INTRODUCTION TO THE FAST FOURIER TRANSFORM ALGORITHM

Transcription

1 Course Outline

2 Course Outline

3 INTRODUCTION TO THE FAST FOURIER TRANSFORM ALGORITHM

4 Introduction Fast Fourier Transforms have revolutionized digital signal processing What is the FFT? A collection of tricks that exploit the symmetry of the DFT calculation to make its execution much faster Speedup increases with DFT size

5 Introduction, continued Some dates: algorithm first described by Gauss algorithm rediscovered (not for the first time) by Cooley and Tukey In 1967, calculation of a 8192-point DFT on the top-of-the line IBM 7094 took. 30 minutes using conventional techniques 5 seconds using FFTs

6 Measures of computational efficiency Could consider Number of additions Number of multiplications Amount of memory required Scalability and regularity For the present discussion we ll focus most on number of multiplications as a measure of computational complexity More costly than additions for fixed-point processors Same cost as additions for floating-point processors, but number of operations is comparable

7 Computational Cost of Discrete-Time Filtering Computational Cost of Discrete-Time Filtering Convolution of an N-point input with an M-point unit sample response. Direct convolution: Number of multiplies MN y[n] = x[k]h[n k] k=

8 Computational Cost of Discrete-Time Filtering Convolution of an N-point input with an M-point Computational Cost of Discrete-Time Filtering unit sample response. Using transforms directly: N 1 X[k] = x[n]e j2πkn / N n=0 Computation of N-point DFTs requires N 2 multiplys Each convolution requires three DFTs of length N+M-1 plus an additional N+M-1 complex multiplys or 3(N + M 1) 2 + (N + M 1) N >> M O(N 2 ) For, for example, the computation is

9 The Cooley-Tukey decimation-in-time algorithm The Consider Cooley-Tukey the DFT algorithm decimation-in-time for an integer algorithm power of 2, N 1 X[k] = x[n]w nk N 1 N = x[n]e j2πnk / N ; W N = e j2π / N k=0 k=0 Create separate sums for even and odd values of n: Letting obtain X[k] = x[n]w 2k N + n even n odd x[n]w N 2k for n even and for n odd, we n = 2r n = 2r +1 ( N /2) 1 X[k] = x[2r]w 2rk ( N /2) 1 N + x[2r +1]W ( 2r+1)k N r=0 r=0

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12 The Cooley-Tukey decimation in time algorithm Splitting indices in time, we have obtained ( N /2) 1 X[k] = x[2r]w 2rk N + x[2r +1]W 2r+1 N r=0 ( N /2) 1 r=0 ( )k 2 But W N = e j2π2/n = e j2π /(N /2) = WN 2rk k /2 and W N WN = k WN WN /2 So (N/2) 1 (N/2) 1 X[k] = x[2r]w rk N /2 + W k N x[2r +1]W rk N /2 n=0 n=0 N/2-point DFT of x[2r] N/2-point DFT of x[2r+1] rk

13 Savings so far We have split the DFT computation into two halves: Savings so far X[k] = = N 1 x[n]w N nk k=0 (N/2) 1 x[2r]w N /2 n=0 Have we gained anything? Consider the nominal number of multiplications for N = = 64 2(4 2 ) + 8 = 40 rk + W N k (N/2) 1 x[2r +1]W N /2 n=0 Original form produces multiplications New form produces multiplications So we re already ahead.. Let s keep going!! rk

14 Signal flowgraph representation of 8-point DFT Signal flowgraph representation of 8-point DFT Recall that the DFT is now of the form The DFT in (partial) flowgraph notation:

15 The complete decomposition into 2-point DFTs

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17 The complete decomposition into 2-point DFTs

18 Now let s take a closer look at the 2-point DFT The expression for the 2-point DFT is: X[k] = 1 1 nk x[n]w 2 = j2πnk /2 x[n]e n=0 n=0 Evaluating for k = 0,1 we obtain X[0] = x[0]+ x[1] X[1] = x[0] + e j2π1/2 x[1] = x[0] x[1] which in signal flowgraph notation looks like... This topology is referred to as the basic butterfly

