r 9

20 x[] X[] x[] x[] x[6] X[] X[] X[3] x[] x[5] x[3] x[7] 3 X[] X[5] X[6] X[7] FFT Total complex multiplication: 5 Total complex addition: DFT Total complex multiplication: 6 Total complex addition: 56

21 Remarks: FFT of the form above is called decimation-in-time (DIT) FFT (or Cooley and Tukey FFT) In general, the complexity of DIT FFT is (/)log complex multiplications ( is considered as multiplication in this expression) log complex additions DIT FFT requires to be a power of If is not a power of, need zero padding to let be a power of before FFT More than a hundred of FFT algorithms after DIT FFT for different applications

22 Exercise. Show why the complexity of DIT FFT is given by (/)log complex multiplications ( is considered as multiplication in this expression) log complex additions. If 56, what are the complexities of DIT FFT and DFT?

23 Exercise (cont) 3. If x[n] has the values { }, use FFT to calculate its spectrum. 3

24 Solution. For a length- DFT, there will be log () stages. For each stage, there will be / butterflies. For each butterfly, there will complex multiplication and complex additions. Hence the total no. of complex multiplications is /log. And the number of additions is log

25 . If 56, DFT requires complex multiplications and ( - ) 56(55) 65 complex additions. By using FFT, it requires /log * complex multiplications and log 56* complex additions 5

26 3. If x[n] has the values { }, its spectrum can be calculated using FFT as follows: After st stage, the output is { 6 } After nd stage, the output is { -j +j 6 -j +j} After 3rd stage, the output is {36 -j j -j j.657 +j +j9.657}