Biomedical Signal processing
Chapter 4 Sampling of ContinuousTime Signals
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)》
http://course.sdu.edu.cn/bdsp.html
1
2017/7/28
1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 4: Sampling of
Continuous-Time Signals
• 4.0 Introduction
• 4.1 Periodic Sampling
• 4.2 Frequency-Domain Representation of
Sampling
• 4.3 Reconstruction of a Bandlimited Signal
from its Samples
• 4.4 Discrete-Time Processing of
Continuous-Time signals
2
4.0 Introduction
• Continuous-time signal processing can be
implemented through a process of
sampling, discrete-time processing, and
the subsequent reconstruction of a
continuous-time signal.
x  n  xc  nT  ,
  n  
T: sampling period
f=1/T: sampling frequency
 s  2 T ,
rad / s 
3
Unit
impulse
train
Continuoustime signal

4.1 Periodic
Sampling

  (t  nT )
n 
impulse train sampling
xs  t  
T:
sampling
period

 x  nT   t  nT 
n 
c
x[n]  xc (nT )
Sampling sequence
4
冲激串的傅立叶变换：
2
S  j  
T

    k 
s
k 
T：sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate
s(t)为冲激串序列，周期为T，可展开傅立叶级数



1
s  t      t  nT    ak e jk  s t   e jk st
T k 
n 
k 
s (t )
1 T /2
1
 jk  s t
1
ak 
 (t )e
dt 
…
…
T /2
T

T
-T
F
e jk st 
 2 (  k  s )
2
S  j  
T

    k 
k 
s
…
0
2
T
t
T
S ( j)
2 0

T
…
2
T

5
4.2 Frequency-Domain Representation of Sampling
s t  

  t  nT 
n  
T：sample period; fs=1/T:sample rate
Ωs=2π/T: sample rate
xs  t   xc  t  s  t   xc  t 


   t  nT    x  nT    t  nT 
n 
x[n]  xc (t ) |t nT  xc (nT )
n 
c
2
S  j  
T

    k 
s
k 
1
1 
X s  j  
X c  j  * S  j  
S  j  X c  j (   )  d


2
2
1  2 
1  

   k  s X c  j (   )  d       k  s  X c  j (   )  d



2
T k 
T k  
1 
  X c  j    ks 
T k 
Representation of
X s  j in terms of
 
X e
jw
6
 
jw
X
e
Representation of
in terms of X s  j , X c  j 
xs  t   xc  t  s  t   xc  t 

X s ( j ) 





n 

  t  nT    x  nT   t  nT 
n 
n 

  x  nT    t  nT e
 n 
 jt
c
 x  nT  e
n 

c
xc  nT e
T  
数字角频率ω，圆频率，rad
模拟角频率Ω, rad/s
j
 X (e ) 
DTFT

dt
x[n]  xc (nT )
 jTn
 j n
c
1
 X s  j    X c  j    k  s  
T k 
X (e
j T
)
2
s 
T
7
采样角频率, rad/s
jw
Representation of X e  in terms of X s  j , X c  j 
j
X (e )  X (e
DTFT
j T
)
1 
 X s  j    X c  j    k  s  
T k 
Continuous FT
  /T
1 

X (e )   X c 
T k  
j
  2 k  
j 

T 
T
if X c  j   0,  
then X (e
j

  T
T


T
1
 
)  Xc  j 
T
 T
 
8
Nyquist Sampling Theorem
• Let X c t  be a bandlimited signal with
X c  j  0 for    N . Then X c t  is
uniquely determined by its samples
2


xn  xc nT , n  0,1,2, , if s T
 2 N
• The frequency  N is commonly referred as
the Nyquist frequency.
• The frequency 2 N is called the Nyquist rate.
9
2
S  j  
T

    k 
k 
s
frequency spectrum of ideal
sample signal
1
X s ( j ) 
T

 X ( j (  k ))
c
s
k  
No aliasing
s  N  N
X (e j )  X s ( j) |

aliasing
frequency
s
2
s  N  N
2 

 T

T
1 
j (  k 2 )
  Xc (
)
T k 
T
aliasing
10
Example 4.1: Sampling and
Reconstruction of a sinusoidal signal
Compare the continuous-time and discrete-time
FTs for sampled signal xc  t   cos  4000 t  ,
if sampling period T  1 6000.
Solution:
x  n  xc  nT   cos  4000 Tn   cos 2  n  cos  w0n 
3


