Chapter 4 Angle Modulation and Multiplexing Contents 4.1 Angle Modulation................... 4-3 4.1.1 Narrowband Angle Modulation....... 4-5 4.1.2 Spectrum of an Angle-Modulated Signal.. 4-7 4.1.3 Power in an Angle-Modulated Signal.... 4-13 4.1.4 Bandwidth of Angle-Modulated Signals... 4-13 4.1.5 Narrowband-to-Wideband Conversion.... 4-19 4.1.6 Demodulation of Angle-Modulated Signals. 4-20 4.2 Feedback Demodulators............... 4-30 4.2.1 Phase-Locked Loops for FM Demodulation. 4-30 4.2.2 PLL Frequency Synthesizers......... 4-50 4.2.3 Frequency-Compressive Feedback...... 4-54 4.2.4 Coherent Carrier Recovery for DSB Demod- ulation.................... 4-56 4.3 Interference and Preemphasis............. 4-60 4.3.1 Interference in Angle Modulation...... 4-60 4.3.2 The Use of Preemphasis in FM........ 4-64 4.4 Multiplexing..................... 4-65 4-1 CONTENTS 4.4.1 Frequency-Division Multiplexing (FDM).. 4-66 4.4.2 Quadrature Multiplexing (QM)........ 4-69 4.5 General Performance of Modulation Systems in Noise4-70 4-2 ECE 5625 Communication Systems I 4.1. ANGLE MODULATION Continuing from Chapter 3, we now focus on the .t/ term (angle) in the general modulated carrier waveform xc.t/ A.t/ cos 2fct .t/ D C 4.1 Angle Modulation A general angle modulated signal is of the form xc.t/ Ac cosŒ!ct .t/ D C Definition: Instantaneous phase of xc.t/ is i .t/ !ct .t/ D C where .t/ is the phase deviation Define: Instantaneous frequency of xc.t/ is di .t/ d.t/ !i .t/ !c D dt D C dt where d.t/=dt is the frequency deviation There are two basic types of angle modulation 1. Phase modulation (PM) .t/ kp m.t/ D „ƒ‚… phase dev. const. which implies xc.t/ Ac cosŒ!ct kpm.t/ D C ECE 5625 Communication Systems I 4-3 CONTENTS – Note: the units of kp is radians per unit of m.t/ – If m.t/ is a voltage, kp has units of radians/volt 2. Frequency modulation (FM) d.t/ kf m.t/ dt D „ƒ‚… freq. dev. const. or Z t .t/ kf m.˛/ d˛ 0 D t0 C – Note: the units of kf is radians/sec per unit of m.t/ – If m.t/ is a voltage, kf has units of radians/sec/volt – An alternative expression for kf is kf 2fd D where fd is the frequency-deviation constant in Hz/unit of m.t/ Example 4.1: Phase and Frequency Step Modulation Consider m.t/ u.t/ v D We form the PM signal xPM.t/ Ac cos !ct kpu.t/ ; kp =3 rad/v D C D We form the FM signal h Z t i xFM.t/ Ac cos !ct 2fd m.˛/ d˛ ; fd 3 Hz/v D C D 4-4 ECE 5625 Communication Systems I 4.1. ANGLE MODULATION π/3 phase step at t = 0 3 Hz frequency step at t = 0 f f f f + 3 Hz c 1 c c 1 c tt 1111 1 1 Phase Modulation Frequency Modulation Phase and frequency step modulation 4.1.1 Narrowband Angle Modulation Begin by writing an angle modulated signal in complex form  à j!ct j.t/ xc.t/ Re Ace e D Expand ej.t/ in a power series Â Ä 2.t/ à j!ct xc.t/ Re Ace 1 j.t/ D C 2Š The narrowband approximation is .t/ 1, then j j  à j!ct j!ct xc.t/ Re Ace jAc.t/e ' C Ac cos.!ct/ Ac.t/ sin.!ct/ D ECE 5625 Communication Systems I 4-5 CONTENTS Under the narrowband approximation we see that the signal is similar to AM except it is carrier plus modulated quadrature carrier − NBFM φ(t) xc(t) + ω o Ac sin( ct) 90 NBFM modulator block diagram Example 4.2: Single tone narrowband FM Consider NBFM with m.t/ Am cos !mt D Z t .t/ 2fd Am cos !m˛ d˛ D 2fd fd Am sin !mt Am sin !mt D 2fm D fm Now,  à fd xc.t/ Ac cos !ct Am sin !mt D C fm  à fd Ac cos !ct Am sin !mt sin !ct ' fm Amfd Ac cos !ct sin.fc fm/t D C 2fm C Amfd sin.fc fm/t 2fm This looks very much like AM 4-6 ECE 5625 Communication Systems I 4.1. ANGLE MODULATION f - f c m f f f + f 0 c c m Single tone NBFM spectra 4.1.2 Spectrum of an Angle-Modulated Signal The development in this obtains the exact spectrum of an angle