2. 数字信号处理
离散时间线性系统 2
Teaching Content
Introduction
绪
论
Sampling in Time
Domain
Discrete time
signal analysis
method
Discrete time
system analysis
method
Time domain
analysis of
discrete-time
systems
Transformation domain
(frequency domain and
z domain) analysis of
discrete time systems
3. 数字信号处理
离散时间线性系统 3
Teaching Content
Introduction
绪
论
Sampling in
Time Domain
Frequency domain
analysis of
discrete time
Systems
Time domain
analysis of discrete
time signals
Time domain
analysis of discrete
time Systems
Discrete Fourier Series
（DFS）
Discrete Time Fourier
Transform （DTFT）
Discrete Fourier
Transform （DFT）
Fast Fourier Transform
（FFT）
7. 数字信号处理
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Classification of signals
Periodic signal Aperiodic signal
Continuous
time signal
Discrete
time signal
( )
x t
t
)
(n
x
n
)
(n
x
n
t
n
xa
( )
x t
8. 数字信号处理
离散时间线性系统 8
Method for finding the spectrum according to
the time domain signal
Periodic signal Aperiodic signal
Continuous
time signal
Discrete time
signal
0
( ) ( )
FS
x t X jk
( ) ( )
FT
x t X j
x t
t
n
xa
0
( )
X jk
k
( )
x t
t
( )
X j
)
(n
x
n
？
)
(n
x
n
？
• How to get the spectrum based on
the time domain signal?
• Spectrum characteristics?
• How to get the spectrum based on
the time domain signal?
• Spectrum characteristics?
9. 数字信号处理
离散时间线性系统 9
Explain the idea
First, we discuss how to obtain the spectrum of
a periodic discrete time signal.
Based on the relationship between periodic
discrete time signal and aperiodic discrete time
signal, analyze the spectrum of aperiodic
discrete time signal.
This part focus on the method to calculate the
spectrum of periodic discrete time signal.
10. 数字信号处理
离散时间线性系统 10
1 1 2 2
( ) ( ) ( )
y n a n a n
1 1 2 2
( ) ( ) ( )
x n a n a n
input ：
Output :
( ) ( ) ( )
y n x n h n
( ) ( )
m n n m
Discrete Fourier Series （DFS）
( )
x n ( )
y n
LTI System
T[●]
?
( )=
m n
Building modules:
11. 数字信号处理
离散时间线性系统 11
Discrete Fourier Series （DFS）
Can a discrete periodic signal be
represented by a linear combination
of complex exponential functions?
( ) ( )
x n x n rN
0
0
( ) ( )
( ) ( )
a a
jk t
a
k
x t x t kT
x t A k e
0
jk t
e
Continuous periodic signal can be
expanded into Fourier series form
0
( ) jk t
k t e
Continuous time
period signal:
Period： 0
T
Building modules：
Base frequency：
K subharmonic
component:
0 0
2 /T
Discrete time
period signal:
is arbitrary integer
r
is period
N
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Discrete Fourier Series （DFS）
0
( ) jk n
k n e
0, 1, 2,
k
A periodic sequence with a period of N:
Its base wave frequencies are:
The base wave is represented by a complex index:
The kth harmonics are:
2
1
j n
N
e e
2
j nk
N
k
e e
0
2
N
0
( ) jk t
k t e
0 0
( ) jk Tn jk n
k n e e
t nT
Time domain sampling
13. 数字信号处理
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Discrete Fourier Series （DFS）
Discrete periodic signal can
be represented by a linear
combination of complex
exponential functions
( ) ( )
x n x n rN
0
0
( ) ( )
( ) ( )
a a
jk t
a
k
x t x t kT
x t A k e
Continuous periodic signal can
be expanded into Fourier series
form
Continuous time
period signal:
Discrete time
period signal:
0
( ) jk t
k t e
0 0
2
( )
jk n
jk Tn jk n N
k n e e e
t nT
Time domain sampling
2
( )
jk n
N
k n e
2
1
( ) ( )
jk n
N
k
x n X k e
N
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Discrete Fourier Series （DFS）
2 2
30 30
1 1
( ) ( )
j n j n
N
N
n e n e
The kth harmonic is
still a sequence of N-
period.
Signal sets only N
signals are not the
same
( ) ( )
k k rN
n n
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( ) ( )
x n x n rN
2
( )
jk n
N
k n e
0, 1, 2,
k
linear combination
represents a general periodic
sequence
( )
k n
Discrete-time Fourier
series (DFS)：
Discrete period Sequence
2
1
( ) ( )
jk n
N
k
x n X k e
N
2
1
0
1
( ) ( )
N jk n
N
k
x n X k e
N
Signal sets only N
signals are not the
same
( ) ( )
k k rN
n n
Discrete Fourier Series （DFS）
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Discrete Fourier Series （DFS）
2
1
0
1
( ) ( )
N jk n
N
k
x n X k e
N
Multiply the coefficient of 1/N for the convenience of the calculation below;
is coefficient of k-th harmonic
( )
X k
?
