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Narrow-band Frequency Modulation KEEE343 Communication Theory Lecture #16, May 3, 2011 Prof.Young-Chai Ko [email protected]

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Page 1: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

Narrow-band Frequency Modulation

KEEE343 Communication Theory

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

Page 2: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

Summary

·•Narrowband Frequency Modulation

·•Wideband Frequency Modulation

Page 3: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

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(2�fmt)

fi(t) = fc + kfAm cos(2�fmt)

= fc +�f cos(2�fmt)

�i(t) = 2⇥

Z t

0fi(⇤) d⇤ = 2⇥

fct+

�f

2⇥fmsin(2⇥fmt)

= 2⇥fct+�f

fmsin(2⇥fmt)

�f = kfAm

Page 4: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• 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

� =�f

fm

s(t) = Ac cos[2⇥fct+ � sin(2⇥fmt)]

fm

s(t) ⇡ Ac cos(2⇥fct)� �Ac sin(2⇥fct) sin(2⇥fmt)

s(t) = Ac cos(2⇥fct) cos(� sin(2⇥fmt))�Ac sin(2⇥fct) sin(� sin(2⇥fmt))

�cos[� sin(2⇥fmt)] ⇡ 1, sin[� sin(2⇥fmt)] ⇡ � sin(2⇥fmt)

Page 5: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

[Ref: Haykin & Moher, Textbook]

Page 6: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Consider the modulated signal

which can be rewritten as

where

and

Polar Representation from Cartesian

s(t) = a(t) cos(2�fct+ ⇥(t))

s(t) = sI(t) cos(2�fct)� sQ(t) sin(2�fct)

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

a(t) =⇥s2I(t) + s2Q(t)

⇤ 12 , and �(t) = tan�1

sQ(t)

sI(t)

Page 7: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Approximated narrow-band FM signal can be written as

• Envelope

• Maximum value of the envelope

• Minimum value of the envelope

s(t) ⇡ Ac cos(2⇥fct)� �Ac sin(2⇥fct) sin(2⇥fmt)

for small�

a(t) = Ac

⇥1 + �2 sin2(2⇥fmt)

⇤⇡ Ac

✓1 +

1

2�2 sin2(2⇥fmt)

Amax

= Ac

✓1 +

1

2�2

Amin = Ac

Page 8: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• 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

2�2

s(t) ⇡ Ac cos(2⇥fct) +1

2

� cos(2⇥(fc + fm)t)� 1

2

�Ac cos(2⇥(fc � fm)t)

Pav =1

2A2

c +

✓1

2�Ac

◆2

+

✓1

2�Ac

◆2

=1

2A2

c(1 + �2)

Pc =1

2A2

c

Page 9: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

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

• Comparison with the AM signal

• Approximated Narrow-band FM signal

• AM signal

Pav

Pc= 1 + �2

s(t) ⇡ Ac cos(2⇥fct) +1

2

� cos(2⇥(fc + fm)t)� 1

2

�Ac cos(2⇥(fc � fm)t)

sAM(t) = Ac cos(2�fct) +1

2

µAc {cos[2�(fc + fm)t)] + cos[2�(fc � fm)t]}

difference between the narrow-band FM and AM waves

Page 10: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Bandwidth of the narrow-band FM:

• Phasor interpretation

2fm

[Ref: Haykin & Moher, Textbook]

Page 11: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Angle

• Using the power series of the tangent function such as

⇥(t) = 2⇤fct+ ⌅(t) = 2⇤fct+ tan�1(� sin(2⇤fmt))

tan�1(x) ⇥ x� 1

3x

3 + · · ·

Page 12: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Angle can be approximated as

• Ideally, we should have

• The harmonic distortion value is

• The maximum absolute value of D(t) is

⇥(t) ⇡ 2⇤fct+ � sin(2⇤fmt)

D(t) =�3

3sin3(2⇥fmt)

⇥(t) ⇡ 2⇤fct+ � sin(2⇤fmt)� 1

3�3 sin3(2⇤fmt)

Dmax

=�3

3

Page 13: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• For example for

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

� = 0.3,

Dmax

=0.33

3= 0.009 ⇡ 1%

Page 14: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

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 frequency

�i(t)fm

Page 15: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

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[2⇥fct+ � sin(2⇥fmt)]

s(t) = < [Ac exp[j2⇥fct+ j� sin(2⇥fmt)]] = <[s̃(t) exp(j2⇥fct)]

s̃(t) = Ac exp [j� sin(2⇥fmt)]

fm

s̃(t) = s̃(t+ kTm) = s̃(t+k

fm)

Tm = 1/fm

Page 16: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Then we can rewrite

• Fourier series form

where

s̃(t) =

1X

n=�1cn exp(j2�nfmt)

cn = fm

Z 1/(2fm)

�1/(2fm)s̃(t) exp(�j2⇥nfmt) dt

= fmAc

Z 1/(2fm)

�1/(2fm)exp[j� sin(2⇥fmt)� j2⇥nfmt] dt

s̃(t) = s̃(t+ k/fm)

= Ac exp[j� sin(2⇥fm(t+ k/fm))]

= Ac exp[j� sin(2⇥fmt+ 2k⇥)]

= Ac exp[j� sin(2⇥fmt)]

Page 17: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Define the new variable:

Then we can rewrite

• nth order Bessel function of the first kind and argument

• Accordingly

which gives

x = 2�fmt

cn =

Ac

2⇥

Z �

��exp[j(� sinx� nx)] dx

Jn(�) =1

2⇥

Z �

��exp[j(� sinx� nx)] dx

cn = AcJn(�)

s̃(t) = Ac

1X

n=�1Jn(�) exp(j2⇥nfmt)

Page 18: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

• Then the FM wave can be written as

• Fourier transform

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

s(t) = <[s̃(t) exp(j2⇥fct)]

= <"Ac

1X

n=�1Jn(�) exp[j2⇥n(fc + fm)t]

#

= Ac

1X

n=�1Jn(�) cos[2⇥(fc + nfm)t]

S(f) =Ac

2

1X

n=�1Jn(�) [⇥(f � fc � nfm) + ⇥(f + fc + nfm)]

f = fc ± nfm n = 0,+1,+2, ...

Page 19: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

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 Modulation Index �

Jn(�) = J�n(�), for n even

Jn(�) = �J�n(�), for n odd

J0(�) ⇡ 1,

J1(�) ⇡ �

2Jn(�) ⇡ 0, n > 2

�1X

n=�1J2n(�) = 1

Page 20: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

[Ref: Haykin & Moher, Textbook]

Page 21: Narrow-band Frequency Modulationelearning.kocw.net/KOCW/document/2011/korea/koyoungchai/...Narrow-Band Frequency Modulation • Narrow-Band FM means that the FM modulated wave has

1. The spectrum of an FM wave contains a carrier component and and an infinite set 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

fm, 2fm, 3fm

fc ± fm

Pav =1

2A2

c

Pav =1

2A2

c

1X

n=�1J2n(�)