1. Electrical Communications Systems ECE.09.433 Spring 2019
Lecture 7a March 5, 2019
Shreekanth Mandayam
ECE Department
Rowan University

2. Plan Angle Modulation Systems Analyzing FM Signals - Battle Plan!!!!
Single-tone FM
Narrowband FM
Wideband FM
Bessel Functions

3. ECOMMS: Topics

4. Analyzing FM Signals - Battle Plan!!!
Instrument Demo
Signals Systems
Time Domain
Complex Envelope
Spectrum
Single-tone FM
Narrowband FM
Wideband FM
Bessel Functions
Power
Performance
Transmitters
Receivers
Standards
Modulation
Index
Efficiency
Bandwidth
Noise

5. Bessel’s Differential Equation
German mathematician and astronomer Friedrich Wilhelm Bessel ( )
Discovered this equation while investigating planetary motion
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 …………!!!!!!!

6. 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');

7. Bessel Functions Matlab Demo Instrument Demo
%ECOMMS Spring 19 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');
Instrument Demo

8. FM Signal & Spectrum Single-tone FM Signal Single-tone FM Spectrum
fc-3fm fc-2fm fc-fm fc fc+fm fc+2fm fc+3fm
|S(f)| / (Ac/2) f
J0(b)
J1(b)
J2(b)
J3(b)

9. Summary