Wide-band Frequency ModulationKEEE343 Communication Theory

Lecture #17, May 12, 2011Prof. Young-Chai Kokoyc@korea.ac.kr

Summary

·•Wideband Frequency Modulation

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

• 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)]

• 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)

• 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, ...

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

[Ref: Haykin & Moher, Textbook]

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 average 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(�)

• Recall the single-tone frequency modulated wave given as

• and its FT is given as

where for the message signal

• To see the bandwidth let us consider two different cases

1. Case 1: Fix and vary (phase deviation is varied but the BW of message signal is fixed.)

2. Case 2: Fix and vary (phase deviation is fixed but the BW of message signal is varied.)

Transmission Bandwidth of FM Waves

s(t) = Ac cos[2⇥fct+ � sin(2⇥fmt)]

S(f) =Ac

2

1X

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

m(t) = Am cos(2�fmt)

fm Am

Am fm

� =kfAm

fm=

�f

fm

Case 1

[Ref: Haykin & Moher, Textbook]

Case 2

[Ref: Haykin & Moher, Textbook]

• In theory, an FM wave contains an infinite number of side-frequencies.

• However, we find that the FM wave is effectively limited to a finite number of significant side-frequencies compatible with a specified amount of distortion.

• Observations of two limiting cases

1. For large values of the modulation index , the bandwidth approaches, and is only slightly greater than the total frequency excursion .

2. For small values of the modulation index, the spectrum of the FM wave is effectively limited to one pair of side-frequencies at so that the bandwidth approaches .

Transmission Bandwidth of the FM Wave

�

2�f

fc ± fm2fm

• Carson’s rule is the approximate rule for the transmission bandwidth of an FM wave

• Single-tone case

• Arbitrary modulating wave

where is deviation ratio.

Carson’s Rule

BT ⇡ 2�f + 2fm = 2�f

✓1 +

1

�

◆

BT ⇡ 2(�f +W ) = 2�f

✓1 +

1

D

◆

D =�f

W

• Carson’s rule is simple but unfortunately it does not always provide a good estimate of the transmission bandwidth, in particular, for the wideband frequency modulation.

Universal Curve for FM Transmission Bandwidth

[Ref: Haykin & Moher, Textbook]

[Ref: Haykin & Moher, Textbook]