( )
X k
Discrete time Fourier series:
2 2
1 1 1 ( )
0 0 0
1
( ) ( )
N N N
j nr j k r n
N N
n n k
x n e X k e
N
2
1 1 ( )
0 0
1
( )
N N j k r n
N
k n
X k e
N
Step 1: Multiply the two sides of the upper formula by
Step 2: Sum from n=0 to N-1 to get:
2
j nr
N
e
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Orthogonity by a complex exponential sequence:
2
1
0
( ) ( )
N j nr
N
n
x n e X r
2
2
2
1
( ) ( )
0
( )
1
0
1
N
N
N
N
j k r n j k r N
n
j k r
N k r
e e
k r
e
2 2
1 1 1 ( )
0 0 0
1
( ) ( )
N N N
j nr j k r n
N N
n k n
x n e X k e
N
k r
Discrete Fourier Series （DFS）
18. 数字信号处理
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Discrete Fourier Series （DFS）
2
1
0
1
( ) ( )
N jk n
N
k
x n X k e
N
( )
X k
?
( )
X k
Discrete time Fourier series:
2
1
0
( ) ( )
N j nr
N
n
x n e X r
2
1
0
( ) [ ( )] ( )
N j nk
N
n
X k DFS x n x n e
Multiply the coefficient of 1/N for the convenience of the calculation below;
is coefficient of k-th harmonic
19. 数字信号处理
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The DFS positive transformation of the periodic sequence:
2
1 1
0 0
( ) [ ( )] ( ) ( )
N N
j nk
nk
N
N
n n
X k DFS x n x n e x n W
2
1 1
0 0
1 1
( ) [ ( )] ( ) ( )
N N
j nk
nk
N
N
k k
x n IDFS X k X k e X k W
N N
2
j
N
N
W e
We have:
Discrete Fourier Series （DFS）
k is a discrete frequency variable; n is a discrete time variable
The DFS inverse transformation of the periodic sequence:
0,1, 1
k N
0,1, 1
n N
( )
X k is called the amplitude characteristic of spectrum ( )
X k
arg ( )
X k
is called the phase characteristic of spectrum ( )
X k
20. 数字信号处理
离散时间线性系统 20
Discrete Fourier Series （DFS）
2 2
1 1
( )
0 0
( ) ( ) ( ) ( )
N N
j k mN n j kn
N N
n n
X k mN x n e x n e X k
2
1
0
1
( ) ( )
N jk n
N
k
x n X k e
N
• The spectrum of the periodic sequence with a period of N has only N independent harmonic
components, and the base rate
• Kth harmonic frequency is ， represents kth harmonic
component
• Spectrum is discrete, and the base wave frequency is the interval between
two discrete spectral lines adjacent
• The spectrum of a periodic sequence is a discrete periodic sequence with a period of N
2 / N
2 /
k N
2 /
( ) ( )
j k N
X k X e
0 2 / N
( )
x nT x n
n
0 1 2 3 4 N-1N
( )
X k
0 1 2 3 4 N-1 N k
21. 数字信号处理
离散时间线性系统 21
Example
1
0
( ) ( )
N
nk
N
n
X k x n W
7
8
0
( ) nk
n
x n W
2 2 2
2 3
8 8 8
1
j k j k j k
e e e
3
8
0
nk
n
W
(0) 4 (1) 1 2 1 (2) 0 (3) 1 2 1
(4) 0 (5) 1 2 1 (6) 0 (7) 1 2 1
X X j X X j
X X j X X j
4
Problem: Given sequence extend to a periodic
sequence , the period is
( ) ( ), ( )
8 Find the DFS for
, ( ).
x n R n x n
N
x(n x
) n
：
22. 数字信号处理
离散时间线性系统 22
2
2
2
2
(1) 1 2 1
(0) 4
(2) 0
(3) 1 2 1
(5) 1 2 1
(4) 0
(6) 0
(7) 1 2 1
X
X
X
X
X
X
X
X
Example
4
Problem: Given sequence extend to a periodic
sequence , the period is
( ) ( ), ( )
8 Find the DFS for
, ( ).
x n R n x n
N
x(n x
) n
：
23. 数字信号处理
离散时间线性系统 23
arg (1) arctan 2 1
arg (0) 0
arg (2) 0 arg (1) arctan 2 1
arg (1) arctan 2 1
arg (4) 0
arg (6) 0 arg (1) arctan 2 1
X
X
X X
X
X
X X
Example
4
Problem: Given sequence extend to a periodic
sequence , the period is
( ) ( ), ( )
8 Find the DFS for
, ( ).
x n R n x n
N
x(n x
) n
：
