1. Chapter 9 Computation of the Discrete Fourier Transform
Biomedical Signal processing
Chapter 9 Computation of the Discrete Fourier Transform
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
2019/1/11 1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 9 Computation of the Discrete Fourier Transform
9.0 Introduction
9.1 Efficient Computation of Discrete Fourier Transform
9.2 The Goertzel Algorithm
9.3 decimation-in-time FFT Algorithms
9.4 decimation-in-frequency FFT Algorithms
9.5 practical considerations（software realization)

3. 9.0 Introduction
Implement a convolution of two sequences by the following procedure:
1. Compute the N-point DFT and of the two sequence and
2. Compute for
3. Compute as the inverse DFT of
Why not convolve the two sequences directly?
There are efficient algorithms called Fast Fourier Transform (FFT) that can be orders of magnitude more efficient than others.

4. 9.1 Efficient Computation of Discrete Fourier Transform
The DFT pair was given as
Baseline for computational complexity:
Each DFT coefficient requires
N complex multiplications;
N-1 complex additions
All N DFT coefficients require
N2 complex multiplications;
N(N-1) complex additions 4 4

5. 9.1 Efficient Computation of Discrete Fourier Transform
Complexity in terms of real operations
4N2 real multiplications
2N(N-1) real additions (approximate 2N2) 5 5

6. 9.1 Efficient Computation of Discrete Fourier Transform
Most fast methods are based on Periodicity properties
Periodicity in n and k; Conjugate symmetry 6 6

7. 9.2 The Goertzel Algorithm
Makes use of the periodicity
Multiply DFT equation with this factor
Define
using x[n]=0 for n<0 and n>N-1
X[k] can be viewed as the output of a filter to the input x[n]
Impulse response of filter:
X[k] is the output of the filter at time n=N 7 7

8. 9.2 The Goertzel Algorithm
Goertzel Filter:
Computational complexity
4N real multiplications; 4N real additions
Slightly less efficient than the direct method
But it avoids computation and storage of 8 8

9. Second Order Goertzel Filter
Multiply both numerator and denominator 9 9

10. Second Order Goertzel Filter
Complexity for one DFT coefficient ( x(n) is complex sequence).
Poles: 2N real multiplications and 4N real additions
Zeros: Need to be implement only once:
4 real multiplications and 4 real additions
Complexity for all DFT coefficients
Each pole is used for two DFT coefficients
Approximately N2 real multiplications and 2N2 real additions 10 10

11. Second Order Goertzel Filter
If do not need to evaluate all N DFT coefficients
Goertzel Algorithm is more efficient than FFT if
less than M DFT coefficients are needed,M < log2N 11 11

12. 9.3 decimation-in-time FFT Algorithms
Makes use of both periodicity and symmetry
Consider special case of N an integer power of 2
Separate x[n] into two sequence of length N/2
Even indexed samples in the first sequence
Odd indexed samples in the other sequence 12 12

13. 9.3 decimation-in-time FFT Algorithms
Substitute variables n=2r for n even and n=2r+1 for odd
G[k] and H[k] are the N/2-point DFT’s of each subsequence 13 13

14. 9.3 decimation-in-time FFT Algorithms
G[k] and H[k] are the N/2-point DFT’s of each subsequence 14 14

15. 8-point DFT using decimation-in-time
Figure 9.3

16. computational complexity
Two N/2-point DFTs
2(N/2)2 complex multiplications
2(N/2)2 complex additions
Combining the DFT outputs
N complex multiplications
N complex additions
Total complexity
N2/2+N complex multiplications
N2/2+N complex additions
More efficient than direct DFT 16 16

17. 9.3 decimation-in-time FFT Algorithms
Repeat same process , Divide N/2-point DFTs into
Two N/4-point DFTs
Combine outputs
N=8 17 17

18. 9.3 decimation-in-time FFT Algorithms
After two steps of decimation in time
Repeat until we’re left with two-point DFT’s 18 18

19. 9.3 decimation-in-time FFT Algorithms
flow graph for 8-point decimation in time
Complexity:
Nlog2N complex multiplications and additions 19 19

20. Butterfly Computation
Flow graph constitutes of butterflies
We can implement each butterfly with one multiplication
Final complexity for decimation-in-time FFT
(N/2)log2N complex multiplications and additions 20 20

21. 9.3 decimation-in-time FFT Algorithms
Final flow graph for 8-point decimation in time
Complexity:
(Nlog2N)/2 complex multiplications and Nlog2N additions 21 21

22. 9.3.1 In-Place Computation同址运算
Decimation-in-time flow graphs require two sets of registers
Input and output for each stage 22 22

23. 9.3.1 In-Place Computation同址运算
Note the arrangement of the input indices
Bit reversed indexing（码位倒置） 23 23

24. cause of bit-reversed order
binary coding for position：
000
001
010
011
100
101
110
111
must padding 0 to
Figure 9.13

25. 9.3.2 Alternative forms Bit reversed indexing（码位倒置）
Note the arrangement of the input indices
Bit reversed indexing（码位倒置） 25 25

26. 9.3.2 Alternative forms Figure 9.14 strongpoint：in-place computations
shortcoming：non-sequential access of data
Figure 9.14

27. Figure 9.15
shortcoming：not in-place computation
non-sequential access of data

28. Figure 9.16
shortcoming：not in-place computation
strongpoint: sequential access of data

29. 9.3 decimation-in-time FFT Algorithms
Substitute variables n=2r for n even and n=2r+1 for odd
Review
G[k] and H[k] are the N/2-point DFT’s of each subsequence 29 29

30. 9.3.1 In-Place Computation同址运算
Bit reversed indexing（码位倒置） 30 30

31. 9.3.2 Alternative forms Figure 9.14 strongpoint：in-place computations
shortcoming：non-sequential access of data
Figure 9.14

32. 9.4 Decimation-In-Frequency FFT Algorithm
The DFT equation
Split the DFT equation into even and odd frequency indexes
Substitute variables 32 32

33. 9.4 Decimation-In-Frequency FFT Algorithm
The DFT equation 33 33

34. decimation-in-frequency decomposition of an N-point DFT to N/2-point DFT34 34

35. decimation-in-frequency decomposition of an 8-point DFT to four 2-point DFT35 35

36. 2-point DFT 36 36

41. Final flow graph for 8-point DFT decimation in frequency41 41

42. 9.4.1 In-Place Computation同址运算
DIF FFT
DIT FFT 42 42

43. 9.4.1 In-Place Computation同址运算
DIF FFT
DIT FFT 43 43

44. 9.4.2 Alternative forms decimation-in-frequecy Butterfly Computation
decimation-in-time Butterfly Computation 44 44

45. The DIF FFT is the transpose of the DIT FFT45 45

46. Alternative forms
DIF FFT
DIT FFT

47. Alternative forms
DIF FFT
DIT FFT

48. Figure 9.24 erratum

49. Alternative forms
DIF FFT
DIT FFT

50. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 9 HW
9.1, 9.2, 9.3,
Zhongguo Liu_Biomedical Engineering_Shandong Univ. 50
2019/1/11
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