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Angle Modulation Frequency Modulation ( ) Tele 2060 Angle Modulation • Contrast to AM ω φ • Generalized sinusoid: v(t)=Vmaxsin( t+ ) ω φ • Instead of Varying Vmax, Vary ( t+ ) Martin B.H. Weiss Angle and Pulse Modulation - 1 University of Pittsburgh Tele 2060 Frequency Modulation • Instantaneous Carrier Frequency fi = fc + kem ω • For Tone Modulation, em = Emaxsin( mt) ∆ • Peak Frequency Deviation: f=kEmax ∆ ω • Therefore, fi = fc + f sin( mt) • The Sine Expression Needs an Angle as an Argument, Therefore ω π θ Recall That i = 2 fi = d /dt θω= Alternately, ()ttdt∫i () =+πω∆ ∫2( ffcmsin tdt) ∆ =−ωωf ct cos( mt) fm Martin B.H. Weiss Angle and Pulse Modulation - 2 University of Pittsburgh Tele 2060 Frequency Modulation ∆ • Define the Modulation Index mf= f/fm • Then the Standard Form of the Equation Becomes =−ωω • vt( ) cos(cf m cos m t ) • From Trigonometry:∞ π αβ−=∑ ++ βn cos(mJman cos )n ( )cos • n=−∞ 2 = ω •Thus: vt( ) J1 ( m )cos(c t ) +++ωω π Jm2( )cos(cm t t 2 ) + Martin B.H. Weiss Angle and Pulse Modulation - 3 University of Pittsburgh Tele 2060 Frequency Composition of FM Tone Moduation • Does Not Lend Itself to Simple Subsititutions, as With AM • Solved with the Bessel Function mm2 4 m6 mn ( ff) ( ) ( f) = f1− 2 + 2 − 2 + Jmnf() 2nn!!()!!()!!()!112233+ n+ n+ mf J0 J1 J2 J3 J4 0.00 1.00 0.25 0.98 0.12 0.5 0.94 0.24 0.03 1.0 0.77 0.44 0.11 0.02 1.5 0.51 0.56 0.23 0.06 0.01 2.0 0.22 0.58 0.35 0.13 0.03 2.4 0.00 0.52 0.43 0.20 0.06 2.5 -0.05 0.50 0.45 0.22 0.07 3.0 -0.26 0.34 0.49 0.31 0.13 4.0 -0.40 -0.07 0.36 0.43 0.28 5.0 -0.18 -0.33 0.05 0.36 0.39 6.0 0.15 -0.28 -0.24 0.11 0.36 Martin B.H. Weiss Angle and Pulse Modulation - 4 University of Pittsburgh Tele 2060 FM Example •Let mf = 1.0 • Look Up the Amplitude of the Spectral Components Carrier (J = 0.77) o st 1 Harmonic (J = 0.44) 1 nd 2 Harmonic (J = 0.11) 2 rd 3 Harmonic (J = 0.02) 3 These are with respect to a E = 1V Cmax Martin B.H. Weiss Angle and Pulse Modulation - 5 University of Pittsburgh Tele 2060 Discussion • Some Indices of Modulation Assign No Power to the Carrier (eg. mf = 2.4, 5.5, 8.65) • By Observation: B = 2nf (n is the Highest Order of the Side Frequency) FM m n~(m +1) f • Therefore, Bandwidth for a Sinusoid Modulated FM Signal is: B = 2(m + 1)f FM f m Martin B.H. Weiss Angle and Pulse Modulation - 6 University of Pittsburgh Tele 2060 Power in Frequency Modulation • Bessel Function Relates Voltage Amplitude to the Unmoduated Carrier (i.e., En = Jn*Ec) • Recall 2 Pn = En /R PT = P0 + 2P1 + 2P2. 2 2 2 •Thus, PT = Pc(J0 + 2(J1 + J2 + . )) Martin B.H. Weiss Angle and Pulse Modulation - 7 University of Pittsburgh Tele 2060 Maximum Bandwidth of a Complex Signal ∆ •M = F/Fm - Deviation ratio ∆F - Maximum frequency deviation Fm - Highest frequency component in modulating signal •Carson’s Rule ∆ Bmax = 2(M+1)Fm = 2( F+Fm) Carson's Rule Underestimates the Bandwidth Requirement Somewhat Martin B.H. Weiss Angle and Pulse Modulation - 8 University of Pittsburgh Tele 2060 Phase Modulation ω φ • Generalized sinusoid: v(t)=Vmaxsin( t+ ) • Vary φ •Mathematics φ (t) = kpvm(t) kp is the Modulator Sensitivity ω For Tone Modulation, vm(t) = Vp sin( mt) ∆φ Define = kpVp=mp as the Modulation Index for Phase Modulation ω φ ω ω Thus, v(t)=Vmaxsin( t+ ) = Vmaxsin( ct+ mpsin( mt)) ω ω Compare with FM: v(t)= Vmaxsin( ct+ mpcos( mt)) o Recall: sin(x) = cos(x+ π/2) = cos(x+90 ) Martin B.H. Weiss Angle and Pulse Modulation - 9 University of Pittsburgh Tele 2060 Spectrum of Phase Modulation • Identical to FM with Tone Modulation • The Difference Lies in Only in the Phase Shift • The Bessel Function Computes Amplitude Spectrum Only Martin B.H. Weiss Angle and Pulse Modulation - 10 University of Pittsburgh Tele 2060 Equivalence between FM and PM ω φ • Recall That i = d (t)/dt ω ω φ • In