Leo Lam © 2010-2012

Signals and Systems

EE235Lecture 28

Leo Lam © 2010-2011

Today’s menu

• Sampling/Anti-Aliasing• Communications (intro)

Leo Lam © 2010-2012

Sampling

• Convert a continuous time signal into a series of regularly spaced samples, a discrete-time signal.

• Sampling is multiplying with an impulse train

3

t

t

t

multiply

=0 TS

Leo Lam © 2010-2012

Sampling

• Sampling signal with sampling period Ts is:

• Note that Sampling is NOT LTI

4

)()()(

n

nsss nTtnTxtx

sampler

Leo Lam © 2010-2012

Sampling

• Sampling effect in frequency domain:

• Need to find: Xs(w)• First recall:

5

timeT

Fourier spectra0

1/T

0 02 03002

Leo Lam © 2010-2012

Sampling

• Sampling effect in frequency domain:

• In Fourier domain:

6

distributive property

Impulse train in time impulse train in frequency,dk=1/Ts

What does this mean?

Leo Lam © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

Quick Recap: Would these alias?

• Remember, no aliasing if• How about:

0 1

0 1 3-3

NO ALIASING!

Leo Lam © 2010-2012

Would these alias?

• Remember, no aliasing if• How about: (hint: what’s the bandwidth?)

Definitely ALIASING!

Y has infinite bandwidth!

Leo Lam © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

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 © 2010-2012

Bandwidth Practice

• Find the Nyquist frequency for:

-100 0 100

200s

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Bandwidth Practice

• Find the Nyquist frequency for:

const[rect(w/200)*rect(w/200)] =

-200 200

400s

Leo Lam © 2010-2012

Bandwidth Practice

• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

300s