Transcript Signals and Systems EE235 Leo Lam © 2010-2013 Futile Q: What did the monsterous voltage source say to the chunk of wire? A: "YOUR RESISTANCE IS.

Signals and Systems
EE235
Leo Lam © 2010-2013
Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"
Leo Lam © 2010-2013
Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)
Leo Lam © 2010-2013
Sampling
• Convert a continuous time signal into a series
of regularly spaced samples, a discrete-time
signal.
• Sampling is multiplying with an impulse train
t
multiply
=
0
t
TS
t
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4
Sampling
• Sampling signal with sampling period Ts is:
xs (t ) 
n 
 x(nT ) (t  nT )
n  
s
s
• Note that Sampling is NOT LTI
sampler
Leo Lam © 2010-2013
5
Sampling
• Sampling effect in frequency domain:
• Need to find: Xs(w)
• First recall:
T
1/T
 2w0
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 w0
0
time
w0
2w0
3w0
Fourier spectra
6
Sampling
• Sampling effect in frequency domain:
• In Fourier domain:
Impulse train in time
 impulse train in
frequency,
dk=1/Ts
distributive
What does
this
property
mean?
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7
Sampling
• Graphically:
1
X s (w ) 
Ts

2

X
w

n
 
Ts
n  


• In Fourier domain:



1
X (w
X()w)
Ts
bandwidth
0
• No info loss if no overlap (fully reconstructible)
• Reconstruction = Ideal low pass filter
Leo Lam © 2010-2013
Sampling
• Graphically:
1
X s (w ) 
Ts

2

X
w

n
 
Ts
n  





• In Fourier domain:
0
• Overlap = Aliasing if
• To avoid Alisasing:
Nyquist Frequency (min. lossless)
• Equivalently:
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Shannon’s Sampling Theorem
Sampling (in time)
• Time domain representation
cos(2100t)
100 Hz
Fs=1000
Fs=500
Fs=250
Fs=125 < 2*100
cos(225t)
Frequency wraparound,
sounds like Fs=25
(Works in spatial
frequency, too!)
Leo Lam © 2010-2013
Aliasing
Summary: Sampling
• Review:
– Sampling in time = replication in frequency domain
– Safe sampling rate (Nyquist Rate), Shannon theorem
– Aliasing
– Reconstruction (via low-pass filter)
• More topics:
– Practical issues:
– Reconstruction with non-ideal filters
– sampling signals that are not band-limited (infinite
bandwidth)
• Reconstruction viewed in time domain: interpolate with
sinc function
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Would these alias?
• Remember, no aliasing if
• How about:
0
1
NO ALIASING!
-3
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0
1
3
Would these alias?
• Remember, no aliasing if
• How about: (hint: what’s the bandwidth?)
Definitely ALIASING!
Y has infinite bandwidth!
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Would these alias?
• Remember, no aliasing if
• How about: (hint: what’s the bandwidth?)
ws  .7  2wB  1.0
wB  0.5
-.5
0
.5
Copies every .7
ALIASED!
-1.5
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-.5
-.5 0
.5.5
1.5
Would these alias?
• Remember, no aliasing if
• How about: (hint: what’s the bandwidth?)
ws  .7  2wB  1.0
wB  0.5
-.5
0
.5
Copies every .7
ALIASED!
-1.5
Leo Lam © 2010-2013
-.5
-.5 0
.5.5
1.5
How to avoid aliasing?
• We ANTI-alias.
time signal
x(t)
Anti-aliasing
filter
X(w)
Sample
Z(w)
B
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ws > 2wc
wc < B
Reconstruct
z(n)
How bad is anti-aliasing?
• Not bad at all.
• Check: Energy in the signal (with example)
lowpass
anti-aliasing
filter
• Sampled at
• Add anti-aliasing (ideal) filter
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sampler
with bandwidth 7
How bad is anti-aliasing?
• Not bad at all.
• Check: Energy in the signal (with example)
lowpass
anti-aliasing
filter
• Energy of x(t)?
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sampler
How bad is anti-aliasing?
• Not bad at all.
• Check: Energy in the signal (with example)
lowpass
anti-aliasing
filter
sampler
1
Ef 
2
7
2
|
X
(
w
)
|
dw

• Energy of filtered x(t)?
7
1
1
1
2
X (w ) 
 X (w ) 

1  jw
(1  jw )(1  jw ) 1  w 2
1
Ef 
2
7
1
1
7 1  w 2 dw   arctan(7)
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~0.455
Bandwidth Practice
• Find the Nyquist frequency for:
-100
ws  200
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0
100
Bandwidth Practice
• Find the Nyquist frequency for:
const[rect(w/200)*rect(w/200)] =
-200
ws  400
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200
Bandwidth Practice
• Find the Nyquist frequency for:
(bandwidth = 100) + (bandwidth = 50)
ws  300
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Summary
• Sampling and the frequency domain
representations
• Sampling frequency conditions
Leo Lam © 2010-2013