Leo Lam © 2010-2013

Signals and Systems

EE235

Leo Lam © 2010-2013

Today’s menu

• Sampling/Anti-Aliasing• Communications (intro)

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

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

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

FDM (Frequency Division Multiplexing)

• Focus on Amplitude Modulation (AM)• From Fourier Transform:

Xx(t)

m(t)=ejw0t

y(t)

Y(w)=X( -w w0)

w0 w

X(w)

w

Time FOURIER

Leo Lam © 2010-2013

FDM (Frequency Division Multiplexing)

• Amplitude Modulation (AM)

• Frequency change – NOT LTI!

w-5 5

w

F( )w

( )* ( 5) ( 5)F

Multiply by cosine!

Leo Lam © 2010-2013

Double Side Band Amplitude Modulation

• FDM – DSB modulation in time domain

( ) [ ( ) ]cos( )cy t x t B t

x(t)+B

x(t)

Leo Lam © 2010-2013

Double Side Band Amplitude Modulation

• FDM – DSB modulation in freq. domain

• For simplicity, let B=0

1( ) ( ) 2 ( ) ( ) ( )

2 c cY X B

1( ) ( ) ( ) ( )

2 c cY X

1 1( ) ( ) ( )

2 2c cY X X

( ) [ ( ) ]cos( )cy t x t B t

!0

X(w)1

!–!C !C0

1/2Y(w)

Leo Lam © 2010-2013

DSB – How it’s done.

• Modulation (Low-Pass First! Why?)

y(t)

!1 !0 !2 !3

1/2Y(w)

!0

!0

!0

X3(w)

X1(w)1

1

1

X2(w) x2(t)

x1(t)

x3(t)

cos(w3t)

cos(w1t)

cos(w2t)

Leo Lam © 2010-2013

DSB – Demodulation

• Band-pass, Mix, Low-Pass

xy(t)=x(t)cos(w0t)

m(t)=cos(w0t) z(t) = y(t)m(t) = x(t)[cos(w0t)]2

= 0.5x(t)[1+cos(2w0t)]

w0-w0

2w0-2w0

LPF

Y(w)Z(w)

X(w)

w

w

w

What assumptions? -- Matched phase of mod & demod cosines -- No noise -- No delay -- Ideal LPF

Leo Lam © 2010-2013

DSB – Demodulation (signal flow)

• Band-pass, Mix, Low-Pass

LPF

LPF

LPF

BPF1

BPF2

BPF3

!1 !0 !2 !3

1/2Y(w)

!0

!0

!0

X3(w)

X1(w)1

1

1

X2(w)

cos(w1t)

cos(w2t)

cos(w3t)

y(t)

x1(t)

x3(t)

x2(t)

Leo Lam © 2010-2013

DSB in Real Life (Frequency Division)

• 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