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ECE 4371, Fall, 2014
Introduction to Telecommunication
Engineering/Telecommunication Laboratory
Zhu Han
Department of Electrical and Computer Engineering
Class 4
Sep. 8th, 2014
Overview

Homework
– 4.2.1, 4.2.3, 4.2.5, 4.2.7, 4.2.9
– 4.3.3, 4.3.8
– 4.4.2, 4.4.4
– 4.5.2
– 4.8.1
– Due 9/22/14

Phase-locked loop

FM basics
Carrier Recover Error

DSB: e(t)=2m(t)cos(wct)cos((wc+ w)t+)
e(t)=m(t) cos((w)t+)
– Phase error: if fixed, attenuation. If not, shortwave radio
– Frequency error: catastrophic beating effect

SSB, only frequency changes, f<30Hz.
– Donald Duck Effect

Crystal oscillator, atoms oscillator, GPS, …

Pilot: a signal, usually a single frequency, transmitted over a
communications system for supervisory, control, equalization,
continuity, synchronization, or reference purposes.
Phase-Locked Loop

Can be a whole course. The most important part of receiver.

Definition: a closed-loop feedback control system that generates and outputs
a signal in relation to the frequency and phase of an input ("reference") signal

A phase-locked loop circuit responds both to the frequency and phase of the
input signals, automatically raising or lowering the frequency of a controlled
oscillator until it is matched to the reference in both frequency and phase.
Voltage Controlled Oscillator (VCO)

W(t)=wc+ce0(t), where wc is the free-running frequency

Example
Ideal Model

Model
LPF
VCO
– Si=Acos(wct+1(t)), Sv=Avcos(wct+c(t))
– Sp=0.5AAv[sin(2wct+1+c)+sin(1-c)]
– So=0.5AAvsin(1-c)=AAv(1-c)

Capture Range and Lock Range
Carrier Acquisition in DSB-SC

Signal Squaring method

Costas Loop
v1 ( t ) 
2
1
2
Ac Al m ( t ) cos  ,
2
v 2 (t ) 
1

1
 1
v 3 ( t )   Ac Al m ( t )  cos  sin    Ac Al m ( t ) 
sin 2
2

2
 2

SSB-SC not working
1
2
Ac Al m ( t ) sin 
v 4 ( t )  K sin 2
Costas receiver
PLL Applications

Clock recovery: no pilot

Deskewing: circuit design

Clock generation: Direct Digital Synthesis

Spread spectrum:

Jitter Noise Reduction

Clock distribution
FM Basics

VHF (30M-300M) high-fidelity broadcast

Wideband FM, (FM TV), narrow band FM (two-way radio)

1933 FM and angle modulation proposed by Armstrong, but
success by 1949.

Digital: Frequency Shift Key (FSK), Phase Shift Key (BPSK,
QPSK, 8PSK,…)

AM/FM: Transverse wave/Longitudinal wave
Angle Modulation vs. AM

Summarize: properties of amplitude modulation
– Amplitude modulation is linear

just move to new frequency band, spectrum shape does not
change. No new frequencies generated.
– Spectrum: S(f) is a translated version of M(f)
– Bandwidth ≤ 2W

Properties of angle modulation
– They are nonlinear

spectrum shape does change, new frequencies generated.
– S(f) is not just a translated version of M(f)
– Bandwidth is usually much larger than 2W
Angle Modulation Pro/Con Application

Why need angle modulation?
– Better noise reduction
– Improved system fidelity

Disadvantages
– Low bandwidth efficiency
– Complex implementations

Applications
– FM radio broadcast
– TV sound signal
– Two-way mobile radio
– Cellular radio
– Microwave and satellite communications
Instantaneous Frequency
•Angle modulation has two forms
- Frequency modulation (FM): message is represented as the
variation of the instantaneous frequency of a carrier
- Phase modulation (PM): message is represented as the
variation of the instantaneous phase of a carrier
s ( t )  Ac cos  i ( t )  ,
w here Ac : carrier am plitude,  i ( t ) : angle ( phase)
f i (t ) 
1 d  i (t )
2
s ( t )  Ac cos  2  f c t   ( t ) 
w here  ( t ) is a function of m essage signa l m ( t ).
dt
Phase Modulation

