1. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2011 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring11/ecomms/ Lecture 8a March 8, 2011

2. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan Analyzing FM Signals Single-tone FM Narrowband FM Wideband FM Bessel Functions

3. S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

4. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Angle Modulation Systems Signal Representation Complex Envelope Time Domain Representation Terminology Phase Sensitivity Frequency Deviation Instantaneous Frequency Phase & Frequency Modulation Indices Phase Modulation (PM) Frequency Modulation(FM) Matlab Demo: anglemod.m Instrument Demo

5. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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

6. 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 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 …………!!!!!!!

7. 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'); »

8. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Bessel Functions %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 Matlab Demo Instrument Demo

9. S. Mandayam/ ECOMMS/ECE Dept./Rowan University FM Signal & Spectrum Single-tone FM Signal Single-tone FM Spectrum 0 f c -3f m f c -2f m f c -f m f c f c +f m f c +2f m f c +3f m |S(f)| / (A c /2) f J0()J0() J1()J1() J2()J2() J3()J3()

10. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary