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
2018/9/12 1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 9 Computation of the Discrete Fourier Transform
9.0 Introduction
9.1 Direct Computation Of The Discrete Fourier Transform
9.2 decimation-in-time FFT Algorithms
9.3 decimation-in-frequency FFT Algorithms
9.4 practical considerations

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?
DFT has efficient FFT(Fast Fourier Transform) algorithms
that can be orders of magnitude more efficient than others.

4. 9.1 Direct Computation of the Discrete Fourier Transform
The DFT pair was given as
DFTcomputational 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 Direct Computation of the Discrete Fourier Transform
Most fast methods are based on Periodicity properties
Periodicity in n and k;
Complex conjugate symmetry 5 5

6. 9.1.1 Direct Evalutation of the Defination of DFT
Complexity in terms of real operations
4N2 real multiplications
N(4N-2) real additions : N (2(N-1) +2N ) 6 6

7. 9.1.1 Direct Evalutation of the Defination of DFT
Conjugate symmetry:
group terms in the summation for n and (N-n):
乘法次数减少一半 7

8. 9.1.1 Direct Evalutation of the Defination of DFT
Conjugate symmetry:
group terms in the summation for n and (N-n):
similarly,
乘法次数减少一半 8

9. 9.1.2 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,
yk[n] can be viewed as output of a filter to input x[n]
Filter impulse response :
X[k] is the output of the filter at time n=N.
DFT of x[n]
Goertzel Filter 9 9

10. 9.1.2 The Goertzel Algorithm
Goertzel Filter:
for causal LTI system
x[n]=0, n<0
Computational complexity for one X[k].
4N real multiplications; 4N real additions;
Slightly less efficient than the direct method(4N, 4N-2);
But it avoids computation and storage of
(N2 个)
(N 个) 10

11. Second Order Goertzel Filter
为便于分析
Goertzel Filter
x[n]=0, n<0
Multiply both numerator and denominator
for causal LTI system
实现极点因式
实现零点因式
每个K只需计算 1次 11 11

12. Second Order Goertzel Filter
除n=0时y[0]=x[0], y[n]对每个n有两个实数乘4个实数加
实现极点因式:
实现零点因式:
same
Complexity for one X[k] ( x[n], y[n] is complex).
Poles: 2N real multiplications and 4N real additions
Zeros: computed only once: 4 real ×and 4 real + .
total: 2(N+2) real ×and 4(N+1) real + .
乘 N /2
Complexity for all N X[k]
Each pole is used for X[k], X[N-k]
Approximately N2 real multiplications and 2N2 real additions.
More efficient than the direct method. 12 12

13. Second Order Goertzel Filter
If we only need to evaluate M X[k] , the total computation is proportional to NM,
Goertzel method is attractive when M is small.
The computation for FFT algorithm is Nlog2N.
if M < log2N, either Goertzel Algorithm or direct method is more efficient than FFT. 13 13

14. 9.2 Decimation-in-Time FFT Algorithms
Makes use of periodicity and symmetry of
Consider special case of N=2ν : integer ν power of 2
Separate x[n] into two sequences of length N/2
Even-numbered sequence ;
Odd-numbered sequences . 14 14

15. 9.2 decimation-in-time FFT Algorithms
Substitute variables n=2r for n even and n=2r+1 for odd
N/2-point DFT
g[r]
h[r]
G[k] and H[k] are the N/2-point DFTs of each subsequence: g[n]=x[2n] and h[n] =x[2n+1] 15 15

16. 9.2 decimation-in-time FFT Algorithms
G[k] and H[k] are the N/2-point DFTs of each subsequence: g[n]=x[2n] and h[n] =x[2n+1] 16 16

17. 2018年9月12日3时8分
8-point DFT using DIT
Figure 9.4

18. 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 when N>2 18 18

19. 9.2 decimation-in-time FFT Algorithms
Repeat same process , Divide N/2-point DFTs into
Two N/4-point DFTs
Combine outputs
N=8 19 19

20. 9.2 decimation-in-time FFT Algorithms
After two steps of decimation in time
Repeat until we’re left with two-point DFT’s 20 20

21. 9.2 decimation-in-time FFT Algorithms
flow graph for 8-point decimation in time
binary code:
N=8=23
3 stages:
N=2m
m=log2N
m级: 每级乘法和加法个数为N
Complexity:
Nlog2N complex multiplications and additions 21 21

