Chapter 4: Sampling of Continuous
Download
Report
Transcript 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 ,
rad / s
3
(t nT )
n
4.1 Periodic
Sampling
Continuoustime signal
T:
sampling
period
4
4.2 Frequency-Domain Representation of Sampling
st
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
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
st 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 jkst
2 ( ks )
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 )
jTn
c
x nT t nT
xc nT t nT e jt dt
x nT e
k
j
xc nT e
T
j n
数字角频率,圆频率,rad
模拟角频率, rad/s
X (e jT )
2
s
T7
1
X s j X c j k s
T k
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
jT
)
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 xn 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
xnh t nT
n
r
sin t nT T
x n
t nT T
n
X r j H r jX e j11T
4.4 Discrete-Time Processing of
Continuous-Time signals
H e jw
xn xc nT
X e
1
Xc
T k
sin t nT T
yr t yn
t nT T
n
, T
Yr j H r j Y e jT
jw
w 2 k
j
T
T
jT
TY e
0, otherwise
12
C/D Converter
• Output of C/D Converter
xn xc nT
Xe
jw
w 2k
1
X c j
T k T
T
13
D/C Converter
• Output of D/C Converter
sin t nT T
yr t yn
t nT T
n
, T
Yr j H r j Y e jT
jT
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 jT X e jT
jT
H e X c j ,
1
2
k
T
H r j H e jT 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 jT X e jT
jT
r
k
If X c j 0 for T ,
2k
c
j T
T ,
H r j
T
0, otherwise
jT
X c j, T
H
e
Yr j
0,
T
Yr j Heff jX c j
jT
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
jT
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,
截止频率
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