Transcript Document

Sampling of Continuous-Time Signals
Quote of the Day
Optimist: "The glass is half full."
Pessimist: "The glass is half empty."
Engineer: "That glass is twice as large as it needs to be."
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
2
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
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
3
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
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
4
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.
pt   e j2 fot
pn  pnT   pn / 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
5
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
-3 -2 -1 0 1 2 3 4
351M Digital Signal Processing
6
Continuous-Time Fourier Transform
• Continuous-Time Fourier transform pair is defined as
Xc j 

 jt


x
t
e
dt
c



1
jt


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)ejot  Xj  o 
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
7
Frequency Domain Representation of Sampling
• Modulate (multiply) continuous-time signal with pulse train:
xs t   xc t st  
•

 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   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 
Xs j 
Xc j  ks 

T k  
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
8
Frequency Domain Representation of Sampling
• Convolution with pulse creates replicas at pulse location:
1 
Xs j 
Xc 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
Xc j
-N
N
Xs j
s>2N
3s
-2s
s -N
N
s
2s
3s
Xs j
s<2N
3s
Copyright (C) 2005 Güner Arslan
-2s
s -N
N
s
2s
351M Digital Signal Processing
3s
9
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 
 2fs  2N
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>2N
3s
-2s
s -N
N
s
2s
3s
Xs j
s<2N
3s
Copyright (C) 2005 Güner Arslan
-2s
s -N
N
s
2s
351M Digital Signal Processing
3s
10
Demo
Aliasing and Folding Demo
Samplemania from John Hopkins University
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
11