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EE354 : Communications System I. Lecture 17,18: Phase Modulations Aliazam Abbasfar. Outline. Summary of amplitude modulations Phase Modulation FM/PM. Amplitudes modulations - summary. Modulates the carrier amplitude Frequency spectrum shifted to f c W< Bandwidth < 2W
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EE354 : Communications System I Lecture 17,18: Phase Modulations Aliazam Abbasfar
Outline • Summary of amplitude modulations • Phase Modulation • FM/PM
Amplitudes modulations - summary • Modulates the carrier amplitude • Frequency spectrum shifted to fc • W< Bandwidth < 2W • Linear modulation • Transmitted signal • xo(t) = Am x(t) cos(wct) + Am x’(t) sin(wct) + Accos(wct) • Coherent demodulation • Strong carrier helps simplify the receiver • AM/SSB • Peak detection for demodulation • Good for broadcasting
Non-linear modulations • Phase and frequency modulation • Transmitted signal • xo(t) = Accos(wct + f(t)) = Re[ Ac exp( jwct + jf(t) ) ] • Constant envelope, but time-varying phase • A(t) = Ac , Q(t)= f(t) • PXo = Pc • Phase modulation (PM) • f(t)= fD x(t) • fD : phase deviation • Frequency modulation (FM) • f(t)= fc + 1/2pdf(t)/dt • f(t) = fc + fD x(t) • fD x(t) << fc • fD : frequency deviation • PM: f(t) =? • PM and FM modulators are interchangeable • Zero crossings are not periodic
Narrowband PM/FM • Narrowband PM/FM (NBPM/NBFM) • f(t) << 1 rad • xI(t) = Ac , xQ(t)= Acf(t) • Xlp(f) = Acd(f) + AcF(f) • NBPM : F(f) = fD X(f)/f • NBFM : F(f) = -j fD X(f)/f • Modulated BW = 2 W • Example : sinc(2Wt) • X(f) = rect(f/2W)/2W • What if we include 2nd terms too? • xI(t) = Ac(1- f2(t)/2 ) , xQ(t)= Acf(t)
Tone modulation (FM) • x(t) = Amcos(wmt) • f(t) = Am fD/fm sin (wmt) = b sin (wmt) • b = Am fD/fm • b indicates maximum phase change • xlp(t) = Ac exp( j b sin (wmt)) • Periodic with fundamental frequency of fm • c(n fm) = Jn(b) • nth order Bessel function with argument b • J-n(b) = (-1)nJn(b) • Modulated signal • Carrier frequency line • Infinite # of sidebands lines
FM bandwidth • If (n/b), then Jn(b) << 1 • BW is a function of b • NBFM • b < 0.2 • Only J0 and J1 • Acceptable distortion • PB / PT > 0.98 • N = (b +1) • B = 2(b +1)fm • Arbitrary signal • b = Am fD/fm • D = xmaxfD/fmax • Carlson’s rule • B = 2(1+ D)W • If |x(t)|< 1, B = 2(fD+W) • NBFM : B = 2W • WBFM : B = 2fD
WBFM spectrum • X(t) is a random signals with pdf of fX(x) • X(t) = x Xo(t) = Accos( 2p(fc +fDx)t ) • f = fc +fDx • dp = PCfX(x) dx • GXo(f) = PC/2 fDfX( (f-fc)/fD) ; f>0 • Example • Gaussian message source • B = 4.66 sfD
Distortion • xo(t) = Accos(wct + f(t)) • Linear distortion • Amplitude distortion • FM to AM conversion • Not a big problem • Phase distortion • Distorts message • Should be equalized • Non-linear distortion • y(t) = A0 + A1cos(wct + f(t)) + A2cos(2wct + 2f(t)) • Distortion can be filtered out • FM/PM is resistant to non-linear distortion • Use clipping to mitigate FM to AM conversion
FM Modulator • Voltage controlled oscillator (VCO) • Oscillation frequency is proportional to input voltage • C = C0 – kx(t) • f = fc + k fc/2C0 • Indirect modulation • Use NBFM modulator • fc1 = 200KHz, fD = 25 Hz • Frequency multiplier • n = 3000 fc1 = 600 MHz, fD = 75 MHz • Mixer • fLO = 500 MHz fc = 100 MHz
FM demodulation • Discriminator • Frequency to amplitude (voltage) conversion • |H(f)| = V0 + k(f-fc) • A differentiator • AM demodulation • Needs a limiter to regulate Ac
FM demodulation • Balanced discriminator • Wider linear range • WBFM • No DC block is needed • Phase locked loop (PLL)
Reading • Carlson Ch. 5.1, 5.2 • Proakis & Salehi 3.3
