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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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