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 Linear modulation Transmitted signal x o (t) = A m x(t) cos( c t) + A m x’(t) sin( c t) + A c cos( c t) 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 x o (t) = A c cos( c t + (t)) = Re[ A c exp( j c t + j(t) ) ] Constant envelope, but time-varying phase A(t) = A c, (t)= (t) P Xo = P c Phase modulation (PM) (t)=   x(t)   : phase deviation Frequency modulation (FM) f(t)= f c + 1/2 d(t)/dt f(t) = f c + f  x(t) f  x(t) << f c f  : frequency deviation PM: f(t) =? PM and FM modulators are interchangeable Zero crossings are not periodic

Narrowband PM/FM Narrowband PM/FM (NBPM/NBFM) (t) << 1 rad x I (t) = A c, x Q (t)= A c (t) X lp (f) = A c (f) + A c (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 2 nd terms too? x I (t) = A c (1-   (t)/2 ), x Q (t)= A c (t)

Tone modulation (FM) x(t) = A m cos( m t) (t) = A m f  /f m sin ( m t) = sin ( m t) = A m f  /f m indicates maximum phase change x lp (t) = A c exp( j sin ( m t)) Periodic with fundamental frequency of f m c(n f m ) = J n () nth order Bessel function with argument  J -n () = (-1) n J n () Modulated signal Carrier frequency line Infinite # of sidebands lines

FM bandwidth If (n/), then J n () << 1 BW is a function of  NBFM < 0.2 Only J 0 and J 1 Acceptable distortion P B / P T > 0.98 N = (+1) B = 2(+1)f m Arbitrary signal = A m f  /f m = x max f  /f max Carlson’s rule B = 2(1+)W If |x(t)|< 1, B = 2(f  +W) NBFM : B = 2W WBFM : B = 2f 

WBFM spectrum X(t) is a random signals with pdf of f X (x) X(t) = x  Xo(t) = A c cos( 2(f c +f  x)t ) f = f c +f  x dp = P C f X (x) dx G Xo (f) = P C /2 f  f X ( (f-fc)/f  ) ; f>0 Example Gaussian message source B = 4.66  f 

Distortion x o (t) = A c cos( c t + (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) = A 0 + A 1 cos( c t + (t)) + A 2 cos(2 c t + 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 = C 0 – kx(t) f = f c + k f c /2C 0 Indirect modulation Use NBFM modulator f c1 = 200KHz, f  = 25 Hz Frequency multiplier n = 3000  f c1 = 600 MHz, f  = 75 MHz Mixer f LO = 500 MHz  f c = 100 MHz

FM demodulation Discriminator Frequency to amplitude (voltage) conversion |H(f)| = V 0 + k(f-f c ) 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