1. Chapter 8 The Discrete Fourier Transform
2018年12月3日1时44分
Biomedical Signal processing
Chapter 8 The Discrete Fourier Transform
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
山东省精品课程《生物医学信号处理(双语)》
2018/12/3 1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 8 The Discrete Fourier Transform
2018年12月3日1时44分
Chapter 8 The Discrete Fourier Transform
8.0 Introduction
8.1 Representation of Periodic Sequence: the Discrete Fourier Series (DFS)
8.2 Properties of the DFS
8.3 The Fourier Transform of Periodic Signal
8.4 Sampling the Fourier Transform
8.5 Fourier Representation of Finite-Duration Sequence: the Discrete Fourier Transform(DFT)
8.6 Properties of the DFT
8.7 Linear Convolution using the DFT
8.8 the discrete cosine transform (DCT)

3. Chapter 8 The Discrete Fourier Transform
8.0 Introduction

4. 8.0 Introduction
Discrete Fourier Transform (DFT) is Transform of finite duration sequence.
DFT corresponds to samples, equally spaced in frequency, of the Discrete-time Fourier transform (DTFT) of the signal.
DFT is a sequence rather than a function of a continuous variable ω.

5. 8.0 Introduction
Derivation and interpretation of DFT is based on relationship between periodic sequence and finite-length sequences：
The Fourier series representation of the periodic sequence corresponds to the DFT of the finite-length sequence.

6. 8.1 Representation of Periodic Sequence: the Discrete Fourier Series
The Fourier series representation can be written as
Fourier series representation of continuous-time periodic signals require infinitely many complex exponentials,
for discrete-time periodic signals:

7. 8.1 Representation of Periodic Sequence: the Discrete Fourier Series
Due to periodicity,we only need N complex exponentials

8. Discrete Fourier Series Pair
The Fourier series:
To obtain the Fourier series coefficients we multiply both sides by for 0nN-1 and then sum both the sides , we obtain

9. Discrete Fourier Series Pair
2018年12月3日1时44分
Discrete Fourier Series Pair
Problem 8.51
orthogonality of the complex exponentials.
r→k
Periodic
DFS
coefficients
The Discrete Fourier Series

10. 8.1 Representation of Periodic Sequence: the Discrete Fourier Series
a periodic sequence with period N,
The Discrete Fourier Series:
Synthesis equation
Coefficients:
Analysis equation

11. 8.1 Representation of Periodic Sequence: the Discrete Fourier Series
The sequence is periodic with period N

12. Discrete Fourier Series (DFS)
Let
Analysis equation:
Synthesis equation:

13. Ex. 8.1 determine the DFS of a impulse train
Consider the periodic impulse train n 1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points
Solution:

14. Ex. 8.1 DFS of a impulse train1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points 1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points k
orthogonality

15. Example 8.2 Duality in the Discrete Fourier Series
If the Discrete Fourier Series coefficients is the periodic impulse train, determine the signal. 1 2 …… N -1 -2 -N
N points
Solution:
DFS

16. Example 8.2 Duality in the Discrete Fourier Series1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points k 1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points n

17. Duality in Discrete Fourier Series1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points k 1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points
周期为N 1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points k 1 2 ……
N-1 N
N+1
N+2 -1 -2
-N+1 -N
N points n

18. Periodic sequence with period N=10
Example 8.3 The Discrete Fourier Series of a Periodic Rectangular Pulse Train
Periodic sequence with period N=10 1
Solution:

19. magnitud of the DFS magnitude phase x denotes indeterminate phase

20. magnitud of the DFS Ex. 8.6 magnitude phase 一个周期的DTFT: sampling
2018年12月3日1时44分
magnitud of the DFS
一个周期的DTFT:
Ex. 8.6
magnitude
sampling
phase

21. 8.2 Properties of the Discrete Fourier Series
2018年12月3日1时44分
8.2 Properties of the Discrete Fourier Series
8.2.1 Linearity
For two periodic sequence, both with period N:

22. 8.2 Properties of the Discrete Fourier Series
Shift of a sequence if
then
and
shift in the Fourier coefficients by an integer l .