The highest frequency of the signal : 0  4000
2
sampling frequency  s 
 12000  20
T
11
Example 4.1: Sampling and
Reconstruction of a sinusoidal signal
continuous-time FT of xc t   cos4000t 
X c  j      4000       4000 

discrete-time FT of

j
X (e )
 4000

0
4000
x  n  cos 2  n  cos  w0 n 
3

1
j T
 X ( e )  X s  j   T  X c  j    k  s  
k 
12

1
j
j T
j  2 k  s  
  xnXc cos
X (xec t )  cos
X ( e4000
) tX s  jsame

n
 
T k 
3

1 t   cos
 16
 ,000
2 
k t
x
 c  Xc  j  
     4000       4000 
T
T
T 

k 
 
T  1 6000

    2

T  T
3


从积分(相同的面积)或冲击函数的定义可证
 
    T   
T 
2 

    
3 


Example 4.2: Aliasing in the Reconstruction
of an Undersampled sinusoidal signal
Compare the continuous-time and discrete-time
FTs for sampled signal xc  t   cos 16,000 t 
if sampling period T  1 6000
Solution:
The highest frequency of the signal : 0  16,000
2
sampling frequency  s 
 12000  20
T
x  n   xc  nT   cos 16, 000 Tn   cos 16, 000 n / 6000 

 cos  2 n  2 n / 3  cos 2  n
3

14
4.3 Reconstruction of a Bandlimited Signal
from its Samples 
xs  t  
 x  n    t  nT 
n 
xr  t   xs  t  * hr t 


 x  n     nT h t    d
  x  n     nT  h  t    d


 n 
n 
Gain: T

sin t T 
hr t  
t T
r

r


 x  n  h  t  nT 
n 
r
sin   t  nT  T 
  x  n
  t  nT  T
n 


X r  j  H r  jX e 15
jT

4.4 Discrete-Time Processing of
Continuous-Time signals
H  e jw 
xn  xc nT 
sin  t  nT  T 
yr t    yn
 t  nT  T
n  
X e   


T
,



H
1 r  j    w 2 kT 
X c 0,
j  otherwise



Tk    T
T 
jw
T

Yr  j   H r  j  Y  e jT 
: a half of the sampling frequency


jT
TY  e  ,  

T

 0, otherwise 16
C/D Converter
• Output of C/D Converter
xn  xc nT 
 
X e
jw
  w 2k  
1
  X c  j  
 
T k    T
T 

17
D/C Converter
• Output of D/C Converter
sin  t  nT  T 
yr t    yn
 t  nT  T
n  



T ,

H r  j   
T
0, otherwise



 

TY e ,  

T
Yr  j   H r  j Y e jT
jT
 0, otherwise
18

4.4.1
Linear
Time-Invariant
1
 
2 k  
jT
jw
X  e  =X  e    X c  j   

Discrete-Time
Systems
T k   
T 
=T
X c  j 
 
X e jw
 
H e jw
 
Y e jw
Yr  j 
Is the system Linear Time-Invariant ?
Y  e jw   H  e jw  X  e jw  , Yr  j   H r  j  Y  e jT 
Yr  j   H r  j  H  e jT  X  e jT 


jT
H  e  X c  j  ,  


1

2

k



T

 H r  j  H  e jT   X c  j   



T k   
T    0,
19

T
Linear and Time-Invariant
• Linear and time-invariant behavior of the system
of Fig.4.11 depends on two factors:
• First, the discrete-time system must be linear and
time invariant.
• Second, the input signal must be bandlimited,
and the sampling rate must be high enough to
satisfy Nyquist Sampling Theorem.(避免频率混叠)
20
   
