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Sampling of Continuous-Time Signals
Quote of the DayOptimist: "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.
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 2
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 3
Periodic (Uniform) Sampling• Sampling is a continuous to discrete-time conversion
• Most common sampling is periodic
• 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
nnTxnx c
-3 -2 2 3 4-1 10
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 4
Periodic Sampling• Sampling is, in general, not reversible• Given a sampled signal one could fit infinite continuous signals
through the samples
0-1
20 40 60 80 100
-0.5
0
0.5
1
• 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 5
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.
• Demo from DSP First: A Multimedia Approach by McClellan, Schafer, Yoder
n
ff
2j
s
tf2j
s
o
o
ef/npnTpnp
etp
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 6
Representation of Sampling• Mathematically convenient to represent in two stages
– Impulse train modulator– Conversion of impulse train to a sequence
Convert impulse train to discrete-time sequence
xc(t) x[n]=xc(nT)x
s(t)
-3T-2T 2T3T4T-T T0
s(t)xc(t)
t
x[n]
-3 -2 2 3 4-1 10n
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 7
Continuous-Time Fourier Transform• Continuous-Time Fourier transform pair is defined as
• We write xc(t) as a weighted sum of complex exponentials
• Remember some Fourier Transform properties– Time Convolution (frequency domain multiplication)
– Frequency Convolution (time domain multiplication)
– Modulation (Frequency shift)
dtetxjX tjcc
dejX21
tx tjcc
)j(Y)j(X)t(y)t(x
)j(Y)j(X)t(y)t(x
otj jXe)t(x o
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 8
Frequency Domain Representation of Sampling• Modulate (multiply) continuous-time signal with pulse train:
• Let’s take the Fourier Transform of xs(t) and s(t)
• 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
n
nTt)t(s
n
ccs nTttxtstxtx
k
skT2
jS
jSjX21
jX cs
k
scs kjXT1
jX
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 9
Frequency Domain Representation of Sampling• Convolution with pulse creates replicas at pulse location:
• 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
k
scs kjXT1
jX
jXc
jXs
jXs
N-N
N-N s 2s 3s -2s s 3s
N-N s 2s 3s -2s s 3s
s<2N
s>2N
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 10
Nyquist Sampling Theorem
• Let xc(t) be a bandlimited signal with
• Then xc(t) is uniquely determined by its samples x[n]= xc(nT) if
N is generally known as the Nyquist Frequency
• The minimum sampling rate that must be exceeded is known as the Nyquist Rate
Nc for 0)j(X
Nss 2f2T2
jXs
jXs
N-N s 2s 3s -2s s 3s
N-N s 2s 3s -2s s 3s
s<2N
s>2N
Low pass filter
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 11
Demo
Aliasing and Folding Demo
Samplemania from John Hopkins University