23. 8.2 Properties of the Discrete Fourier Series
Duality 1 2 ……
N-1 n 1 2 ……
N-1 n 1 2 ……
N-1 k 1 2 ……
N-1 k N

24. 8.2 Properties of the Discrete Fourier Series
Duality
Proof：
according
defination
interchang the roles of n and k
N×time inverse
defination
Symmetry Properties

25. 8.2.4 Symmetry
Problem 8.53, HW

26. 8.2.5 Periodic Convolution If then Proof:
For two periodic sequence, both with period N: If
then
Proof:

27. 8.2.5 Periodic Convolution
major differences between periodic convolutions and aperiodic convolutions:
The sum is over the finite interval
The value of in the interval repeat periodically for m outside of the interval.
duality

28. Example 8.4 Periodic Convolution
,再周期拓展 类推

29. 8.3 The Fourier Transform (DTFT) of discrete-time Periodic Signal
Periodic sequences are neither absolutely summable nor square summable,
hence they don’t have a strict Fourier Transform (DTFT) : 定义
r是整数
满足反变换定义

30. 8.3 The Fourier Transform of Periodic Signal
2018年12月3日1时44分
8.3 The Fourier Transform of Periodic Signal
We can represent Periodic sequences as sums of complex exponentials: DFS
DFS can be incorporated within framework of DTFT.
DTFT of periodic sequences: Periodic impulse train with values proportional to DFS coefficients
Periodic, N FT
DTFT
proof
next
IDTFT ?

31. 8.3 The Fourier Transform of Periodic Signal
2018年12月3日1时44分
8.3 The Fourier Transform of Periodic Signal
DTFT of periodic sequences: Periodic impulse train with values proportional to DFS coefficients ?
r是顺序整数时
k'是顺序整数

32. 8.3 The Fourier Transform of Periodic Signal
是周期为N的脉冲串函数, 脉冲面积为周期信号离散傅里叶级数的系数

33. 8.3 The Fourier Transform of Periodic Signal
The inverse transform can be written as

34. Ex. 8.5 determine the Fourier Transform of a periodic impulse train:1 2 …… N -1 -2 -N
N points
Sol: n
The DFS coefficients was calculated previously :
N points K 1 2 ……
N-1 N -1 -2 -N
Therefore the Fourier transform is
N points ω 1 2 ……
N-1 N -1 -2 -N

35. Relation between Finite-length and Periodic Signals1 2 N …… -1 -2 -N
Consider finite length signal x[n] spanning from 0 to N-1
Convolve with periodic impulse train
有限长度信号的周期拓展
The Fourier transform of the periodic sequence is
是周期为N的脉冲串函数

36. Relation between Finite-length and Periodic Signals1 2 N …… -1 -2 -N
The Fourier transform of the periodic sequence is
N points ω 1 2 ……
N-1 N -1 -2 -N
是周期为N的脉冲串函数, 脉冲面积为有限长度信号DTFT的采样

37. Relation between Finite-length and Periodic Signals
DTFT
DTFT
This implies that
DFS coefficients of a periodic signal
can be thought as equally spaced samples of
the Fourier transform of one period of

38. Relation between Finite-length and Periodic Signals——verification:
If is periodic with period N, the DFS are
(1)
If is one period of , i.e.
then
(2)
compare (1) and (2),we get:

39. the Periodic sequence: and one period:
Ex. 8.6 Relation between DFS coefficients and Fourier transform of one period of Periodic Signal
Verify the equality.
N=10
the Periodic sequence:
and one period:
The Fourier transform
Solution:

40. Ex. 8.6 Relation between DFS coefficients and Fourier transform of one period x[n] of
2018年12月3日1时44分
N=10
The DFS coefficients
The Fourier transform

