Angle Modulation
ECE460Spring, 2012
2
Angle Modulation Techniques1. A nonlinear process2. Transmission bandwidth is theoretically infinite3. Additive noise affects performance less than in
Amplitude Modulation
( ) cos 2c cs t A f t t
1
2i c
df t f t
dt
Instantaneous Frequency:
2
2
cos 2 cos 2 2
t
p f
p f
t
c c p c c f
PM FM
t
dt
dt
s
k m t k m d
dk m t k m tdt
A f t k m t A f k m dt t
3
FM and PM are similar
4
Narrowband Angle Modulation
Angle Modulation by a Sinusoidal Signal
( ) cos 2
cos 2 cos sin 2 sin
c c
c c c c
s t A f t t
A f t t A f t t
( ) cos 2m mm t A f t
( ) cos 2 sin 2c c ms t A f t f t
( ) cos 2c cc t A f t
PM
FM
cos 2
sin 2
p m m
f mm
m
t k A f t
k At f t
f
FM:
5
Angle Modulation by a Sinusoidal Signal
( ) cos 2 sin 2c c ms t A f t f t
6
Properties of the Bessel Function1. Bessel Function of the first order n
2. For small values of the modulation index
3. The equality holds exactly for arbitrary
for even( )
for oddn
nn
J nJ
J n
0
1
( ) 1,
,20, 2n
J
J
J n
2 1nn
J
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Properties of Single-Tone FM
1. Spectrum contains a carrier component and an infinite set of side frequencies
2. For the special case where << 1, only the Bessel coefficients have significant values.
3. Amplitude of the carrier component varies with according to
( ) cos 2 sin 2
2
c c m
cn c m c m
n
s t A f t f t
AS f J f f nf f f nf
0 1( ) and J J
0 ( )J
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Example 3.3.2
Find the expression for the modulated signal and determine how many harmonics should be selected to contain 99% of the modulated signal power.
Total carrier power:
Modulated signal:
Modulation Index:
Find:
10cos 2 2 cos 20
10cos 2 5sin 20
t
c f
c
s t f t k d
f t t
10cos 2
cos 20
50
c
f
c t f t
m t t
k
2
502c
c
AP
max5f
m
m tk
f
2
2 0.99 502c
nn
AP J
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Example 3.3.2 (Continued)
For k = 6, P > 49.5. Therefore, the effective bandwidth about the carrier frequency isIs
In general, for a single-tone information signal, the effective bandwidth with at least 98% of the signal power is given by
2
2 20
1
5 2 5 49.52
kc
nn
AP J J
where 6 6.c mf n f n 120 Hz.cB
2 1c mB f
10
Expand for Arbitrary Periodic Message
Rewriting using exponentials
Following the logic used for a single-tone message
Carson’s Rule
( ) cos 2c cs t A f t m t
2( ) Re c j m tj f tcs t A e e
2 2
0
( ) cos 2
1
2m
c n c m nn
uj m nu
f
n
s t A c f nf t c
c e du
max , PM
max, FM
p
f
k m t
k m t
W
2 1cB W