Transcript Signals and Systems EE235 Lecture 29 Leo Lam © 2010-2012 Today’s menu • The lost Sampling slides • Communications (intro) Leo Lam © 2010-2012 Sampling • Convert a continuous.

Signals and Systems
EE235
Lecture 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
t
multiply
=
0
t
TS
t
Leo Lam © 2010-2012
3
Sampling
• Sampling signal with sampling period Ts is:
xs (t ) 
n 
 x(nT ) (t  nT )
n  
s
s
• Note that Sampling is NOT LTI
sampler
Leo Lam © 2010-2012
4
Sampling
• Sampling effect in frequency domain:
• Need to find: Xs(w)
• First recall:
T
1/T
 2w0
Leo Lam © 2010-2012
 w0
0
time
w0
2w0
3w0
Fourier spectra
5
Sampling
• Sampling effect in frequency domain:
• In Fourier domain:
Impulse train in time
 impulse train in
frequency,
dk=1/Ts
distributive
What does
this
property
mean?
Leo Lam © 2010-2012
6
Sampling
• Graphically:
1
X s (w ) 
Ts

2

X
w

n
 
Ts
n  


• In Fourier domain:



1
X (w
X()w)
Ts
bandwidth
0
• No info loss if no overlap (fully reconstructible)
• Reconstruction = Ideal low pass filter
Leo Lam © 2010-2012
Sampling
• Graphically:
1
X s (w ) 
Ts

2

X
w

n
 
Ts
n  





• In Fourier domain:
0
• Overlap = Aliasing if
• To avoid Alisasing:
Nyquist Frequency (min. lossless)
• Equivalently:
Leo Lam © 2010-2012
Shannon’s Sampling Theorem
Sampling (in time)
• Time domain representation
cos(2100t)
100 Hz
Fs=1000
Fs=500
Fs=250
Fs=125 < 2*100
cos(225t)
Frequency wraparound,
sounds like Fs=25
(Works in spatial
frequency, too!)
Leo Lam © 2010-2012
Aliasing
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:
X
x(t)
y(t)
X(w)
m(t)=ejw0t
w
Time
Leo Lam © 2010-2012
Y(w)=X(ww0)
w0
FOURIER
w
FDM (Frequency Division Multiplexing)
• Amplitude Modulation (AM)
F(w)
w
-5
5
w
 F (w)* (w  5)   (w  5)
Multiply by cosine!
• Frequency change – NOT LTI!
Leo Lam © 2010-2012
Double Side Band Amplitude Modulation
• FDM – DSB modulation in time domain
x(t)
x(t)+B
y(t )  [ x(t )  B]cos(wct )
Leo Lam © 2010-2012
Double Side Band Amplitude Modulation
• FDM – DSB modulation in freq. domain
y(t )  [ x(t )  B]cos(wct )
• For simplicity, let B=0
1
Y (w ) 
 X (w )  2 B (w )    (w  wc )   (w  wc )
2
1
Y (w ) 
X (w )    (w  wc )   (w  wc ) 
2
1
1
Y (w )  X (w  wc )  X (w  wc )
2
2
X(w)
0
Leo Lam © 2010-2012
Y(w)
1
!
–!C
0
1/2
!C
!
DSB – How it’s done.
• Modulation (Low-Pass First! Why?)
X1(w)
x1(t)
1
!
0
cos(w1t)
X2(w)
y(t)
x2(t)
1
!
0
0
cos(w2t)
X3(w)
x3(t)
1
0
!
cos(w3t)
Leo Lam © 2010-2012
Y(w)
!1
1/2
!2
!3
!
DSB – Demodulation
• Band-pass, Mix, Low-Pass
m(t)=cos(w0t)
y(t)=x(t)cos(w0t)
x
Y(w)
z(t) = y(t)m(t) = x(t)[cos(w0t)]2
= 0.5x(t)[1+cos(2w0t)]
Z(w)
2w0
w0
w0
w
What assumptions?
-- Matched phase of mod & demod
cosines
-- No noise
-- No delay
-- Ideal LPF
Leo Lam © 2010-2012
2w0
LPF
X(w)
w
w
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(w3t)
Leo Lam © 2010-2012
!
X2(w)
1
!
0
cos(w2t)
BPF3
1
0
cos(w1t)
Y(w)
X1(w)
x3(t)
X3(w)
0
1
!
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
Leo Lam © 2010-2012