leo lam © 2010-2013 signals and systems ee235. leo lam © 2010-2013 futile q: what did the...
Post on 24-Dec-2015
218 Views
Preview:
TRANSCRIPT
Leo Lam © 2010-2013
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
4
t
t
t
multiply
=0 TS
Leo Lam © 2010-2013
Sampling
• Sampling signal with sampling period Ts is:
• Note that Sampling is NOT LTI
5
)()()(
n
nsss nTtnTxtx
sampler
Leo Lam © 2010-2013
Sampling
• Sampling effect in frequency domain:
• Need to find: Xs(w)• First recall:
6
timeT
Fourier spectra0
1/T
0 02 03002
Leo Lam © 2010-2013
Sampling
• Sampling effect in frequency domain:
• In Fourier domain:
7
distributive property
Impulse train in time impulse train in frequency,dk=1/Ts
What does this mean?
Leo Lam © 2010-2013
Sampling
• Graphically:
• In Fourier domain:
• No info loss if no overlap (fully reconstructible)• Reconstruction = Ideal low pass filter
n sss T
nXT
X 21
)(
0
1( )
s
XT
X(w) bandwidth
Leo Lam © 2010-2013
Sampling
• Graphically:
• In Fourier domain:
• Overlap = Aliasing if • To avoid Alisasing:
• Equivalently:
n sss T
nXT
X 21
)(
0
Shannon’s Sampling TheoremNyquist Frequency (min. lossless)
Leo Lam © 2010-2013
Sampling (in time)
• Time domain representation
cos(2100t)100 Hz
Fs=1000
Fs=500
Fs=250
Fs=125 < 2*100
cos(225t)
Aliasing
Frequency wraparound, sounds like Fs=25
(Works in spatial frequency, too!)
Leo Lam © 2010-2013
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
Leo Lam © 2010-2013
Would these alias?
• Remember, no aliasing if• How about:
0 1
0 1 3-3
NO ALIASING!
Leo Lam © 2010-2013
Would these alias?
• Remember, no aliasing if• How about: (hint: what’s the bandwidth?)
Definitely ALIASING!
Y has infinite bandwidth!
Leo Lam © 2010-2013
Would these alias?
• Remember, no aliasing if• How about: (hint: what’s the bandwidth?)
.7s 2 1.0B
-.5 0 .5
0.5B
-.5 0 .5
Copies every .7
-1.5 -.5 .5 1.5
ALIASED!
Leo Lam © 2010-2013
Would these alias?
• Remember, no aliasing if• How about: (hint: what’s the bandwidth?)
.7s 2 1.0B
-.5 0 .5
0.5B
-.5 0 .5
Copies every .7
-1.5 -.5 .5 1.5
ALIASED!
Leo Lam © 2010-2013
How to avoid aliasing?
• We ANTI-alias.
Sample Reconstruct
B
ws > 2wc
time signal
x(t)
X(w)
Anti-aliasingfilter
wc < B
Z(w) z(n)
Leo Lam © 2010-2013
How bad is anti-aliasing?
• Not bad at all.• Check: Energy in the signal (with example)
• Sampled at • Add anti-aliasing (ideal) filter with bandwidth
7
samplerlowpass
anti-aliasingfilter
Leo Lam © 2010-2013
How bad is anti-aliasing?
• Not bad at all.• Check: Energy in the signal (with example)
• Energy of x(t)?
samplerlowpass
anti-aliasingfilter
Leo Lam © 2010-2013
How bad is anti-aliasing?
• Not bad at all.• Check: Energy in the signal (with example)
• Energy of filtered x(t)?
samplerlowpass
anti-aliasingfilter
72
7
1| ( ) |
2fE X d
2
2
1 1 1( ) ( )
1 (1 )(1 ) 1X X
j j j
7
27
1 1 1arctan(7)
2 1fE d
~0.455
Leo Lam © 2010-2013
Bandwidth Practice
• Find the Nyquist frequency for:
-100 0 100
200s
Leo Lam © 2010-2013
Bandwidth Practice
• Find the Nyquist frequency for:
const[rect(w/200)*rect(w/200)] =
-200 200
400s
Leo Lam © 2010-2013
Bandwidth Practice
• Find the Nyquist frequency for:
(bandwidth = 100) + (bandwidth = 50)
300s
Leo Lam © 2010-2013
Summary
• Sampling and the frequency domain representations
• Sampling frequency conditions
top related