PM (phase modulation) signal
s ( t )  Ac cos  2  f c t  k p m ( t ) 
 ( t )  k p m ( t ),
k p : phase sensitivity
instantanous frequency f i ( t )  f c 
k p dm ( t )
2
dt
Frequency Modulation

FM (frequency modulation) signal
s ( t )  Ac cos  2  f c t  2  k f


m
(

)
d

0

t
k f : freq u en cy sen sitivity
in stan tan o u s freq u en cy f i ( t )  f c  k f m ( t )
an g le  i ( t )  2 

t
0
f i ( ) d 
 2 f c t  2 k f
(Assume zero initial phase)

t
0
m ( ) d 
FM Characteristics

Characteristics of FM signals
– Zero-crossings are not regular
– Envelope is constant
– FM and PM signals are similar
Relations between FM and PM
FM o f m ( t )
P M of m ( t )


PM of
FM of

t
m ( ) d 
0
dm ( t )
dt
FM/PM Example (Time)
FM/PM Example (Frequency)
Matlab
fc=1000; Ac=1; % carrier frequency (Hz) and magnitude
fm=250; Am=0.1; % message frequency (Hz) and magnitude
k=4;
% modulation parameter
% generage single tone message signal
t=0:1/10000:0.02;
% time with sampling at 10KHz
mt=Am*cos(2*pi*fm*t); % message signal
% Phase modulation
sp=Ac*cos(2*pi*fc*t+2*pi*k*mt);
% Frequency modulation
dmt=Am*sin(2*pi*fm*t);
% integration
sf=Ac*cos(2*pi*fc*t+2*pi*k*dmt); % PM
% Plot the signal
subplot(311), plot(t,mt,'b'), grid, title('message m(t)')
subplot(312), plot(t,sf,'r'), grid, ylabel('FM s(t)')
subplot(313), plot(t,sp,'m'), grid, ylabel('PM s(t)')
Matlab
% spectrum
w=((0:length(t)-1)/length(t)-0.5)*10000;
Pm=abs(fftshift(fft(mt))); % spectrum of message
Pp=abs(fftshift(fft(sp))); % spectrum of PM signal
Pf=abs(fftshift(fft(sf))); % spectrum of FM signal
% plot the spectrums
figure(2)
subplot(311), plot(w,Pm,'b'),
axis([-3000 3000 min(Pm) max(Pm)]), grid, title('message spectrum M(f)'),
subplot(312), plot(w,Pf,'r'),
axis([-3000 3000 min(Pf) max(Pf)]), grid, ylabel('FM S(f)')
subplot(313), plot(w,Pp,'m'),
axis([-3000 3000 min(Pp) max(Pp)]), grid, ylabel('PM S(f)')
Frequency Modulation

FM (frequency modulation) signal
s ( t )  Ac cos  2  f c t  2  k f


m
(

)
d

0

t
k f : freq u en cy sen sitivity
in stan tan o u s freq u en cy f i ( t )  f c  k f m ( t )
an g le  i ( t )  2 

t
0
f i ( ) d 
 2 f c t  2 k f
m ( t )  Am cos(2 f m t )
(Assume zero initial phase)

t
m ( ) d 
0
f i  f c  k f Am cos(2  f m t )
d  2 k f
1 d
1 d  2 f c t 
1

fi 


2  dt
2
dt
2
 fc 
1
2
2  k f  Am cos(2  f m ) 
Let   t

t
0
Am cos(2  f m ) d  

dt
Example
Consider m(t)- a square wave- as shown. The FM wave for this m(t) is
t
shown below.
 FM ( t )  A cos(  c t  k f
 m(  )d  ).
-
t
Assume m(t) starts at t  0. For 0  t  T
m(t)  1 ,
2
 m(  )d   t
and
0
t
for
T
2
 t  T
m(t)  - 1
,
ous frequency
t
2
 m(  )d    m(  )d    m(  )d  
0
The instantane
T
T
2
- (t - T )  T - t.
2
T
0
2
is  i ( t )   c  k f m ( t )   c  k f
for
and  i ( t )   c  k f for T  t  T .
2
 i max   c  k f
and
 i min   c  k f
m(t)
0
T
2T
t 
 FM ( t )
t 
0  t  T
2
Frequency Deviation

