### Chapter 4: Sampling of Continuous

```Biomedical Signal processing
Chapter 4 Sampling of ContinuousTime Signals
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
1
2015/4/13
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 ,
3


 (t  nT )
n  
4.1 Periodic
Sampling
Continuoustime signal
T:
sampling
period
4
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  
n 
x[n]  xc (t ) |t nT  xc (nT )

 x  nT   t  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
5

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 st   e jk st
T n 
n  
n 
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

6
jw
Representation of X e  in terms of X s  j , X c  j

xs  t   xc  t  s  t   xc  t     t  nT  
n 

X s ( j ) 


 n 

X (e ) 
DTFT

n 
n 
c
x[n]  xc (nT )
 jTn
c

 x  nT   t  nT 
xc  nT    t  nT e jt dt
 x  nT  e
k 
j


xc  nT e

T  
 j n

 X (e jT )
2
s 
T7
1
 X s  j    X c  j    k  s  
T k 

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,  

T
1
 
then X (e )  X c  j 
T
 T
j
 
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 xn  xc nT , n  0,1,2, ,if
2
s 
 2 N
T
• The frequency  N is commonly referred as
the Nyquist frequency.
• The frequency 2 N is called the Nyquist rate.
9
frequency spectrum of ideal
sample signal
s   N   N
1
X s ( j ) 
T

 X ( j(  k ))
c
s
k  
No aliasing
s   N   N
1

T
s / 2
aliasing frequency
X (e j )  X s ( j) |  / T

2
  T

 X ( j(  k 2 ) / T )
c
k  
aliasing
10
4.3 Reconstruction of a Bandlimited Signal
from its Samples
sin t T 
hr t  
t T
Gain: T
xr t  

 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 j11T

4.4 Discrete-Time Processing of
Continuous-Time signals
H  e jw 
xn  xc nT 
X e  
1 

Xc 

T k  
sin  t  nT  T 
yr t    yn
 t  nT  T
n  

 

,   T
Yr  j   H r  j Y e jT
jw
 w 2 k  
j 

T 
T


jT
TY e

 0, otherwise
12
C/D Converter
• Output of C/D Converter
xn  xc nT 
 
Xe
jw
  w 2k  
1
  X c  j 
 
T k    T
T 

13
D/C Converter
• Output of D/C Converter
sin  t  nT  T 
yr t    yn
 t  nT  T
n  

 

,   T
Yr  j   H r  j Y e jT


jT
TY e



 0, otherwise H j  T ,




r
T

0, otherwise
14
4.4.1 Linear Time-Invariant
Discrete-Time Systems
X c  j
 
X e jw
 
 
H e jw
 
Y e jw
   
Yr  j 
Y e jw  H e jw X e jw
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,
15

T
Linear and Time-Invariant
• Linear and time-invariant system behavior
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.
16
   
1
 H  j H e   X
T
Yr  j   H r  j H e jT X e jT
jT
r

k  
If X c  j  0 for    T ,
 
2k  
c
 j    T  

 


T ,

H r  j   
T
0, otherwise




jT

X c  j,   T
H
e

Yr  j  

 0,

T

Yr  j  Heff  jX c  j


jT
H e ,   T
H eff  j  

 0,

T

17
4.4.2 Impulse Invariance
Given:
Design:
X c  j
H
jw
e
 
 
X e jw
Hc  j ,
 
H e jw
h  n
 
Y e jw
hc  nT 
Yr  j 


jT
h n  Thc nT
H e  ,   T
H c  j   H eff  j   

 18

impulse-invariant version of the continuous-time system  0,

T
 
 
4.4.2 Impulse Invariance
 Two constraints
1.
H  e j   H c  j T  ,
2.
T is chosen such that
H c  j  0,
 

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
 
X (e )  X c  j 
T
 T
 
j
 19 
X (e )  X c  j 
 T
j
```