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
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Periodic (Uniform) Sampling
• Sampling is a continuous to discrete-time conversion
-3 -2 -1 0 1 2 3 4
• Most common sampling is periodic
xn  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=2fs 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
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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
1
0.5
0
-0.5
-1
0
20
40
60
80
100
• 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
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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)
x
xc(t)
x[n]=xc(nT)
x[n]
s(t)
-3T-2T-T 0 T 2T3T4T
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Convert impulse
train to discretetime sequence
t
n
-3 -2 -1 0 1 2 3 4
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Continuous-Time Fourier Transform
• Continuous-Time Fourier transform pair is defined as
X c j  

 jt


x
t
e
dt
 c


1
jt


x c 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)ejot  Xj  o 
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Frequency Domain Representation of Sampling
• Modulate (multiply) continuous-time signal with pulse train:
x s t   x c t st  

 x t t  nT 
c
s(t) 
 t  nT 
n  
n  
•

Let’s take the Fourier Transform of xs(t) and s(t)
1
Xs j 
Xc j  Sj
2
2 
Sj  
  k s 

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  = 2f hence s = 2fs
1 
X s j  
X c j  k s 

T k  
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Frequency Domain Representation of Sampling
• Convolution with pulse creates replicas at pulse location:
1 
X s j  
X c j  k s 

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
X c j 
-N
N
X s j 
s>2N
3s
-2s
s -N
N
s
2s
3s
X s j 
s<2N
3s
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s -N
N
s
2s
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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 
T
 2fs  2N
• 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
X s j 
s>2N
3s
-2s
s -N
N
s
2s
3s
X s j 
s<2N
3s
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-2s
s -N
N
s
2s
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Sample Question I
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Sample Question II
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