1. Chapter 8 The Discrete Fourier Transform
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)》
2019/5/4 1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.

2. Chapter 8 The Discrete Fourier Transform
8.0 Introduction
8.1 Representation of Periodic Sequence: the Discrete Fourier Series
8.2 Properties of the Discrete Fourier Series
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
8.6 Properties of the Discrete Fourier Transform
8.7 Linear Convolution using the Discrete Fourier Transform
8.8 the discrete cosine transform (DCT)

3. Filter Design Techniques
8.0 Introduction

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

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
Given a periodic sequence with period N so that
The Fourier series representation can be written as
Fourier series representation of continuous-time periodic signals require infinitely many complex exponentials
Not that for discrete-time periodic signals, since

7. 8.1 Representation of Periodic Sequence: the Discrete Fourier Series
Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series
No need

8. Discrete Fourier Series Pair
The Fourier series representation of a periodic sequence:
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
Problem 8.51, HW

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

15. N points n …… N points k …… 1 2 N-1 N N+1 N+2 -1 -2 -N+1 -N 1 2 N-1 N1 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

16. 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 [ ] å - = 1 ~ N n kn W x k X

17. …… …… 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 -11 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. 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 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

19. 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

20. magnitude
phase

21. magnitude
phase

22. 8.2 Properties of the Discrete Fourier Series
2019年5月4日9时51分
8.2 Properties of the Discrete Fourier Series
8.2.1 Linearity
two periodic sequence, both with period N:

23. 8.2 Properties of the Discrete Fourier Series
Shift of a sequence

24. 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
Symmetry Properties

25. 8.2.4 Symmetry
Problem 8.53, HW

26. 8.2.5 Periodic Convolution
and are two periodic sequences, each with period N and with discrete Fourier series and , if
then
Proof:

27. 8.2.5 Periodic Convolution The sum is over the finite interval
The value of in the interval repeat periodically for m outside of that interval

28. Example 8.4 Periodic Convolution

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) :

30. 8.3 The Fourier Transform of Periodic Signal
We can represent Periodic sequences as sums of complex exponentials: DFS
We can combine DFS and Fourier transform
Fourier transform of periodic sequences:
Periodic impulse train with values proportional to DFS coefficients

31. 8.3 The Fourier Transform of Periodic Signal
This is periodic with 2 since DFS is periodic with period N, and the impulses are spaced at integer multiples of 2/N.
The inverse transform can be written as

32. Ex. 8.5 determine the Fourier Transform of a periodic impulse train
the periodic impulse train 1 2 …… N -1 -2 -N
N points n
Solution:
The DFS was calculated previously to be
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

33. 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
The Fourier transform of the periodic sequence is

34. 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

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

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

37. 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:

38. Ex. 8.6 Relation between FS coefficients and Fourier transform of one period of Periodic Signal
Consider the sequence
and one period:
The Fourier transform
Solution:

39. Ex. 8.6 Relation between FS coefficients and Fourier transform of one period x[n] of
The DFS coefficients
The Fourier transform

40. 8.4 Sampling the Fourier Transform
x[n] Length M, may be ∞
Consider an aperiodic sequence with Fourier transform ,and assume that a sequence is obtained by sampling at frequency
Length M, N may not be the same
could be Fourier series coefficients of a periodic sequence with period N .

41. Sampling the Fourier Transform to recover x[n]1 2 …… N -1 -2 -N
N points ？

42. Sampling the Fourier Transform to recover x[n]1 2 …… N -1 -2 -N
N points
x[n]
= x[n]

43. Sampling the Fourier Transform to recover x[n]

44. 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
obtained through summing periodic replicas of original sequence x[n].
If the original sequence 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 original sequence x[n] can be recovered by
infinite shifted replicas

45. Discrete Fourier Transform or DFT
Given a finite-length sequence x[n], we can form a periodic sequence , which in turn can be represented by a DFS
Given a sequence of Fourier coefficients , we can find , and then obtain x[n] .

46. Discrete Fourier Transform or DFT
to represent finite length x[n] by FS :
When the Fourier series is used in this way to represent finite-length sequences, it is called the Discrete Fourier Transform or DFT.
It is not necessary to know the DTFT X(e jw) at all frequencies, to recover the discrete-time sequence x[n] in time domain.

47. 2019年5月4日9时51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.

