Chapter 4. Angle Modulation

4.4 Narrow-Band Frequency Modulation

– We first consider the simple case of a single-tone modulation that produces a narrow-band FM wave

– We next consider the more general case also involving a single-tone modulation, but this time the FM wave is wide-band

– The two-stage spectral analysis described above provides us with enough insight to propose a useful solution to the problem

– An FM signal is

)9.4()2cos()( tfAtm mm

(4.10) )2cos(

)2cos()(

tfff

tfAkftf

mc

mmfci

(4.11) Amfkf

The frequency deviation

)12.4()2sin(2)( tff

ftft m

m

ci )13.4(

mf

fModulation index

The phase deviation)14.4()2sin(2)( tftft mci

– The FM wave is

– If the modulation index is small compared to one radian, the approximate form of a narrow-band FM wave is

1. The envelope contains a residual amplitude modulation that varies with time

2. The angel θi(t) contains harmonic distortion in the form of third- and higher order harmonics of the modulation frequency fm

)15.4()2sin(2cos)( tftfAts mcc

BABABA sinsincoscos)cos(

)16.4()2sin(sin)2sin()2sin(cos)2cos()( tftfAtftfAts mccmcc

1)2sin(cos tfm )2sin()2sin(sin tftf mm

)17.4()2sin()2sin()2cos()( tftfAtfAts mcccc

– We may expand the modulated wave further into three frequency components

– The basic difference between and AM wave and a narrow-band FM wave is that the algebraic sign of the lower side-frequency in the narrow-band FM is reversed

– A narrow-band FM wave requires essentially the same transmission bandwidth as the AM wave.

)18.4()(2cos)(2cos2

1)2cos()( tfftffAtfAts mcmcccc

)19.4()(2cos)(2cos2

1)2cos()( tfftffAtfAts mcmccccAM

• Phasor Interpretation– A resultant phasor representing the narrow-band FM

wave that is approximately of the same amplitude as the carrier phasor, but out of phase with respect to it.

– The resultant phasor representing the AM wave has a different amplitude from that of the carrier phasor, but always in phase with it.

4.5 Wide-Band Frequency Modulation

– Assume that the carrier frequency fc is large enough to justify rewriting Eq. 4.15) in the form

– The complex envelope is

(4.20) )2exp()(Re

))2sin(2exp(Re)(~

tfjts

tfjtfjAts

c

mcc

(4.21) ] )2sin(exp[)(~

tfjAts mc

] )2sin(exp[

] )22sin(exp[

] ))/(2sin(exp[)(~

tfjA

ktfjA

fktfjAts

mc

mc

mmc

(4.22))2exp()(~

tnfjcts mn

n

• The complex Fourier coefficient

(4.23)]2)2sin(exp[

)2exp()(

)2/(1

)2/(1

)2/(1

)2/(1

~

m

m

m

m

f

fmmcm

m

f

fmn

dttnfjtfjAf

dttnfjtsfc

)24.4(2 tfx m

)25.4()]sin(exp[2

dxnxxj

Ac c

n

)26.4()]sin(exp[2

1)(

dxnxxjJ n

)27.4()(ncn JAc

(4.28))2exp()()(~

tnfjJAts mn

nc

(4.29)])(2exp[)(Re)(

tnffjJAts mcn

nc

• In the simplified form of Eq. (4.29)

(4.30)])(2cos[)()( tnffJAts mcn

nc

(4.31))]()()[(2

)( mcmcn

nc nfffnfffJ

AfS

• Properties of single-tone FM for arbitrary modulation index β

1. For different integer values of n,

2. For small values of the modulation index β

3. The equality holds exactly for arbitrary β

)32.4(even for ),()( nJJ nn

)33.4(odd for ),()( nJJ nn

)34.4(

2,0)(

,2

)(

,1)(

1

0

nJ

J

J

n

)35.4(1)(2

nnJ

1.The spectrum of an FM wave contains a carrier component and an infinite set of side frequencies located symmetrically on either side of the carrier at frequency separations of fm,2fm, 3fm….

2.The FM wave is effectively composed of a carrier and a single pair of side-frequencies at fc±fm

3.The amplitude of the carrier component of an FM wave is dependent on the modulation index βThe average power of such a signal developed across a 1-ohm resistor is also constant.

• The average power of an FM wave may also be determined form

2

av 2

1cAP

)36.4()(2

1 22

nc JAP

4.6 Transmission Bandwidth of FM waves

• Carson’s Rule– The FM wave is effectively limited to a finite number of

significant side-frequencies compatible with a specified amount of distortion

– Two limiting cases1. For large values of the modulation index β, the bandwidth

approaches, and is only slightly greater than the total frequency excursion 2∆f,

2. For small values of the modulation index β, the spectrum of the FM wave is effectively limited to the carrier frequency fc and one pair of side-frequencies at fc±fm, so that the bandwidth approaches 2fm

– An approximate rule for the transmission bandwidth of an FM wave

)37.4(1

1222

fffB mT

• Universal Curve for FM Transmission Bandwidth– A definition based on retaining the maximum number of

significant side frequencies whose amplitudes are all greater than some selected value.

– A convenient choice for this value is one percent of the unmodulated carrier amplitude

– The transmission bandwidth of an FM waves• The separation between the two frequencies beyond which

none of the side frequencies is greater than one percent of the carrier amplitude obtained when the modulation is removed.

• As the modulation index β is increased, the bandwidth occupied by the significant side-frequencies drops toward that value over which the carrier frequency actually deviates.

• Arbitrary Modulating Wave– The bandwidth required to transmit an FM wave

generated by an arbitrary modulating wave is based on a worst-case tone-modulation analysis

– The deviation ratio D

– The generalized Carson rule is

)38.4(W

fD

)39.4()W(2 fBT