 
1
2k  
 H  jH e   X  j  
 
T
T 
 
Yr  j  H r  jH e jT X e jT
jT
r

k  
c


T ,

If X c  j  0 for    T , H r  j   
T
0,
otherwise




jT
H
e
X
j

,


 c 
 
T
Yr  j   
 0,



T
Yr  j  Heff  j X c  j


jT
 H  e  ,   T
H eff  j   

 0,


T
effective
frequency response
of the overall LTI
continuous-time
system
21
4.4.2 Impulse Invariance
Given:
Design:
X c  j
H
e 
jw
 
X e
jw
H c  j , i.e. h  n
H  e jw 
 
Y e jw
Heff  j  Hc  j
hc  t 
hc  nT 
Yr  j


jT
h n  Thc nT H  j   H  j    H  e  ,   T
c
eff


0,
 22
impulse-invariant version of the continuous-time system
T
 
 
4.4.2 Impulse Invariance
=T
 Two constraints
1.
H e
2.
T is chosen such that
j
  H  j T  ,
c
H c  j   0,
 
 
截止频率

T
C   / T
  T
h  n  Thc  nT 
The discrete-time system is called an impulseinvariant version of the continuous-time system
h  n  hc  nT 
h  n  Thc  nT 
1
  ,  
H (e )  H c  j 
T
T
 T
   ,  
j
H (e )  H c  j 
23
 T
j
4.5 Continuous-time Processing of
Discrete-Time Signal
 
X c  j
X e jw
Yc  j
sin   t  nT  T 
xc  t    x  n 
  t  nT  T
n 
 
Y e jw

sin   t  nT  T 
yc  t    y  n
  t  nT  T 24
n 

4.5 Continuous-time Processing of Discrete-Time Signal
X c  j   TX  e jT  ,
  T
Yc  j  Hc  j X c  j ,    T
w=T
w
j
T 

1
w
1
w




jw
Y  e   Yc  j   H c  j  TX  e T  , w  
T  T T
 T 

 
 w
H e  H c  j , w  
 T
H c  j  H e jT ,    T
jw
25
4.5 Continuous-time Processing of Discrete-Time Signal
Errata
Figure 4.18 Illustration of moving-average filtering. (a) Input signal
x[n] = cos(0.25πn). (b) Corresponding output of six-point movingaverage filter.
26
 What is Nyquist rate？
 What is Nyquist frequency？
Review
 The Nyquist rate is two times the bandwidth
of a bandlimited signal.
 The Nyquist frequency is half the sampling
frequency of a discrete signal processing
system.( The Nyquist frequency is one-half
the Nyquist rate)
27
 What is the physical meaning for the equation:
DTFT of a discrete-time signal is equal to the
FT of a impulse train sampling .
Review
xs  t   xc  t  s  t  


 x  nT   t  nT 
n 
c
1
X s  j    X c  j    k  s   
T k 


 x  ne
n 
 j n
j
 X (e )

 x  nT  e
n 
 jTn
c
T  
x[n]  xc (nT )
 DTFT derived from the equation.
 impulse train sampling xs(t) and x[n] have the
28
same frequency component.
Review
 How many factors does the linear and time-invariant
behavior of the system of Fig.4.11 depends on ?
 First, the discrete-time system must be linear and
time invariant.
 Second, the input signal must be bandlimited, and
the sampling rate must be high enough to satisfy
29
Nyquist Sampling Theorem.(避免频率混叠)
 Assume that we are given a desired continuous-time
system that we wish to implement in the form of the
following figure, how to decide h[n] and H(ejw)?
Review
h  n  Thc  nT 


jT
H e  ,   T
H c  j   H eff  j   

 0,


T 30
Chapter 4 HW
• 4.5
2017/7/28
31
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