General, (t) = ct+ (t) • Applying the Definition ω ω φ ieq= c+d (t)/dt ω πφ ω φ c + 2 eq(t) = c+d (t)/dt π φ • Therefore, feq(t)=[1/(2 )][d (t)/dt] Martin B.H. Weiss Angle and Pulse Modulation - 11 University of Pittsburgh Tele 2060 Comparison of FM and PM • PM Requires Coherent Receivers • PM has a Constant Modulation Index with Frequency Changes This Results in Better Signal to Noise at the Demodulator Eliminates the Need for Preemphasis • In PM, the Modulation Occurs After the Carrier Signal is Generated This Results in a Stable Carrier Frequency Carrier Drift is Reduced • In FM, High Index Signals can be Generated Easily • FM can be Demodulated Cheaply Martin B.H. Weiss Angle and Pulse Modulation - 12 University of Pittsburgh Tele 2060 FM Transmitters • Direct FM Simple in Concept Use a Voltage Controlled Oscillator (VCO) Common Circuit Device Frequency Output is a Function of the Input Voltage • Indirect FM Use a Phase Modulator Integrate Modulating Signal Frequency multiplier: can use PLL Use Heterodyne to translate to carrier freq. • Limiters Remove any AM Component from the Carrier Avoids Confounding the Receiver Martin B.H. Weiss Angle and Pulse Modulation - 13 University of Pittsburgh Tele 2060 FM Receiver •Requires Non-Linear Devices Use of Limiters • Frequency Discriminator: The Output is a (Linear) Function of Frequency • Phase-Locked Loop (PLL) Martin B.H. Weiss Angle and Pulse Modulation - 14 University of Pittsburgh Tele 2060 Broadcast FM • Pre-emphasis/De-emphasis Used to Improve the Signal to Noise Ratio Pre-Emphasis: High Pass Filter Resides in the Transmitter De-Emphasis Simple Low-PassFilter Fesides in the Receiver Compensates for Pre-Emphasis N |H(f)| 0 FM Noise Detector Pre-emphasis Output Spectrum Filter In the US, f1 = 2.12 kHz f f1 f2 f Martin B.H. Weiss Angle and Pulse Modulation - 15 University of Pittsburgh Tele 2060 Broadcast FM • Capture Effect FM Receivers Suppress Weaker Signals and are “Captured” by Stronger Signals at the Same Frequency Result of the Non-Linearity of Demodulation • Automatic Frequency Control (AFC) Negative Feedback Network Used to Control Frequency Drift Martin B.H. Weiss Angle and Pulse Modulation - 16 University of Pittsburgh Tele 2060 Noise in Modulated Systems •Signal Signal = VMaxv(t) 2 2 Signal Power = V Maxv (t) = SR x(t)=<v(t)> 2 2 Mean Square: x =<v(t)> •Noise Additive noise, n(t) Noise Power , η (Watts/Hz) η Received Noise = BT, i.e. the Noise Power in the Received Band Martin B.H. Weiss Angle and Pulse Modulation - 17 University of Pittsburgh Tele 2060 Noise in AM •vr(t)=v(t)+n(t) η • Signal-to-Noise ratio = S/N = SR/NR = SR/ BT γ η •Let =SR/W, W = Bandwidth of Baseband Channel W = BT/2 for DSB W = BT for SSB γ is the Maximimum Value of the Destination S/N Baseband Martin B.H. Weiss Angle and Pulse Modulation - 18 University of Pittsburgh Tele 2060 Noise in AM •Signal Power: Signal at the Output of the Detector EMmax = mECmax\ PT = PC[1+(1/2)meff] • Noise power: PN = FkTB •Thus, SNR = Signal Power/Noise Power AM 22 mEeff C SNR = max AM 2FkTB mP2 = eff c 2FkTB mP2 = eff T m2 + eff 21( 2)FkTB Martin B.H. Weiss Angle and Pulse Modulation - 19 University of Pittsburgh Tele 2060 Noise in Angle Modulation • The Analysis is Complex • For Wideband FM 2 2γ S/N=(3/4)(BT/W) x Signal to Noise Performance Improves with the Square of BT/W • Noise in Wideband PM S/N= ∆φ2x2 γ Martin B.H. Weiss Angle and Pulse Modulation - 20 University of Pittsburgh Tele 2060 Comparison of AM, FM, PM •Bandwidth AM: Low FM and PM: High • Noise performance AM: Poor FM: Very Good PM: Somewhat Better than AM • Receiver Implementation AM: Generally Easy FM: Generally Easy PM: More Difficult Because Synchronous Detection is Required Martin B.H. Weiss Angle and Pulse Modulation - 21 University of Pittsburgh.
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