Angle Modulation Contrast to AM Generalized sinusoid: v(t)=v max sin(ωt+φ) Instead of Varying V max, Vary (ωt+φ) Angle and Pulse Modulation - 1 Frequency Modulation Instantaneous Carrier Frequency f i = f c + ke m For Tone Modulation, e m = E max sin(ω m t) Peak Frequency Deviation: f=ke max Therefore, f i = f c + f sin(ω m t) The Sine Expression Needs an Angle as an Argument, Therefore Recall That ω i = πf i = dθ/dt Alternately, θ() t = ωi() t dt = π( fc + f sinωmt) dt f = ωct cos( ωmt) f m Angle and Pulse Modulation -
Frequency Modulation Define the Modulation Index m f = f/f m Then the Standard Form of the Equation Becomes vt ( ) = cos( ωc mf cos ωmt) From Trigonometry: π cos( α cos β) = ( )cos n m + β+ Jn m a n n= Thus: vt ( ) = J1( m)cos( ωct) + J ( m)cos( ω t + ω t+ π c m ) + Angle and Pulse Modulation - 3 Frequency Composition of FM Tone Moduation Does Not Lend Itself to Simple Subsititutions, as With AM Solved with the Bessel Function n mf mf mf mf ( ) ( ) ( ) Jn( mf ) = 4 6 1 + + n! 1!( n+ 1)!!( n+ )! 3!( n+ 3)! m f 0.00 0.5 0.5 1.0 1.5.0.4.5 3.0 4.0 5.0 6.0 J 0 J 1 J J 3 J 4 1.00 0.98 0.94 0.77 0.51 0. 0.00-0.05-0.6-0.40-0.18 0.15 0.1 0.4 0.44 0.56 0.58 0.5 0.50 0.34-0.07-0.33-0.8 0.03 0.11 0.3 0.35 0.43 0.45 0.49 0.36 0.05-0.4 0.0 0.06 0.13 0.0 0. 0.31 0.43 0.36 0.11 0.01 0.03 0.06 0.07 0.13 0.8 0.39 0.36 Angle and Pulse Modulation - 4
FM Example Let m f = 1.0 Look Up the Amplitude of the Spectral Components Carrier (J o = 0.77) 1 st Harmonic (J 1 = 0.44) nd Harmonic (J = 0.11) 3 rd Harmonic (J 3 = 0.0) These are with respect to a E Cmax = 1V Angle and Pulse Modulation - 5 Discussion Some Indices of Modulation Assign No Power to the Carrier (eg. m f =.4, 5.5, 8.65) By Observation: B FM = nf m (n is the Highest Order of the Side Frequency) n~(m f +1) Therefore, Bandwidth for a Sinusoid Modulated FM Signal is: B FM = (m f + 1)f m Angle and Pulse Modulation - 6
Power in Frequency Modulation Bessel Function Relates Voltage Amplitude to the Unmoduated Carrier (i.e., E n = J n *E c ) Recall P n = E n /R P T = P 0 + P 1 + P... Thus, P T = P c (J 0 + (J 1 + J +... )) Angle and Pulse Modulation - 7 Maximum Bandwidth of a Complex Signal M = F/F m - Deviation ratio F - Maximum frequency deviation F m - Highest frequency component in modulating signal Carson s Rule B max = (M+1)F m = ( F+F m ) Carson's Rule Underestimates the Bandwidth Requirement Somewhat Angle and Pulse Modulation - 8
Phase Modulation Generalized sinusoid: v(t)=v max sin(ωt+φ) Vary φ Mathematics φ(t) = k p v m (t) k p is the Modulator Sensitivity For Tone Modulation, v m (t) = V p sin(ω m t) Define φ = k p V p =m p as the Modulation Index for Phase Modulation Thus, v(t)=v max sin(ωt+φ) = V max sin(ω c t+ m p sin(ω m t)) Compare with FM: v(t)= V max sin(ω c t+ m p cos(ω m t)) Recall: sin(x) = cos(x+ π/) = cos(x+90 o ) Angle and Pulse Modulation - 9 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 Angle and Pulse Modulation - 10
Equivalence between FM and PM Recall That ω i = dφ(t)/dt In General, ω(t) = ω c t+ φ(t) Applying the Definition ω ieq = ω c +dφ(t)/dt ω c + πφ eq (t) = ω c +dφ(t)/dt Therefore, f eq (t)=[1/(π)][dφ(t)/dt] Angle and Pulse Modulation - 11 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 Angle and Pulse Modulation - 1
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 Angle and Pulse Modulation - 13 FM Receiver Requires Non-Linear Devices Use of Limiters Frequency Discriminator: The Output is a (Linear) Function of Frequency Phase-Locked Loop (PLL) Angle and Pulse Modulation - 14
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 0 FM Noise Detector Output Spectrum H(f) Pre-emphasis Filter In the US, f 1 =.1 khz f f 1 f f Angle and Pulse Modulation - 15 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 Angle and Pulse Modulation - 16
Noise in Modulated Systems Signal Signal = V Max v(t) Signal Power = V Max v (t) = S R x(t)=<v(t)> Mean Square: x =<v(t)> Noise Additive noise, n(t) Noise Power, η (Watts/Hz) Received Noise = ηb T, i.e. the Noise Power in the Received Band Angle and Pulse Modulation - 17 Noise in AM v r (t)=v(t)+n(t) Signal-to-Noise ratio = S/N = S R /N R = S R /ηb T Let γ=s R /ηw, W = Bandwidth of Baseband Channel W = B T / for DSB W = B T for SSB γ is the Maximimum Value of the Destination S/N Baseband Angle and Pulse Modulation - 18
Noise in AM Signal Power: Signal at the Output of the Detector E Mmax = me Cmax \ P T = P C [1+(1/)m eff ] Noise power: P N = FkTB Thus, SNR AM = Signal Power/Noise Power meff EC max SNR AM = FkTB meff Pc = FkTB m = 1 eff P T m ( + eff ) FkTB Angle and Pulse Modulation - 19 Noise in Angle Modulation The Analysis is Complex For Wideband FM S/N=(3/4)(B T /W) x γ Signal to Noise Performance Improves with the Square of B T /W Noise in Wideband PM S/N= φ x γ Angle and Pulse Modulation - 0
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 Angle and Pulse Modulation - 1