Lecture 17,18: Phase Modulations Aliazam Abbasfar.
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Transcript of Lecture 17,18: Phase Modulations Aliazam Abbasfar.
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