Frequency deviation Δf
– difference between the maximum instantaneous and carrier frequency
 f  k f Am  k f m ax | m ( t ) |
– Definition:
– Relationship with instantaneous frequency
single-tone m ( t ) case: f i  f c   f cos(2  f m t )
general case:
fc  f  fi  fc  f
– Question: Is bandwidth of s(t) just 2Δf?
No, instantaneous frequency is not
equivalent to spectrum frequency
(with non-zero power)!
S(t) has ∞ spectrum frequency
(with non-zero power).
Modulation Index

Indicate by how much the modulated variable (instantaneous
frequency) varies around its unmodulated level (message
frequency)
A M (envelope):
m ax | k a m ( t ) |
,
1
A
FM (frequency):  
m ax | k f m ( t ) |
fm

Bandwidth
a (t ) 

t

m ( ) d 
2
2


kf 2
kf 3
 ( t )  Re(  ( t ))  A  cos w c t  k f a ( t ) sin w c t 
a ( t ) cos w c t 
a ( t ) sin w c t ... 
2!
3!


Narrow Band Angle Modulation
Definition
k f a ( t )  1
Equation  ( t )  A cos w c t  k f a ( t ) sin w c t 
Comparison with AM
Only phase difference of Pi/2
Frequency: similar
Time: AM: frequency constant
FM: amplitude constant
Conclusion: NBFM signal is
similar to AM signal
NBFM has also bandwidth
2W. (twice message signal
bandwidth)
Example
Block diagram of a method for generating a narrowband FM
signal.
A phasor comparison of narrowband FM and AM waves for
sinusoidal modulation. (a) Narrowband FM wave. (b) AM wave.
Wide Band FM

Wideband FM signal
m ( t )  Am cos(2  f m t )
s ( t )  Ac cos  2  f c t   sin(2  f m t ) 

Fourier series representation

s ( t )  Ac

J n (  ) cos  2  ( f c  nf m ) t 
n  
S( f ) 
Ac
2


J n (  )   ( f  f c  nf m )   ( f  f c  nf m ) 
n  
J n (  ) : n -th order Bessel function of the firs t kind
Example
Bessel Function of First Kind
1 . J n (  )  (  1) J  n (  )
n
2 . If  is sm all, th en J 0 (  )  1,
J1(  ) 

Jn( )  0

3.

n  
J n ( )  1
2
,
2
fo r all n  2
Spectrum of WBFM (Chapter 5.2)

Spectrum when m(t) is single-tone
s ( t )  Ac cos  2  f c t   sin(2  f m t )   Ac


J n (  ) cos  2  ( f c  nf m ) t 
n  
S( f ) 
Ac
2



n  
Example 2.2
J n (  )   ( f  f c  nf m )   ( f  f c  nf m ) 
Spectrum Properties
1. frequencies: f c , f c  f m , f c  2 f m ,
(for all n ).
, f c  nf m ,
T heoretically infinite bandw idth.
2. For  << 1 (N BFM ), frequency: f c , f c  f m
J 0 (  )  1, J 1 (  )   J  1 (  ), J n (  )  0 for all n  2
3. M agnitude of  f c  nf m :
Ac
2
J n (  ), depend on 
4. C arrier ( f c ) magnitude J 0 (  ) can be 0 for some 

5. A verage pow er: P 

n  
A 
2
c
1
2
J ( ) 
2
n
1
2
2
Ac
Bandwidth of FM

Facts
– FM has side frequencies extending to infinite frequency 
theoretically infinite bandwidth
– But side frequencies become negligibly small beyond a point
 practically finite bandwidth
– FM signal bandwidth equals the required transmission
(channel) bandwidth

Bandwidth of FM signal is approximately by
– Carson’s Rule (which gives lower-bound)
Carson’s Rule

Nearly all power lies within a bandwidth of
– For single-tone message signal with frequency fm
BT  2  f  2 f m  2(   1) f m
– For general message signal m(t) with bandwidth (or highest
frequency) W
BT  2  f  2W  2( D  1)W
w h ere D 
f
is d eviatio n ratio (eq u ivalen t to  ),
W
 f  m ax  k f m ( t ) 
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