41. 8.4 Sampling the Fourier Transform
an aperiodic sequence
Length M, may be ∞
Relation? same?
assume a sequence is obtained by sampling at frequency
period N
M, N may not be the same
could be Fourier series coefficients of a periodic sequence with period N .
1 period

42. Sampling the Fourier Transform to recover x[n]?？ N≥
DFS M
频域(时域)取样，时域(频域) 作周期延拓

43. Sampling the Fourier Transform to recover x[n]?
2018年12月3日1时44分
Sampling the Fourier Transform to recover x[n]? 1 2 …… N -1 -2 -N
N points
x[n] 1 2 …… N -1 -2 -N
= x[n] ] (N=12)
≠ x[n] (N=7)

44. Sampling the Fourier Transform to recover x[n]
2018年12月3日1时44分
Sampling the Fourier Transform to recover x[n]
Samples of DTFT X(e jw) of an aperiodic sequence x[n],
are thought of as DFS coefficients of a periodic sequence , N
obtained through summing periodic replicas of original sequence x[n].
每隔N点
If x[n] is of finite length M, and we take sufficient number N (N≥M) of samples of its DTFT X(e j w),
then X(e jw) can be recovered from these samples,
and the x[n] can be recovered by

45. derivation of Discrete Fourier Transform or DFT
2018年12月3日1时44分
derivation of Discrete Fourier Transform or DFT
relation between and in recovering x[n].
一个周期采样,再频域周期延拓
sampling
length M
length N
时域周期N延拓
It is no necessary to know the DTFT X(e jw) at all frequencies, to recover x[n] .
只需一个周期采样
When the Fourier Series is used in this way to represent finite-length sequences, it is called the Discrete Fourier Transform or DFT.

46. 2018年12月3日1时44分
8.5 Fourier Representation of Finite-Duration Sequence: Discrete Fourier Transform
Consider a finite-length sequence of length N samples such that outside the range
To each finite-length sequence of length N, we can associate a period sequence
-∞<n<∞;
notation,用有限长度序列x[m](0<m<N)表示
；((n))N 表示n以N为模.

47. Discrete Fourier Transform
用有限长度序列表示周期序列
DFS coefficients of is with period N.
-∞<k<∞
DFT:
有限长度N
It is called Discrete Fourier Transform of

48. Discrete Fourier Transform
2018年12月3日1时44分
Discrete Fourier Transform
DFS:
DFT:
Usually DFT is written as:
Analysis equation:
Synthesis equation:
Inverse DFT:
IDFT:

49. Discrete Fourier Transform pairs
Analysis equation
Synthesis equation

50. Recall different transforms between Time-Frequency domains
Fourier transform (FT)
continuous
Fourier series (FS)
periodic
discrete
Discrete-time Fourier transform (DTFT)
Discrete Fourier series (DFS)
discrete,
Discrete Fourier transform (DFT)
finite

51. 四种傅立叶变换
DFT is one period
of DFS

52. Ex. 8.7 Calculate the DFT of a Rectangular Pulse
DFS→DFT
x[n]
Solution:
DFT eqution
To form from x[n], x[n] can be seen as a finite-duration sequence of length N ≥ 5.
Let’s pick N=5
Calculate the DFS of

53. Ex. 8.7 The DFT of a Rectangular Pulse
2018年12月3日1时44分
Ex. 8.7 The DFT of a Rectangular Pulse
x[n]
If we consider x[n] of length 10(N), form
We get a different set of DFS coefficients
It’s still samples of , but in different places.
Different N results in
Different

54. 8.6 Properties of the Discrete Fourier Transform
8.6.1 Linearity
N-point DFT
If has length and has length ,
for DFT to be meaningful,

55. 8.6.2 Circular Shift of a Sequence
Ex. 8.8 ？
Circular Shift

56. Ex. 8.8 Circular Shift of a Sequence？
circular shift
Figure 8.12

57. 8.6.3 Duality 1 period 1 period DFT 1 period 用有限长度序列表示
周期序列
Time domain
Frequency domain
1 period
1 period
DFT