22. Butterfly Computation
Flow graph of basic butterfly computation
We can implement each butterfly with one multiplication
Final complexity for decimation-in-time FFT
(N/2)log2N complex multiplications and Nlog2N additions 22 22

23. 9.2 decimation-in-time FFT Algorithms
DIT flow graph for 8-point decimation in time
N=8=23
generalized to
N=2m
m=log2N
m级: 每级乘法和加法个数为N
Complexity:
Nlog2N complex multiplications and Nlog2N additions 23 23

24. 9.2.1 Generalization and Programming the FFT
Final flow graph for 8-point decimation in time
N=8=23
generalized to
N=2m
m=log2N -1
m级: 每级乘法N/2个,加法N个 -1
the figure serves as proof,
and for FFT programming. -1
Complexity:
(Nlog2N)/2 complex multiplications and Nlog2N additions 24 24

25. 9.2.2 In-Place Computation同址运算
DIT flow graphs require two sets of registers: Input, output
or memory units
In-Place Computation needs one set of registers. 25 25

26. 9.2.2 In-Place Computation同址运算
Note the arrangement of the input indices
Bit reversed indexing（码位倒置） 26 26

27. normal order binary coding for position： 对应x[n]的序号 从上向下出现的顺序 000 001
2018年9月12日3时8分
normal order
binary coding for position：
对应x[n]的序号
从上向下出现的顺序
000
001
010
011
100
101
110
111
出现的顺序不是输入序号
Figure 9.13

28. cause of bit-reversed order
binary coding for position：
对应x[n]的序号
从上向下出现的顺序
是把二进制序号翻转了
序号的3次奇偶分解
000
001
010
011
100
101
110
111
Figure 9.13

29. 9.2.3 Alternative forms Bit reversed indexing（码位倒置）
Note the arrangement of the input indices
Bit reversed indexing（码位倒置） 29 29

30. input in normal order and output in bit-reversed order
Alternative forms
Figure 9.14
input in normal order and output in bit-reversed order
strongpoint：in-place computations
shortcoming：non-sequential access of elements of intermediate arrays.

31. strongpoint： Input, output same normal order
Figure 9.15
strongpoint： Input, output same normal order
shortcoming：not in-place computation; non-sequential access of elements of intermediate arrays.

32. Same structure, Input bit-reversed order , output normal order
Figure 9.16
Same structure, Input bit-reversed order , output normal order
shortcoming：not in-place computation
strongpoint: sequential access of data

33. 9.3 Decimation-In-Frequency FFT Algorithm
The DFT equation
Split the DFT equation into even and odd frequency indexes
N/2-point DFT 33 33

34. 9.3 Decimation-In-Frequency FFT Algorithm
The DFT equation
N/2-point DFT 34 34

35. DIF decomposition: N-point DFT to N/2-point DFT35 35

36. DIF decomposition: N-point DFT to N/2-point DFT2点
DFT
8=23
N=2m
m=log2N 36 36

37. 2-point DFT in the last stage of decimation-in-frequency decomposition37 37

38. (Nlog2N)/2 complex multiplications and Nlog2N additions
Final flow graph of decimation-in-frequency decomposition of 8-point DFT
N=8=23
generalized to
N=2m
m=log2N
m级: 每级乘法N/2个,加法N个
N/2=4-point DFT
N/4=2-point DFT
Complexity:
(Nlog2N)/2 complex multiplications and Nlog2N additions 42 42

39. 9.3 .1 In-Place Computation同址运算
DIF FFT
结构相同,乘数排列区别大
DIT FFT 43 43

40. 9.3 .1 In-Place Computation同址运算
DIF FFT
transpose
DIT FFT 44 44

41. 9.3 .2 Alternative forms transpose
decimation-in-frequecy Butterfly Computation
transpose
decimation-in-time Butterfly Computation 45 45

42. The DIF FFT is the transpose of the DIT FFT46 46

43. Alternative forms
DIF FFT
transpose
DIT FFT

44. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 9 HW
9.1, 9.6, 9.7,
Zhongguo Liu_Biomedical Engineering_Shandong Univ. 50
2018/9/12
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