Lecture 17,18: Phase Modulations

Aliazam Abbasfar

OutlineSummary 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(ct) + Am x’(t) sin(ct) + Ac cos(ct)

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) = Ac cos(ct + (t)) = Re[ Ac exp( jct + j(t) ) ]

Constant envelope, but time-varying phase A(t) = Ac , (t)= (t) PXo = Pc

Phase modulation (PM) (t)= x(t) : phase deviation

Frequency modulation (FM) f(t)= fc + 1/2 d(t)/dt f(t) = fc + f x(t)

f x(t) << fc f : frequency deviation

PM: f(t) =? PM and FM modulators are interchangeable

Zero crossings are not periodic

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Narrowband PM/FMNarrowband PM/FM (NBPM/NBFM)

(t) << 1 rad xI(t) = Ac , xQ(t)= Ac (t) Xlp(f) = Ac (f) + Ac (f)

NBPM : (f) = X(f)/f NBFM : (f) = -j f 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- (t)/2 ) , xQ(t)= Ac (t)

Tone modulation (FM) x(t) = Am cos(mt)

(t) = Am f/fm sin (mt) = sin (mt) = Am f/fm indicates maximum phase change

xlp(t) = Ac exp( j sin (mt)) Periodic with fundamental frequency of fm c(n fm) = Jn()

nth order Bessel function with argument J-n() = (-1)n Jn()

Modulated signal Carrier frequency line Infinite # of sidebands lines

FM bandwidth If (n/), then Jn() << 1

BW is a function of

NBFM < 0.2 Only J0 and J1

Acceptable distortion PB / PT > 0.98 N = (+1) B = 2(+1)fm

Arbitrary signal = Am f/fm = xmax f/fmax

Carlson’s rule B = 2(1+)W If |x(t)|< 1, B = 2(f+W)

NBFM : B = 2W WBFM : B = 2f

WBFM spectrumX(t) is a random signals with pdf of fX(x)

X(t) = x Xo(t) = Ac cos( 2(fc +fx)t )f = fc +fxdp = PC fX(x) dx

GXo(f) = PC/2 f fX( (f-fc)/f) ; f>0

ExampleGaussian message sourceB = 4.66 f

Distortion xo(t) = Ac cos(ct + (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 + A1 cos(ct + (t)) + A2 cos(2ct + 2(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, f = 25 Hz Frequency multiplier

n = 3000 fc1 = 600 MHz, f = 75 MHz Mixer

fLO = 500 MHz fc = 100 MHz

FM demodulationDiscriminator

Frequency to amplitude (voltage) conversion

|H(f)| = V0 + k(f-fc) A differentiator

AM demodulation

Needs a limiter to regulate Ac

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FM demodulationBalanced discriminator

Wider linear rangeWBFMNo DC block is needed

Phase locked loop (PLL)

ReadingCarlson Ch. 5.1, 5.2

Proakis & Salehi 3.3