Electrical Communications Systems ECE.09.433 Spring 2019 Lecture 6b February 28, 2019 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring19/ecomms/

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

ECOMMS: Topics

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

Bessel’s Differential Equation German mathematician and astronomer Friedrich Wilhelm Bessel (1784 - 1846) 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 …………!!!!!!!

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');

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'); http://engineering.rowan.edu/~shreek/spring19/ecomms/demos/besselfun.m Instrument Demo

FM Signal & Spectrum Single-tone FM Signal Single-tone FM Spectrum 0 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)

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