58. Ex.8.9 The Duality Relationship for the DFT
2018年12月3日1时44分
Ex.8.9 The Duality Relationship for the DFT

59. Ex.8.9 The Duality Relationship for the DFT
2018年12月3日1时44分
Ex.8.9 The Duality Relationship for the DFT

60. 8.6.4 Symmetry Properties periodic conjugate symmetric components
of x[n]
periodic conjugate-antisymmetric components
of x[n]

61. 8.6.4 Symmetry Properties

62. 8.6.5 Circular Convolution
For two finite-duration sequences and , both of length N, with DFTs and If
IDFT
Then
Circular Convolution

63. 8.6.5 Circular Convolution

64. 8.6.5 Circular Convolution

65. Ex. 8.10 find Circular Convolution of x2[n] with x1[n](Delayed Impulse Sequence).或者
N=5
Solution1： 类推

66. Ex. 8.10 Circular Convolution with a Delayed Impulse Sequence
2018年12月3日1时44分
Ex Circular Convolution with a Delayed Impulse Sequence
Solution2：
IDFT

67. Example 8.11 Circular Convolution of Two Rectangular Pulses
DFT length:
Solution: case1,
DFT length
Circular convolution length 或者
ǂ linear convolution

68. Ex. 8.11 Circular Convolution of Two Rectangular Pulses
Solution:case 2,
DFT length
Circular convolution length 或者
= linear convolution
用FFT算法可使计算量减小

69. 8.6.6 Summary of Properties of the Discrete Fourier Transform

70. 8.7 Linear Convolution using the Discrete Fourier Transform
FFT algorithms are available for computing the DFT of a finite-duration sequence.
It’s computationally efficient to do convolution of two sequences by the procedure:
1. Compute the N-point DFT and
of the two sequence and ;
2. Compute for ;
3. Compute by the inverse
DFT of
Result: circular convolution

71. 8.7 Linear Convolution using the Discrete Fourier Transform
In most applications, we are interested in implementing a linear convolution of two sequence.
To obtain a linear convolution, we will discuss the relationship between linear convolution and circular convolution.

72. 8.7.1 Linear Convolution of Two Finite-Length SequencesP
length
8.7.1 Linear Convolution of Two Finite-Length Sequences P
length L
for
is maximum length of

73. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
Is circular convolution corresponding to DFT: ，
same as linear convolution ? ?
This depends on the length N of the DFT in relation to the length of and

74. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
For finite sequence
周期延拓
1 period
N sampling
1 period
IDFT
DFT
The inverse DFT of is one period of :
If N≥length of x[n], then
If N≥length of x[n],
xp[n]= x[n] ？

75. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
可能混叠
周期延拓
DTFT
DFS
N sampling
1 period
N 点
DFT
DFT
IDFT ？
If N≥length of x3[n], then

76. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
From
DFT
And ？
Linear convolution:
since
The circular convolution of two-finite sequences is equivalent to linear convolution of the two sequences, followed by time aliasing as above.
If N < length of x3[n]

77. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
If , has length L and P, then linear convolution has maximum length
If N, the length of the DFTs, satisfies
then circular convolution corresponding to
is equal to linear convolution
corresponding to
DFT
DTFT
N sampling
1 period

78. 6 points shift right of the linear convolution
Ex Analyse Circular Convolution as Linear Convolution with Aliasing.
linear convolution
Solution:
N=6,12
2L-1
DFT
6 points shift right of the linear convolution
N=6
6 points shift left of the linear convolution
6 points circular convolution= linear convolution with aliasing
N=6
12 points circular convolution
= linear convolution,
N=12
no aliasing 12

79. 2018年12月3日1时44分
Which sequence values in L-point Circular Convolution (with Aliasing ) is equal that of Linear Convolution?
Fig.8.19
Linear Convolution
Consider L-point circular convolution of xl[n] (length L ) with x2[n] (length P ), where P < L.
Fig.8.20
Circular Convolution
with Aliasing
equal to Linear Convolution

80. circular convolution “ wraps around ".
View the process of forming the circular convolution x3p[n] through linear convolution plus aliasing,
L+ P -1
as taking the (P - 1) values of x3[n] from n=L to n=L+P -2 and adding them to the first (P - 1) values of x3[n].