48. Discrete Fourier Transform
For , the DFS is with period N
The Discrete Fourier Transform of is
DFT:
-∞<k<∞

49. Discrete Fourier Transform
DFS:
DFT:
Usually DFT pairs is written as:
Analysis equation:
Synthesis equation:

50. Discrete Fourier Transform pairs
Analysis equation
Synthesis equation

51. 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)

52. 四种傅立叶变换

53. Ex. 8.7 Calculate the DFT of a Rectangular Pulse
x[n] is of length 5
x[n]
Solution:
To form from x[n], we can consider x[n] of any length N greater than 5.
Let’s pick N=5
Calculate the DFS of

54. Ex. 8.7 The DFT of a Rectangular Pulse
2019年5月4日9时51分
Ex. 8.7 The DFT of a Rectangular Pulse
x[n]
If we consider x[n] of length 10, form
We get a different set of DFS coefficients
Still samples of the DTFT but in different places
Different N
Different

55. 8.6 Properties of the Discrete Fourier Transform
8.6.1 Linearity
If has length and has length ,

56. 8.6.2 Circular Shift of a Sequence

57. 8.6.2 Circular Shift of a Sequence--proof
linear shift
Recall:
Problem: Given,
Circular Shift ？
Proof:

58. 8.6.2 Circular Shift of a Sequence？

59. 8.6.2 Circular Shift of a Sequence

60. Ex. 8.8 Circular Shift of a Sequence
Figure 8.12

61. 8.6.3 Duality
Time domain
Frequency domain

62. Ex.8.9 The Duality Relationship for the DFT
2019年5月4日9时51分
Ex.8.9 The Duality Relationship for the DFT

63. 8.6.4 Symmetry Properties of x[n]
periodic conjugate symmetric components
periodic conjugate-antisymmetric components
of x[n]

64. 8.6.4 Symmetry Properties

65. xep[n] and xop[n] are not equivalent to xe[n] and xo[n] as in:
8.6.4 Symmetry Properties
xep[n] and xop[n] are not equivalent to xe[n] and xo[n] as in:
conjugate symmetric components
conjugate-antisymmetric components
It can be shown in Problem 56

66. 8.6.4 Symmetry Properties
Defination as before: so
compare

67. 8.6.4 Symmetry Properties

68. 8.6.4 Symmetry Properties

69. 8.6.5 Circular Convolution
For two finite-duration sequences and , both of length N, with DFTs and If
IDFT
Then
Circular Convolution

70. 8.6.5 Circular Convolution

71. 8.6.5 Circular Convolution

72. Ex. 8.10 find Circular Convolution of x2[n] and x1[n](Delayed Impulse Sequence).
Solution1：
N=5

73. Ex. 8.10 Circular Convolution with a Delayed Impulse Sequence
Solution2：

74. Example 8.11 Circular Convolution of Two Rectangular Pulses
Solution1：
DFT length
Circular convolution length
ǂ linear convolution

75. Ex. 8.11 Circular Convolution of Two Rectangular Pulses
Solution2：
DFT length
Circular convolution length
= linear convolution

76. 8.6.6 Summary of Properties of the Discrete Fourier Transform

77. 8.6.6 Summary of Properties of the Discrete Fourier Transform

78. 8.7 Linear Convolution using the Discrete Fourier Transform
FFT algorithms are available for computing the DFT of a finite-duration sequence.
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

79. 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.

80. 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

81. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
circular convolution corresponding to DFTs: ，
as linear convolution
, Whether they are same?
depends on the length of the DFT in relation to the length of and

82. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
For finite sequence
N sampling
The inverse DFT of is one period of :
If N≥length of x[n], then xp[n]= x[n]

83. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
DTFT of is
N sampling
Define a DFT:
The inverse DFT
of is ：
If N≥length of x3[n], then

84. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
From
And
Linear convolution:
The circular convolution of two-finite sequences is equivalent to linear convolution of the two sequences, followed by time aliasing as above.

