Slide 1

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 2

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 3

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 4

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 5

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 6

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 7

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 8

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 9

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 10

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 11

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 12

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 13

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012


Slide 14

Signals and Systems
EE235

Leo Lam © 2010-2011

Futile
Q: What did the monsterous voltage source say
to the chunk of wire?
A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © 2010-2011

Today’s menu
• Sampling/Anti-Aliasing
• Communications (intro)

Leo Lam © 2010-2011

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
NO ALIASING!

-3
Leo Lam © 2010-2012

0

1

3

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?)

s  .7  2B  1.0
B  0.5
-.5

0

.5

Copies every .7

ALIASED!
-1.5
Leo Lam © 2010-2012

-.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

Leo Lam © 2010-2012

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

Leo Lam © 2010-2012

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)?

Leo Lam © 2010-2012

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
(

)
|
d


• Energy of filtered x(t)?
7
1
1
1
2
X ( ) 
 X ( ) 

1  j
(1  j )(1  j ) 1   2

1
Ef 
2

7

1
1
7 1   2 d   arctan(7)

Leo Lam © 2010-2012

~0.455

Bandwidth Practice
• Find the Nyquist frequency for:

-100

s  200
Leo Lam © 2010-2012

0

100

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

s  400
Leo Lam © 2010-2012

200

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

s  300
Leo Lam © 2010-2012