Electronic modulation

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Narrow-band Frequency Modulation

KEEE343 Communication Theory

Lecture #16, May 3, 2011Prof. Young-Chai [email protected]



Summary

·Narrowband Frequency Modulation

· Wideband Frequency Modulation



Narrow-Band Frequency Modulation

• Narrow-Band FM means that the FM modulated wave has narrow

bandwidth.

• Consider the single-tone wave as a message signal:

• FM signal

• Instantaneous frequency

• Phase

m(t) = Am cos(2πf mt)

f i(t) = f c + kf Am cos(2πf mt)

= f c +∆f cos(2

π

f mt)

θi(t) = 2π

Z t

0

f i(τ ) dτ = 2π

f ct +

∆f

2πf msin(2πf mt)

= 2πf ct + ∆f

f msin(2πf mt)

∆f = kf Am



• Definitions

• Phase deviation of the FM wave

• Modulation index of the FM wave:

• Then, FM wave is

• To be narrow-band FM, should be small.

• For small compared to 1 radian, we can rewrite

β = ∆f f m

s(t) = Ac cos[2πf ct + β sin(2πf mt)]

f m

s(t) ≈ Ac cos(2πf ct)− β Ac sin(2πf ct) sin(2πf mt)

s(t) = Ac cos(2πf ct)cos(β sin(2πf mt))−Ac sin(2πf ct)sin(β sin(2πf mt))

β cos[β sin(2πf mt)] ≈ 1, sin[β sin(2πf mt)] ≈ β sin(2πf mt)

β



[Ref: Haykin & Moher, Textbook]



• Consider the modulated signal

which can be rewritten as

where

and

Polar Representation from Cartesian

s(t) = a(t) cos(2πf ct + φ(t))

s(t) = sI (t) cos(2πf ct)−

sQ(t) sin(2πf ct)

sI (t) = a(t) cos(φ(t)), and, sQ(t) = a(t)sin(φ(t))

a(t) =s2

I (t) + s2

Q(t) 12 , and φ(t) = tan−1

sQ(t)

sI (t)



• Approximated narrow-band FM signal can be written as

• Envelope

• Maximum value of the envelope

• Minimum value of the envelope

(t) ≈ Ac cos(2πf ct)− β Ac sin(2πf ct) sin(2πf mt)

for smallβ

a(t) = Ac

1 + β 2 sin2(2πf mt)

≈ Ac

1 +

1

2β 2 sin2(2πf mt)

Amax = Ac

1 +

1

2β 2

Amin = Ac



• The ratio of the maximum to minimum value

• Rewriting the approximated narrow-band FM wave is

• Average power

• Average power of the unmodulated wave

Amax

Amin

=

1 +1

2β 2

s(t) ≈ Ac cos(2πf ct) +2β cos(2π(f c + f m)t)−

2β Ac cos(2π(f c − f m)t)

P av = 1

2A2

c +

1

2β Ac

2

+

1

2β Ac

2

= 1

2A2

c(1 + β 2)

P c =1

2A2

c



• Ration of the average power to the power of the unmodulated wave

• Comparison with the AM signal

• Approximated Narrow-band FM signal

• AM signal

P av

P c= 1+ β 2

s(t) ≈ Ac cos(2πf ct) +2β cos(2π(f c + f m)t)−

2β Ac cos(2π(f c − f m)t)

sAM(t) = Ac cos(2πf ct) +2µAc {cos[2π(f c + f m)t)] + cos[2π(f c − f m)t]}

difference between the narrow-band FM and AM waves



• Bandwidth of the narrow-band FM:

• Phasor interpretation

2f m




• Angle

• Using the power series of the tangent function such as

θ(t) = 2

πf ct +

φ(t) = 2

πf ct + tan

−1

(β

sin(2πf mt

))

tan−1(x) ≈ x−

1

3x3 + · · ·



• Angle can be approximated as

• Ideally, we should have

• The harmonic distortion value is

• The maximum absolute value of D(t) is

θ(t) ≈ 2πf ct + β sin(2πf mt)

D(t) = β 3

3 sin3(2πf mt)

θ(t)≈

2πf ct + β sin(2πf mt)−

1

3β 3

sin3

(2πf mt)

Dmax =β 3

3



• For example for

which is small enough for it to be ignored in practice.

