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Tele 2060 Pulse Modulation • What if the Carrier Signal were a Pulse Train Instead of a Sinusoid? • Three Approaches Pulse Amplitude Modulation (PAM) Pulse Width Modulation (PWM) Pulse Position Modulation (PPM) • Each of these Approaches Uses a Discrete Signal to Carry an Analog Signal Martin B.H. Weiss Pulse Modulation - 1 University of Pittsburgh Tele 2060 Bandwidth of a Pulse Train •Vf Vττ[sin(ππfττ)/(ππfττ)] () = • ττ is the Duration of the Pulse •V is the Amplitude of the Pulse th •The N Harmonic of a Pulse Train Vn = (Vττ/T)[sin(nx)/(nx)] = (Vττ/T)sinc(ττ/T) T is the Interval Between Pulses (i.e., the Period) x = πτπτ/T Martin B.H. Weiss Pulse Modulation - 2 University of Pittsburgh Tele 2060 Bandwidth of a Pulse Train • Find the “Zeros” nx We Need the Location of sin( ) = 0 nπτπτ T ππ This Occurs When / = nT ττ That is, When / =1/ (By Substitution) Let f =1/T, So Zeros Occurs at nf0 = 1/ττ 0 n f ττ The First Zero Occurs at =1, or at the Frequency =1/ • Note The Following Most of the Signal Energy (92%) is Contained in the f ττ Frequency Between 0 and =1/ Thus, we can Ignore the Higher Frequency Components Martin B.H. Weiss Pulse Modulation - 3 University of Pittsburgh Tele 2060 Example •V = 5V •T = 25µsec ττ • = 5µsec •Spectrum st f ττ Calculate 1 zero: =1/ =1/5µsec=200KHz 0 Calculate 2nd zero: f =2/ττ =400KHz 1 Martin B.H. Weiss Pulse Modulation - 4 University of Pittsburgh Tele 2060 Pulse Amplitude Modulation (PAM) • Modulate a Pulse Stream with a Signal Used in Dimension PBX’s Type of a AM system • Categories Natural PAM Top of Pulse Conforms to the Signal Shape Makes Mathematics Easy Flat Top PAM More Practical Approaches Natural PAM for Narrow Pulses Martin B.H. Weiss Pulse Modulation - 5 University of Pittsburgh Tele 2060 Bandwidth of PAM •vt mtpt () = ( ) ( ) m t ( ) is the Modulating Waveform p t ( ) is the Pulse Train • Fourier Equivalent of a Pulse Train p t Vττ T Vττ T x ωωt x ωωt ( ) = / + (2 / )[sinc( )cos + sinc(2 )cos(2 ) + . . .] x πτπτ T Where = / •Thus, v(t) = m(t)Vττ/T + m(t)(2Vττ/ΤΤ)[sinc(x)cosωωt + sinc(2x)cos(2ωωt) + . . .] This is in the Same General Form of the AM Signal: ( )= ( ) / + [(2 / )sinc( )] ( )cos + [(2 / )sinc(2 )] ( )cos(2 ) + . . . v t mtV T V x m t t V x m t t ττ ττ ΤΤ ωω ττ ΤΤ ωω ( )= ( ) + [ ( )+ ( )] + [ ( -2 )+ ( +2 )] + . . . X f c M f c M f-f Mf+f c Mf f Mf f s 0 1 s s 2 s s Where = 2 [sin(n )/n )] = sinc( ) c Vf ττ ππ ττ ππ ττ f ττ nf ττ f f n s s s s s Martin B.H. Weiss Pulse Modulation - 6 University of Pittsburgh Tele 2060 Example •V =5V •T =5µsec, or fs=200,000/sec τ τ • =1µsec f ττ -6 • First Zero, 0=1/ =1/10 =1MHz •fττ f ττ s = .2, s = .628 •c = (10)(.2)[sinc(0)]=2 0 •c = (10)(.2)[sin(.628)/.628]=2(.935)=1.87 1 •c = 2[sinc(.4)]=2(.757)=1.514 2 Martin B.H. Weiss Pulse Modulation - 7 University of Pittsburgh Tele 2060 Form of the Harmonic Terms •Vmt nωωt n ()cos( s ), Where ωω ππ T s = 2 / s, or the Angular Sampling Frequency V VττT nx nx x πτπτ T and =( / )[sin( )/( )], = / n • Observation: m t ( ) is Completely Contained in DC Component Thus, Low Pass Filtering can be Used for Demodulation Martin B.H. Weiss Pulse Modulation - 8 University of Pittsburgh
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