19 The complete 8-point decimation-in-time FFT

20 Number of multiplys for N-point FFTs Let N = 2 ν where ν = log 2 (N) (log 2 (N) columns)(n/2 butterflys/column)(2 mults/butterfly) or ~ N log 2 (N) multiplys

21 Computational complexity

22 Additional timesavers: reducing multiplications in the basic butterfly As we derived it, the basic butterfly is of the form W N r W N N /2 = 1 W N r+n /2 Since we can reduce computation by 2 by premultiplying by W r N 1 W N r 1

23 Bit reversal of the input Recall the first stages of the 8-point FFT: Consider the binary representation of the indices of the input: If these binary indices are time reversed, we get the binary sequence representing 0,1,2,3,4,5,6,7 Hence the indices of the FFT inputs are said to be in bit-reversed order

24 Some comments on bit reversal In the implementation of the FFT that we discussed, the input is bit reversed and the output is developed in natural order Some other implementations of the FFT have the input in natural order and the output bit reversed In some situations it is convenient to implement filtering applications by Use FFTs with input in natural order, output in bit-reversed order Multiply frequency coefficients together (in bit-reversed order) Use inverse FFTs with input in bit-reversed order, output in natural order Computing in this fashion means we never have to compute bit reversal explicitly

25 Decimation in Freq FFT

26 Alternate FFT structures We developed the basic decimation-in-time (DIT) FFT structure, but other forms are possible simply by rearranging the branches of the signal flowgraph Consider the rearranged signal flow diagrams on the following panels..

27 Alternate DIT FFT structures (continued) DIT structure with input bit-reversed, output natural (OSB 9.10):

28 Alternate DIT FFT structures (continued) DIT structure with input natural, output bit-reversed (OSB 9.14):

29 Alternate DIT FFT structures (continued) DIT structure with both input and output natural (OSB 9.15):

30 Alternate DIT FFT structures (continued) DIT structure with same structure for each stage (OSB 9.16):

31 Comments on alternate FFT structures A method to avoid bit-reversal in filtering operations is: Compute forward transform using natural input, bitreversed output (as in OSB 9.10) Multiply DFT coefficients of input and filter response (both in bit-reversed order) Compute inverse transform of product using bitreversed input and natural output (as in OSB 9/14) Latter two topologies (as in OSB 9.15 and 9.16) are now rarely used

32 Using FFTs for inverse DFTs We ve always been talking about forward DFTs in our discussion about FFTs. what about the inverse FFT? N 1 x[n] = N 1 kn X[k]W N ; X[k] = k=0 N 1 x[n]w N kn n=0 One way to modify FFT algorithm for the inverse DFT computation is: Replace W k N by W k N wherever it appears Multiply final output by 1/ N This method has the disadvantage that it requires modifying the internal code in the FFT subroutine

33 A better way to modify FFT code for inverse DFTs Taking the complex conjugate of both sides of the IDFT equation and multiplying by N: Nx *[n] = N 1 N 1 N 1 X *[k]w kn N ; or x[n] = 1 N X *[k]w kn N k=0 k=0 This suggests that we can modify the FFT algorithm for the inverse DFT computation by the following: Complex conjugate the input DFT coefficients Compute the forward FFT Complex conjugate the output of the FFT and multiply by 1/N This method has the advantage that the internal FFT code is undisturbed; it is widely used. *

34 Principle

35 Sinusoidal signals

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38 Windowing: resolution and leakage (I)

39 Windowing: resolution and leakage (II)

40 Windowing: resolution and leakage (III)

41 Windowing: resolution and leakage (IV)

42 Frequencies matched to sampling points

43 N.B: if Hamming window: Frequencies matched to sampling points

44 Not by increasing L, the DFT size: Increasing the resolution

45 Increasing the resolution(ii)

46 Increasing the resolution(iii)

47 Increasing the resolution(iv)

48 Increasing the resolution(v)

49 Increasing the resolution(vi)

50 Effect of window type Use of other windows can reduce the contamination between peaks:

51 Effect of window type (II)

52 Effect of window type (III) Kaiser window: prediction formulae for N and β from the values of main lobe width relative attenuation of side lobes

53 Summary We developed the structure of the basic decimation-in-time FFT Use of the FFT algorithm reduces the number of multiplys required to perform the DFT by a factor of more than 100 for 1024-point DFTs, with the advantage increasing with increasing DFT size