β = 0.3,

Dmax = 0.33

3 = 0.009 ≈ 1%



Amplitude Distortion of Narrow-band FM

• Ideally, FM wave has a constant envelope

• But, the modulated wave produced by the narrow-band FM differ from

this ideal condition in two fundamental respects:

•The envelope contains a residual amplitude modulation that varies

with time

• The angle contains harmonic distortion in the form of third-

and higher order harmonics of the modulation frequencyθi(t)

f m



Wide-Band Frequency Modulation

• Spectral analysis of the wide-band FM wave

or

where is called “complex envelope”.

Note that the complex envelope is a periodic function of time with a

fundamental frequency which means

where

s(t) = Ac cos[2πf ct + β sin(2πf mt)]

s(t) = < [Ac exp[ j2πf ct + jβ sin(2πf mt)]] = <[s̃(t)exp( j2πf ct)]

s̃(t) = Ac exp [ jβ sin(2πf mt)]

m

s̃(t) = s̃(t + kT m) = s̃(t + kf m

)

T m = 1/f m



• Then we can rewrite

• Fourier series form

where

s̃(t) =∞X

n=−∞

cn exp( j2πnf mt)

cn = f mZ 1/(2f m)

−1/(2f m) s̃(t) exp(− j2πnf mt) dt

= f mAc

Z 1/(2f m)

−1/(2f m)

exp[ jβ sin(2πf mt)− j2πnf mt] dt

s̃(t) = s̃(t + k/f m)

= Ac exp[ jβ sin(2πf m(t + k/f m))]

= Ac exp[ jβ sin(2πf mt + 2kπ)]

= Ac exp[ jβ sin(2πf mt)]



• Define the new variable:

Then we can rewrite

• nth order Bessel function of the first kind and argument

• Accordingly

which gives

x = 2πf mt

cn = Ac

2π

Z π

−π

exp[ j(β sinx− nx)] dx

β

J n(β ) = 1

2π

Z π

−π

exp[ j(β sinx− nx)] dx

cn = AcJ n(β )

s̃(t) = Ac

∞X

n=−∞

J n(β ) exp( j2πnf mt)



• Then the FM wave can be written as

• Fourier transform

which shows that the spectrum consists of an infinite number of deltafunctions spaced at for

s t = < s̃ t exp j2πf ct

= <

"Ac

∞Xn=−∞

J n(β ) exp[ j2πn(f c + f m)t]

#

= Ac

∞Xn=−∞

J n(β ) cos[2π(f c + nf m)t]

S (f ) = Ac

2

∞X

n=−∞

J n(β ) [δ (f − f c − nf m) + δ (f + f c + nf m)]

f = f c ± nf m n = 0, +1, +2, ...



1. For different values of n

2. For small value of

6. The equality holds exactly for arbitrary

Properties of Single-Tone FM for Arbitrary ModulationIndex β

J n(β ) = J −n(β ), for n even

J n(β ) = −J

−n(β ), for n odd

J 0(β ) ≈ 1,

J 1(β ) ≈

β

2

J n(β ) ≈

0, n > 2

β

β

∞X

n=−∞

J 2

n(β ) = 1






1. The spectrum of an FM wave contains a carrier component and and an infiniteset of side frequencies located symmetrically on either side of the carrier at

frequency separations of ...

2. The FM wave is effectively composed of a carrier and a single pair of side-

frequencies at .

3. The amplitude of the carrier component of an FM wave is dependent on the

modulation index . The averate power of such as signal developed across a 1-ohm resistor is also constant:

The average power of an FM wave may also be determined from

f m, 2

f m, 3

f m

f c ± f

m

β

P av =1

2A2

c

P av = 1

2A2

c

∞X

n=−∞

J 2

n(β )

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Electronic modulation