81. 8.7.3 Implementing Linear Time-Invariant Systems Using the DFT
Linear time-invariant systems can be implemented by linear convolution
Linear convolution can be obtained from the circular convolution
So, circular convolution can be used to implement linear time-invariant systems
if length N is big enough

82. Zero-Pading
Consider an L-point input sequence and a P-point impulse response
The linear convolution of these two sequence has finite duration with length (L+P-1 ).
For the circular convolution and linear convolution to be identical, the circular convolution must have a length N of at least (L+P-1) points.

83. Zero-Pading The circular convolution
can be achieved by multiplying the DFTs of and
IDFT L P N
Since the length of the linear convolution is (L+P-1) points, the DFTs that we compute must also be of at least that length,
i.e., both and must augmented with sequence values of zero.
The process is called Zero-Pading (补零)。

84. 2018年12月3日1时44分
Block Convolution
If the input signal is of indefinite duration, the input signal to be processed is segmented into sections of length L. L 3L 2L
Each section can be convolved with the finite- length impulse response
and output sections are fitted together in an appropriate way.
衔接，拼接
The processing of each section can then be implemented using the DFT.

85. Block Convolution overlap-add method L 3L 2L 做L+P-1点FFT2L
overlap-add method
做L+P-1点FFT
(1)Segment into sections of length L;
(2) fill 0 into and some section of , do(至少) L+P-1 points FFT ;
(3) calculate

86. overlap-add method (1)Segment into sections of length L; 做L+P-1点FFT
(2) fill 0 into and some section of , then do L+P-1 points FFT ;
(3) calculate
L+P-1 points
(4)add the points n=0…P-2 in yr[n] to the last P-1 points in the former section yr-1[n]，the output for this section is the points n=0…L-1

87. overlap-add method 详细过程 (1)Segment into sections of length L; L=16
(2) fill 0 into and some section of , then do L+P-1 points FFT ;
(3) calculate
L+P-1 points
P-1 points
P-1 points
(4)add the points n=0…P-2 in yr[n] to the last P-1 points in the former section yr-1[n]，the output for this section is the points n=0…L-1
不等 于原 卷积
L-(P-1) points
L-(P-1) points
即原 卷积

88. Circular Convolution as Linear Convolution with Aliasing is related to
overlap-save method
of Block Convolution
Fig.8.20
L-Point Circular Convolution 混叠

89. overlap-discard重叠丢弃 overlap-save method input Block Convolution
2018年12月3日1时44分
Block Convolution
overlap-save method
input
(1) segment into sections of length L, overlap P-1 points;
(2) fill 0 into and some section of , then do L points FFT L
L=25
P-1
points
(3) calculate
P-1
points
P-1
(4) the output for this section is L-(P-1) points of y[n]
n=P-1,…L-1
points
L-(P-1) points
混叠丢弃 丢弃
圆周卷积中后L-(P-1)个点结果与线性卷积相等

90. 8.9 SUMMARY requirements：
definition, calculation and properties of DFS;
concepts of spectral sampling，time-domain periodic extension of finite-length se­quences;
derivation of definition of DFT：DFS or spectral sampling;
properties of DFT：linearity、circular shift , circular convolution;
relationship between linear and circular convolution;
definition DCT and comparison with DFT.
key and difficulty：spectral sampling and properties of DFT

91. Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 8 HW
8.4, 8.7, 8.10, 8.14, 8.17
8.51,8.52, 8.53,
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
112
2018/12/3
返 回
上一页
下一页