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Leo Lam © 2010-2012
Signals and Systems
EE235Lecture 29
Leo Lam © 2010-2012
Today’s menu
• The lost Sampling slides• 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
Onto…
• Communications (intro)
Leo Lam © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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 © 2010-2012
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
