m7. introduction to sampling of continuous-time signals
TRANSCRIPT
3/22/2011 I. Discrete-Time Signals and Systems 1
M7. Introduction to Sampling of
Continuous-Time Signals
Reading material: p.140-160
3/22/2011 I. Discrete-Time Signals and Systems 2
Content
What’s the sampling mechanism?
What’s the aliasing problem?
Nyquist sampling theorem
Frequency analysis of sampling
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Introduction
Discrete-time signals can be commonly obtained by sampling
the continuous-time signals
Digital computers can only deal with digital signals, which is
a special class of discrete-time signals
Digital signal processing, digital control systems …
Plant
Computer
input
Input noise Output disturbance
output
A/DA/DD/A
Sampling and holdingSignal processingconstruction
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Periodic Sampling A sequence of samples x[n], a continuous-time signal xc(t)
x[n]=xc(nT), -∞∞∞∞<n < ∞∞∞∞
Sampling period T, sampling frequency f=1/T (Herz); or sampling frequency ΩΩΩΩs=2ππππ/T (rad/sec)
Practically, the operation of sampling is implemented by analog-to-digital (A/D) converters
Quantization of the sampled outputs
Linearity of quantization steps
Sample-and-hold circuits
Limitations on the sampling rate
Principle of sampling
Generating a impulse train from the continuous-time signal
Converting the impulse train to a discrete-time sequence
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Principle of Sampling
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Frequency Analysis of Sampling
Generation of a impulse train xs(t) through modulation of the
uniformed periodic impulse train
The Fourier transform of a periodic impulse train is a
periodic impulse train in frequency domain
∑∞
−∞=
−=
n
nTtts )()( σ
)()()()()()()( nTtnTxnTttxtstxtxn
c
n
ccs −=−== ∑∑∞
−∞=
∞
−∞=
σσ
Tn
TjS s
n
s
πσ
π 2,)(
2)( =ΩΩ−Ω=Ω ∑
∞
−∞=
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Frequency Analysis of Sampling (Cont’d)
The frequency-domain relation between xs(t) and xc(t)
Xs(jΩΩΩΩ) consists of periodically repeated copies of Xc(jΩΩΩΩ)!!!
xc(t) could be recovered possibly from xs(t) with a ideal
lowpass filter if …
)()()()()()()( nTtnTxnTttxtstxtxn
c
n
ccs −=−== ∑∑∞
−∞=
∞
−∞=
σσ
Tn
TjS s
n
s
πσ
π 2,)(
2)( =ΩΩ−Ω=Ω ∑
∞
−∞=
∑∞
−∞=
Ω−Ω=ΩΩ=Ω
n
sccs njXT
jSjXjX ))((1
)(*)(2
1)(
π
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Effect of Sampling in Frequency Domain
Consider a bandlimited signal with highest nonzero frequency ΩΩΩΩN
If ΩΩΩΩs >2 ΩΩΩΩN, there is no overlap with replicas of Xc(jΩΩΩΩ), so,
xc(t) could be recovered
possibly from xs(t)
If ΩΩΩΩs<2 ΩΩΩΩN, there is overlap with replicas of Xc(jΩΩΩΩ), then,
Aliasing distortion (aliasing)
Exists in the recovered
signal through …
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Exact Recovery from Sampling
Consider bandlimited signals
Under condition ΩΩΩΩs>2 ΩΩΩΩN
Use an ideal lowpass filter
Hr(jΩΩΩΩ) with gain T and cutoff
frequency ΩΩΩΩc such that
ΩΩΩΩN <ΩΩΩΩc < (ΩΩΩΩs-ΩΩΩΩN)
The continuous signal can be
exactly recovered as
Xr(jΩΩΩΩ)= Xc(jΩΩΩΩ)
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Alias and Nyquist Sampling Theorem
If ΩΩΩΩs<2 ΩΩΩΩN, aliasing phenomenon occurs…
Consider a cosine signal xc(t)=cos ΩΩΩΩ0t …