modulated carrier for the case of .t/ ˇ sin !mt D where ˇ is the modulation index for sinusoidal angle modula- tion The transmitted signal is of the form xc.t/ Ac cos !ct ˇ sin !mt D C ˚ j!ct jˇ sin !mt « AcRe e e D jˇ sin !mt Note that e is periodic with period T 2=!m, thus D we can obtain a Fourier series expansion of this signal, i.e., 1 jˇ sin !mt X j n!mt e Yne D n D1 ECE 5625 Communication Systems I 4-7 CONTENTS The coefficients are ! Z =!m m jˇ sin !mt j n!mt Yn e e dt D 2 =!m ! Z =!m m j.n!mt ˇ sin !mt/ e dt D 2 =!m Change variables in the integral by letting x !mt, then dx D D !mdt, t =!m x , and t =!m x D ! D D ! D With the above substitutions, we have Z 1 j.nx ˇ sin x/ Yn e dx D 2 1 Z cos.nx ˇ sin x/ dx Jn.ˇ/ D 0 D which is a Bessel function of the first kind order n with argu- ment ˇ Jn.ˇ/ Properties Recurrence equation: 2n Jn 1.ˇ/ Jn.ˇ/ Jn 1.ˇ/ C D ˇ n – even: J n.ˇ/ Jn.ˇ/ D n – odd: J n.ˇ/ Jn.ˇ/ D 4-8 ECE 5625 Communication Systems I 4.1. ANGLE MODULATION J (β) 1 0 0.8 J (β) 0.6 1 β J2( ) β J3( ) 0.4 0.2 β 246810 0.2 0.4 Bessel function of order 0–3 plotted The zeros of the Bessel functions are important in spectral analysis First five Bessel function zeros for order 0 – 5 β J0( ) = 0 2.40483, 5.52008, 8.65373, 11.7915, 14.9309 β J1( ) = 0 3.83171, 7.01559, 10.1735, 13.3237, 16.4706 β J2( ) = 0 5.13562, 8.41724, 11.6198, 14.796, 17.9598 β J3( ) = 0 6.38016, 9.76102, 13.0152, 16.2235, 19.4094 β J4( ) = 0 7.58834, 11.0647, 14.3725, 17.616, 20.8269 β J5( ) = 0 8.77148, 12.3386, 15.7002, 18.9801, 22.2178 ECE 5625 Communication Systems I 4-9 CONTENTS Spectrum cont. We obtain the spectrum of xc.t/ by inserting the series repre- sentation for ejˇ sin !mt " # 1 j!ct X j n!mt xc.t/ AcRe e Jn.ˇ/e D n D1 X1 Ac Jn.ˇ/ cos .!c n!m/t D n C D1 We see that the amplitude spectrum is symmetrical about fc due to the symmetry properties of the Bessel functions β β |AcJ-1( )| |AcJ1( )| β |A J (β)| |AcJ-2( )| β c 2 |AcJ0( )| β |A J (β)| |AcJ-3( )| c 3 β β |A J ( )| |AcJ4( )| (one-sided) β c -4 β |AcJ-5( )| |AcJ5( )| Amplitude Spectrum f m m m m m m m m m m f f f f f fc f f f f f - + c c - 5 - 4 - 3 - 2 f + 2 + 3 + 4 + 5 c c c c f c c c c f f f f f f f f For PM ˇ sin !mt kp .A sin !mt/ D „ ƒ‚ … m.t/ ˇ kpA ) D For FM Z t fd ˇ sin !mt kf A cos !m˛ d˛ A sin !mt D D fm ˇ .fd =fm/A ) D 4-10 ECE 5625 Communication Systems I 4.1. ANGLE MODULATION When ˇ is small we have the narrowband case and as ˇ gets larger the spectrum spreads over wider bandwidth 1 β = 0.2, Ac = 1 0.8 (narrowband case) 0.6 0.4 Spectrum Amplitude 0.2 (f - f )/f -5 0 5 10 c m 1 β = 1, A = 1 0.8 c 0.6 0.4 Spectrum Amplitude 0.2 (f - f )/f -5 0 5 10 c m 1 β 0.8 = 2.4048, Ac = 1 0.6 (carrier null) 0.4 Spectrum Amplitude 0.2 (f - f )/f -5 0 5 10 c m 1 β 0.8 = 3.8317, Ac = 1 0.6 (1st sideband null) 0.4 Spectrum Amplitude 0.2 (f - f )/f -5 0 5 10 c m 1 β 0.8 = 8, Ac = 1 0.6 (spectrum becoming 0.4 wide) Spectrum Amplitude 0.2 (f - f )/f -5 0 5 10 c m The amplitude spectrum relative to fc as ˇ increases ECE 5625 Communication Systems I 4-11 CONTENTS Example 4.3: VCO FM Modulator Consider again single-tone FM, that is m.t/ A cos.2fmt/ D We assume that we know fm and the modulator deviation con- stant fd Find A such that the spectrum of xc.t/ contains no carrier com- ponent An FM modulator can be implemented using a voltage con- trolled oscillator (VCO) sensitivity fd MHz/v m(t) VCO Center Freq = fc A VCO used as an FM modulator The carrier term is AcJ0.ˇ/ cos !ct We know that J0.ˇ/ 0 for ˇ 2:4048; 5:5201; : : : D D The smallest ˇ that will make the carrier component zero is f ˇ 2:4048 d A D D fm which implies that we need to set f A 2:4048 m D fd 4-12 ECE 5625 Communication Systems I 4.1.