Transcript Document
Sampling of Continuous-Time Signals
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Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Signal Types
• Analog signals: continuous in time and amplitude
– Example: voltage, current, temperature,…
• Digital signals: discrete both in time and amplitude
– Example: attendance of this class, digitizes analog signals,…
• Discrete-time signal: discrete in time, continuous in amplitude
– Example:hourly change of temperature in Austin
• Theory for digital signals would be too complicated
– Requires inclusion of nonlinearities into theory
• Theory is based on discrete-time continuous-amplitude signals
– Most convenient to develop theory
– Good enough approximation to practice with some care
• In practice we mostly process digital signals on processors
– Need to take into account finite precision effects
• Our text book is about the theory hence its title
– Discrete-Time Signal Processing
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Periodic (Uniform) Sampling
• Sampling is a continuous to discrete-time conversion
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• Most common sampling is periodic
xn xc nT n
•
•
•
•
•
T is the sampling period in second
fs = 1/T is the sampling frequency in Hz
Sampling frequency in radian-per-second s=2fs rad/sec
Use [.] for discrete-time and (.) for continuous time signals
This is the ideal case not the practical but close enough
– In practice it is implement with an analog-to-digital converters
– We get digital signals that are quantized in amplitude and time
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Periodic Sampling
• Sampling is, in general, not reversible
• Given a sampled signal one could fit infinite continuous signals
through the samples
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0.5
0
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-1
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• Fundamental issue in digital signal processing
– If we loose information during sampling we cannot recover it
• Under certain conditions an analog signal can be sampled without
loss so that it can be reconstructed perfectly
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Sampling Demo
• In this movie the video camera is sampling at a fixed rate of
30 frames/second.
• Observe how the rotating phasor aliases to different speeds
as it spins faster.
pt e j2 fot
pn pnT pn / fs e
•
j2
fo
n
fs
Demo from DSP First: A Multimedia Approach by McClellan, Schafer, Yoder
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Representation of Sampling
• Mathematically convenient to represent in two stages
– Impulse train modulator
– Conversion of impulse train to a sequence
s(t)
xc(t)
Convert impulse
train to discretetime sequence
x
xc(t)
x[n]=xc(nT)
x[n]
s(t)
-3T-2T-T 0 T 2T3T4T
Copyright (C) 2005 Güner Arslan
t
n
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Continuous-Time Fourier Transform
• Continuous-Time Fourier transform pair is defined as
Xc j
jt
x
t
e
dt
c
1
jt
xc t
X
j
e
d
c
2
• We write xc(t) as a weighted sum of complex exponentials
• Remember some Fourier Transform properties
– Time Convolution (frequency domain multiplication)
x(t) y(t) X(j)Y(j)
– Frequency Convolution (time domain multiplication)
x(t)y(t) X(j) Y(j)
– Modulation (Frequency shift)
x(t)ejot Xj o
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Frequency Domain Representation of Sampling
• Modulate (multiply) continuous-time signal with pulse train:
xs t xc t st
•
x t t nT
n
c
s(t)
t nT
n
Let’s take the Fourier Transform of xs(t) and s(t)
1
X s j
X c j Sj
2
2
Sj
ks
T k
• Fourier transform of pulse train is again a pulse train
• Note that multiplication in time is convolution in frequency
• We represent frequency with = 2f hence s = 2fs
1
Xs j
Xc j ks
T k
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Frequency Domain Representation of Sampling
• Convolution with pulse creates replicas at pulse location:
1
Xs j
Xc j ks
T k
• This tells us that the impulse train modulator
– Creates images of the Fourier transform of the input signal
– Images are periodic with sampling frequency
– If s< N sampling maybe irreversible due to aliasing of images
Xc j
-N
N
Xs j
s>2N
3s
-2s
s -N
N
s
2s
3s
Xs j
s<2N
3s
Copyright (C) 2005 Güner Arslan
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s -N
N
s
2s
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Nyquist Sampling Theorem
• Let xc(t) be a bandlimited signal with
Xc(j) 0
for N
• Then xc(t) is uniquely determined by its samples x[n]= xc(nT)
if
2
s
2fs 2N
T
• N is generally known as the Nyquist Frequency
• The minimum sampling rate that must be exceeded is known
as the Nyquist Rate
Low pass filter
Xs j
s>2N
3s
-2s
s -N
N
s
2s
3s
Xs j
s<2N
3s
Copyright (C) 2005 Güner Arslan
-2s
s -N
N
s
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351M Digital Signal Processing
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Demo
Aliasing and Folding Demo
Samplemania from John Hopkins University
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