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8/13/2019 Chapter4 Sampling of Continuous-Time Signals
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12014/1/25 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ
Biomedical Signal processingChapter 4 Sampling of Continuous-
Time SignalsZhongguo Liu
Biomedical EngineeringSchool of Control Science and Engineering, Shandong University
8/13/2019 Chapter4 Sampling of Continuous-Time Signals
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Chapter 4: Sampling ofContinuous-Time Signals
4.0 Introduction 4.1 Periodic Sampling
4.2 Frequency-Domain Representation ofSampling 4.3 Reconstruction of a Bandlimited Signal
from its Samples 4.4 Discrete-Time Processing of
Continuous-Time signals
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4.0 Introduction Continuous-time signal processing can be
implemented through a process ofsampling , discrete-time processing , andthe subsequent reconstruction of a
continuous-time signal.
f=1/T: sampling frequency ,c x n x nT n srad T s /,2
T: sampling period
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n
nT t )( 4.1 Periodic
Sampling
Continuous-time signal
T:samplingperiod
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4.2 Frequency-Domain Representation of Sampling
[ ] ( ) | ( )c t nT c x n x t x nT
T sample period; fs=1/T :sample rates=2/T : sample rate
n
nT t t s
n
s c cn
c x t t nT x t x t s t x nT t nT
2 sk
S j k T
1 c sk
X j k T
1( )
21 2 1
1*
2
( ) ( )
2
c
s c s
k
s c
c
k
S j X j d
k X j
X j X j S j
d k X j d
T T
Representation of
in terms of jwe X j X s
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T sample period; fs=1/T :sample rate; s=2/T :sample rate
n
nT t t s
2 sk
S j k T
2 sk
S j k T
s jk t k
n
a e s(t) T
1 s jk t
n
eT
2 ( ) s jk t F se k -T
1
t T0
( ) s t
/2
/2
1 1( ) s
T jk t k T
a t e dt T T
0
( )S j
2T
2T
2T
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s c c cn n
x t x t s t x t t nT x nT t nT
ck
j Tn x nT e
( ) j t s cn
X j x nT t nT e dt
[ ] ( )c x n x nT
( ) j cn
j n X e x nT e DTFT
Representation of in terms of , jwe X j X s c X j
( ) j T X e
T rad
, rad/s
2 s T
1
s c sk X j X j k T
, rad/s
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( ) ( ) j j T X e X e
1 2( )
ck
j k X X jT T T
e
0,cif X jT
1( ) c
jthen X X jT T
e
1 s c sk
X j X j k T
DTFT
Representation of in terms of , jwe X j X s c X j
Continuous FT/ T
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Nyquist Sampling Theorem Let be a bandlimited signal
with . Then isuniquely determined by itssamples ,if
The frequency is commonly referred asthe Nyquist frequency .
The frequency is called the Nyquist rate .
t X c N c for j X 0 t X c
,2,1,0, nnT xn x c N s T
22
N
N 2
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N N s
N N s
aliasing frequencyT
2/ s
2
No aliasing
aliasing
k sc s k j X
T j X ))((
1)(
k c
T s j
T k j X T
j X e X
)/)2((1
|)()( /
frequency spectrum of idealsample signal
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4.3 Reconstruction of a Bandlimited Signalfrom its Samples
n
r r nT t hn xt x
T t T t
t hr sin
sin
n
t nT T x n
t nT T
T jr r e X j H j X
Gain: T
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4.4 Discrete-Time Processing ofContinuous-Time signals
nT xn x c
1 2 jw
ck
X e w k X j
T T T
nr T nT t T nT t
n yt y
sin
otherwiseT
eTY eY j H jY
T j
T j
r r
,0
,
jw H e
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C/D Converter Output of C/D Converter
nT xn x c
k c
jw
T k
T w
j X T
e X 21
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D/C Converter Output of D/C Converter
nr
T nT t
T nT t n yt y
sin
otherwise
T eTY
eY j H jY
T j
T jr r
,0
,
,0,
r T H j T
otherwise
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,
0,
j T c H e X j
T
T
4.4.1 Linear Time-InvariantDiscrete-Time Systems
j X c jwe X jY r jweY jwe H
jw jw jw
e X e H eY 1 2
j T j T r r
j T r c
k
Y j H j H e X ek
H j H e X j
T T
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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 mustbe high enough to satisfy NyquistSampling Theorem .
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k c
T j
r
T jT jr r
T
k
j X T e H j H
e X e H j H jY
21
0 ,c If X j for T
T
T j X e H
jY c
T j
r
,0
,
j X j H jY ceff r
T
T e H
j H
T j
eff
,0
,
,0,
r T H j T
otherwise
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4.4.2 Impulse InvarianceGiven:
jw H eDesign:
,
0,
j T
c eff
H eT H j H j
T
ch n Th nT
h n ch nT
j X c jwe X jY r
jw
eY jwe H
impulse-invariant version of the continuous-time system
,c H j
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4.4.2 Impulse InvarianceTwo constraints
, j
c H e H j T
T j H that suchchosenisT
c ,0
1.2.
ch n T T h nThe discrete-time system is called an impulse-invariant version of the continuous-time system
ch n h nT ( ) 1 c j X j T T X e
ch n T T h n
( ) c j
X X jTe
/C T
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