85. 8.7.2 Circular Convolution as Linear Convolution with Aliasing
If has length L and has length P, then has maximum length
if N, the length of the DFTs, satisfies
The circular convolution corresponding to
is equal to linear convolution
corresponding to
DFT
DTFT

86. Ex. 8.12 Circular Convolution as Linear Convolution with Aliasing.
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
N=12
12 points circular convolution
= linear convolution

87. Which sequence values in L-point Circular Convolution (when Aliasing occurs )equal that of Linear Convolution?
Fig.8.19
Linear Convolution
Consider L-point circular convolution of xl[n] (length L ) and x2[n] (length P ), where P < L.
Fig.8.20
Circular Convolution
N点圆周卷积没有时间混叠的点

88. circular convolution “ wraps around ".
View the process of forming the circular convolution x3p[n] through linear convolution plus aliasing, 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].

89. 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.

90. 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 of at least (L+P-1) points.

91. Zero-Pading
The circular convolution can be achieved by multiplying the DFTs of and 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

92. Block Convolution
If the input signal is of indefinite duration, the input signal to be processed is segmented into sections of length L.
Each section can be convolved with the finite- length impulse response and output sections are fitted together in an appropriate way.
The processing of each section can then be implemented using the DFT.

93. Block Convolution overlap-add method
(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

94. overlap-add method (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
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

95. Circular Convolution as Linear Convolution with Aliasing

96. 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=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
圆周卷积中后L-P+1个点结果与线性卷积相等

97. 8.8 the discrete cosine transform (DCT)
The DFT is a class of finite-length transform representations of the form
, basis sequences is complex
Are there real basis sequences that would yield a real-valued transform ?

98. 8.8 the discrete cosine transform(DCT)
An orthogonal transform for real sequences is the discrete cosine transform (DCT).
The DCT is closely related to DFT and has become especially useful in signal- processing applications, particularly speech and image compression.

99. 8.8.1 Definitions of the DCT DFT DCT-1 DCT-2
just as the DFT involves an implicit assumption of periodicity, the DCT involves implicit assumptions of both periodicity and even symmetry.
symmetric and periodic extension of signal, then do DFS and get DCT by taking the dominant period。
DFT
DCT-1
DCT-2

100. 8.8.1 Definitions of the DCT
DCT-1
DCT-2
DCT-3
DCT-4

101. 8.8.2 Definitions of the DCT-1, DCT-2

102. 8.8 the discrete cosine transform(DCT)
other Definitions of DCT-1, DCT-2 , DCT-3 , DCT-4

103. other Definitions of DCT-1, DCT-2 , DCT-3 , DCT-4

104. relationship between 2N-poinsts DFT of extended sequence and N-points DCT of original sequence

105. 8.8.3 Relationship between the DFT and the DCT-1
one period of the periodic sequence 𝑥 1[n] defines the finite-length sequence
(2N-2)-point DFT of the (2N-2)-point sequence x1[n]:

106. 8.8.4 Relationship between the DFT and the DCT-2
one period of the periodic sequence 𝑥 2[n] defines the finite-length sequence
2N-point DFT of the 2N-point sequence x2[n]:

107. 8.8.4 Relationship between the DFT and the DCT-2

108. 8.8 the discrete cosine transform(DCT)
8.8.5 Energy Compaction Property of the DCT-2
Parseval's theorem for
DCT-1
DCT-2

109. 8.8.5 Energy Compaction Property of the DCT-2
Ex Energy Compaction in the DCT-2

110. Ex. 8.1.3 Energy Compaction in the DCT-2
(a) Real part of 32-point DFT (symetric)
(b) Imaginary part of 32-point DFT
(c) 32-point DCT-2 of the test signal

111. Ex. 8.1.3 Energy Compaction in the DCT-2
(a) Real part of 32-point DFT (symetric)
(b) Imaginary part of 32-point DFT

112. Ex. 8.1.3 Energy Compaction in the DCT-2
(c) 32-point DCT-2 of the test signal

113. Ex. 8.1.3 Energy Compaction in the DCT-2
Comparison of truncation errors for DFT and DCT-2

114. 8.8.6 Applications of the DCT
DCT is useful in a number of signal-processing applications, particularly speech and image compression.

115. 8.9 SUMMARY requirements：
definition, calculation and properties of DFS;
concepts of spectral sampling，time-domain periodic extension;
derivation of definition of DFT：DFS or spectral sampling;
properties of DFT：linearity、circular shift , symmetry, circular convolution、Parseval’s theory;
relationship between linear and circular convolution;
definition DCT and comparison with DFT.
key and difficulty：spectral sampling and properties of DFT

116. 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.
124
2019/5/4
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