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

7

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

8

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