Nyquist Sampling
Theorem
Let xc(t) be a bandlimited
signal with Xr(jΩΩΩΩ)= 0 for
|ΩΩΩΩ|> ΩΩΩΩN, then xc(t) is
uniquely determined by its
samples x[n]=xc(nT),
-∞∞∞∞<n < ∞∞∞∞, if ΩΩΩΩs= 2ππππ/T ≥≥≥≥2 ΩΩΩΩN
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Relation between X(ejωωωω) and Xc(jΩΩΩΩ) Xc(jΩΩΩΩ) is the (continuous) Fourier transform of xc(t)
X(ejωωωω) is the (discrete) Fourier transform of x[n]= xc(nT)
The Fourier transform of impulse train function xs(t)
Then there is
i.e., a frequency scaling or normalization …ωωωω=ΩΩΩΩT
dtetxjXtj
∫∞
∞−
Ω−=Ω )()(
jwk
k
jekxeX
−
∞
−∞=
∑= ][)( ω
nTj
n
cs
n
ccs
enTxjX
nTtnTxtstxtx
Ω−
∞
−∞=
∞
−∞=
∑
∑
=Ω
−==
)()(
)()()()()( σ
)(|)()( Tj
T
j
s eXeXjXΩ
Ω===Ω
ω
ω ∑∞
−∞=
Ω−Ω=Ω
n
scs njXT
jX ))((1
)(
∑∞
−∞=
−=
n
c
j
Tn
TjX
TeX ))
2((
1)(
πωω
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Examples: Sampling and Aliasing
Example 4.1 (p.147) sampling and reconstruction …
Example 4.2, 4.3 (p.148-9) Aliasing in reconstruction …
! Should be ππππ/T
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Reconstruction of Bandlimited Sampled Signals
Principle of reconstruction
Generating a impulse train from
discrete-time sequence
Converting the impulse train to a
continuous-time signal
Reconstruction (lowpass) filter
∑∞
−∞=
−=
n
s nTtnxtx )(][)( σ
∑∞
−∞=
−=
n
rr nTthnxtx )(][)(
Tt
Ttth r
/
)/sin()(
π
π=
∑∞
−∞= −
−=
n
rTnTt
TnTtnxtx
/)(
)/)(sin(][)(
π
π
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Discrete-Time Processing of Continuous-Time Signals
C/D converter produces a discrete-time sequence
D/C converter creates a continuous-time signal
LTI discrete-time systems
The overall system is equivalent to a LTI continuous-time
system
∑∞
−∞=
−==
n
c
j
cT
nT
jXT
eXnTxnx ))2
((1
)(),(][πωω
<Ω
=Ω=Ω−
−=
Ω
Ω
∞
−∞=
∑otherwsie
TeTYeYjHjY
TnTt
TnTtnyty
Tj
Tj
rr
n
r,0
/||),()()()(,
/)(
)/)(sin(][)(
π
π
π
)()()(],[*][][ ωωω jjjeXeHeYnxnhny ==
≥Ω
<Ω=ΩΩΩ=Ω
Ω
T
TeHjHjXjHjY
Tj
effceffr/||,0
/||),()(),()()(
π
π
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Example of Discrete Processing
Example 4.4 p.155…
LTI discrete-time system: a ideal lowpass filter
Bandlimited input signal
Sampling rate satisfies Nyquist Theorem
The effective system is a LTI continuous-time system with …
The effective cutoff frequency
ΩΩΩΩc=ωωωωc/T
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Practical Issues in ImplementationIn practice, …
Continuous-time signals are not precisely bandlimited
Ideal lowpas filters can not be realized
Ideal C/D and D/C converters can only be approximated by devices called A/D and D/A converters
Therefore, …
Prefiltering to avoid aliasing
A/D conversion: sample and hold circuits, quantization…
D/A conversion ….
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Exercise Seven
Problem 4.4 on page214 of the textbook;
Problem 4.5 on page214 of the textbook;
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Course Summary M1. Discrete-Time Signal and System
M2. The Z-Transform and its Properties
M3. The Inverse Z-Transform and Frequency-
Domain Representation
M4. Fourier Transform and Its Properties
M5. LTI Systems Described by Linear Constant
Coefficient Different Equations
M6. Frequency Response of LTI Systems
M7. Introduction to Sampling of Continuous-Time
Signals