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