Slide 1

Signals and Systems
EE235

Leo Lam © 2010-2013


Slide 2

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

Leo Lam © 2010-2013


Slide 3

How to avoid aliasing?
• We ANTI-alias.
time signal

x(t)

Anti-aliasing
filter

X(w)

Sample

Z(w)

B

Leo Lam © 2010-2013

ws > 2wc

wc < B

Reconstruct

z(n)


Slide 4

Bandwidth Practice
• Find the Nyquist frequency for:

-100

 s  200
Leo Lam © 2010-2013

0

100


Slide 5

Bandwidth Practice
• Find the Nyquist frequency for:

const[rect(/200)*rect(/200)] =

-200

 s  400

Leo Lam © 2010-2013

200


Slide 6

Bandwidth Practice
• Find the Nyquist frequency for:

(bandwidth = 100) + (bandwidth = 50)

 s  300

Leo Lam © 2010-2013


Slide 7

Communications
• Practical problem
– One wire vs. hundreds of channels
– One room vs. hundreds of people

• Dividing the wire – how?
– Time
– Frequency
– Orthogonal signals (like CDMA)

Leo Lam © 2010-2013


Slide 8

FDM (Frequency Division Multiplexing)
• Focus on Amplitude Modulation (AM)
• From Fourier Transform:

X

x(t)

y(t)

X()
m(t)=ej0t


Time

Leo Lam © 2010-2013

Y()=X(-0)

0
FOURIER




Slide 9

FDM (Frequency Division Multiplexing)
• Amplitude Modulation (AM)
F()



-5

5



 F ( ) *   (  5)   ( - 5) 
Multiply by cosine!

• Frequency change – NOT LTI!
Leo Lam © 2010-2013


Slide 10

Double Side Band Amplitude Modulation
• FDM – DSB modulation in time domain
x(t)

x(t)+B

y ( t )  [ x ( t )  B ] cos( c t )

Leo Lam © 2010-2013


Slide 11

Double Side Band Amplitude Modulation
• FDM – DSB modulation in freq. domain
y ( t )  [ x ( t )  B ] cos( c t )

• For simplicity, let B=0
Y ( ) 
Y ( ) 
Y ( ) 
X(w)

1
2
1
2
1
2

 X ( )  2  B  ( )      (   c )   ( -  c ) 
X ( )     (   c )   ( -  c ) 

X (   c ) 

0
Leo Lam © 2010-2013

1
2

X ( -  c )
Y(w)

1
!

–!C

0

1/2
!C

!


Slide 12

DSB – How it’s done.
• Modulation (Low-Pass First! Why?)
X1()

x1(t)

1
!

0

cos(w1t)

X2()

y(t)

x2(t)

1
!

0

0
cos(w2t)

X3()

x3(t)

1
0

!
cos(w3t)

Leo Lam © 2010-2013

Y()
!1

1/2
!2

!3

!


Slide 13

DSB – Demodulation
• Band-pass, Mix, Low-Pass

m(t)=cos(0t)
y(t)=x(t)cos(0t)
x
Y()

z(t) = y(t)m(t) = x(t)[cos(0t)]2
= 0.5x(t)[1+cos(20t)]
Z()

-20
-0

0



What assumptions?
-- Matched phase of mod & demod
cosines
-- No noise
-- No delay
-- Ideal LPF
Leo Lam © 2010-2013

20
LPF

X()






Slide 14

DSB – Demodulation (signal flow)
• Band-pass, Mix, Low-Pass
BPF1

LPF

x1(t)

0

!1

1/2 y(t)
!2

!3

BPF2

LPF

x2(t)

!
LPF

cos(3t)

Leo Lam © 2010-2013

!

X2()

1
!

0

cos(2t)
BPF3

1

0

cos(1t)
Y()

X1()

x3(t)

X3()
0

1

!


Slide 15

DSB in Real Life (Frequency Division)
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KARI 550 kHz Day DA2 BLAINE WA US 5.0 kW
KPQ 560 kHz Day DAN WENATCHEE WA US 5.0 kW
KVI 570 kHz Unl ND1 SEATTLE WA US 5.0 kW
KQNT 590 kHz Unl ND1 SPOKANE WA US 5.0 kW
KONA 610 kHz Day DA2 KENNEWICK-RICHLAND-P WA
US 5.0 kW
KCIS 630 kHz Day DAN EDMONDS WA US 5.0 kW
KAPS 660 kHz Day DA2 MOUNT VERNON WA US 10.0
kW
KOMW 680 kHz Day NDD OMAK WA US 5.0 kW
KXLX 700 kHz Day DAN AIRWAY HEIGHTS WA US 10.0
kW
KIRO 710 kHz Day DAN SEATTLE WA US 50.0 kW
Leo Lam © 2010-2013