S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Electrical
Communications Systems
ECE.09.331
Spring 2011
Lecture 7b
March 2, 2011
Shreekanth Mandayam
ECE Department
Rowan University
http://engineering.rowan.edu/~shreek/spring11/ecomms/
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Plan
• Angle Modulation Systems
• Definitions - Phase and Frequency
Modulation
• Complex Envelope
• Time Domain
• Analyzing FM Signals - Battle Plan!!!!
• Single-tone FM
• Narrowband FM
• Wideband FM
• Bessel Functions
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
ECOMMS: Topics
E le c trica l C o m m u n ica tio n S ys te m s
S ign a ls
D is cre te
S ys te m s
C o n tin u o u s
An a lo g
P ro b a b ility
P o w e r & E ne rg y S ig n a ls
AM
S w itc h in g M o d u la tor
E n ve lo p D e te c tor
In fo rm a tio n
C o n tin u o u s F ou rie r T ra n s fo rm
D S B -S C
P ro d u c t M o d u la tor
C o h e re n t D e te c tor
C o s ta s L o op
E n tro p y
D is c re te F o u rie r Tra n s fo rm
SSB
W e a ve r's M e th od
P h a s in g M e th od
F re q u e n cy M e th od
C h a n n e l C a p a c ity
B a s e ba n d a n d Ba n d pa s s S ig n a ls
F re q u e nc y & P ha s e M od u la tion
N a rro w b a n d /W id e b a nd
V C O & S lo pe D e te c tor
PLL
D ig ital
D ig ita l Co m m T ran s c e iver
B a s eb a nd
CODEC
B a n dp a ss
M O DEM
S o u rc e E n c o d ing
H u ffm a n c o d es
AS K
PSK
FSK
E rro r-c o n tro l E n c o d ing
H a m m in g C o d es
BPSK
S a m p ling
P AM
QPSK
Q u a n tiza tion
PCM
M -a ry P S K
L in e E nc o d ing
Q AM
C o m ple x E nve lo pe
G a u s sia n No is e & S NR
T im e Divis ion M ux
T 1 (D S 1 ) S ta n d a rds
R a n d om V aria b les
N o is e C a lc ula tio ns
P a c ke t S w itch ing
E th e rn et
IS O 7 -La ye r P ro to c ol
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Angle Modulation Systems
Phase Modulation (PM)
Frequency Modulation(FM)
• Signal Representation
Instrument Demo
• Complex Envelope
• Time Domain Representation
• Terminology
•
•
•
•
Phase Sensitivity
Frequency Deviation
Instantaneous Frequency
Phase & Frequency Modulation Indices
Matlab Demo:
anglemod.m
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Analyzing FM Signals Battle Plan!!!
Instrument Demo
Signals
Systems
Time Domain
Performance
Complex Envelope
Transmitters
Modulation
Index
Efficiency
Spectrum
Receivers
• Single-tone FM
• Narrowband FM
Standards
• Wideband FM
• Bessel Functions
Power
Bandwidth
Noise
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Bessel’s Differential Equation
• German mathematician and astronomer Friedrich Wilhelm
Bessel (1784 - 1846)
• Discovered this equation while investigating planetary motion
2
x
2 d y
dx
2
x
dy
2
2
 (x  n ) y  0
dx
• 2nd order ODE, Nonlinear, Variable Coefficients, Homogeneous
• Very important in applied mathematics and engineering
• Governing equation for problems with cylindrical geometries, e.g.
waveguides, vibrating strings, and …………!!!!!!!


2
 E     E

E  E  ˆ  E  ˆ  E z zˆ
2
2

2 d E
d
2

dE 
d
2
 ( 
2
2
 n )E  0
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Bessel Functions
Matlab Demo
» help besselj
BESSELJ Bessel function of the first kind.
J = BESSELJ(NU,Z) is the Bessel function of the
first kind, J_nu(Z).The order NU need not be an
integer, but must be real.The argument Z can be
complex. The result is real where Z is positive.
»
»
»
» x=0:0.1:10;
» plot(x,besselj(0,x));
» title('Bessel Function of Order Zero, J_0(x)');
» xlabel('x');
»
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Bessel Functions
Matlab Demo
%ECOMMS Spring 11 Classroom Demo
%S. Mandayam, ECE, Rowan University
clear;close all;
n=0:6;
beta=0:0.1:10;
Jn=besselj(n,beta');
plot(beta',Jn);
grid on;
xlabel('Frequency Modulation Index: \beta');
ylabel('J_n(\beta)');
legend('J_0(\beta)','J_1(\beta)','J_2(\beta)',
'J_3(\beta)','J_4(\beta)','J_5(\beta)','J_6(\beta)');
title('J_n(\beta): Spectral Amplitudes of an FM signal at
f_c \pm nf_m');
http://engineering.rowan.edu/~shreek/spring11/ecomms/demos/besselfun.m
Instrument Demo
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
FM Signal & Spectrum
Single-tone FM Signal
s(t)  Ac cos[ 2f c t   f sin( 2f mt )]

g (t )  Ac  J n (  )e
j 2nf mt
n  

s (t )  Ac  J n (  ) cos[ 2 ( f c  nf m )t ]
n  
Single-tone FM Spectrum
J1()
|S(f)| / (Ac/2)
J2()
J0()
0
J3()
fc-3fm fc-2fm fc-fm fc fc+fm fc+2fm fc+3fm
f
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Summary