1

TEL312 Electronic Communications Fundamentals

Angle Modulation – Basic Concepts

Reference: Tomasi, Chapters 7 - 8

TEL312 Electronic Communications Fundamentals

General Angle-Modulated Signal

If the modulating signal is proportional to the phase deviation, then we have phase modulation (PM):

lt.radians/voin modulator,theof

ysensitivitdeviation phasetheiswhere pk

( ))(2cos)( ttfVts cc θπ +=

frequency ousinstantane)(2)()( === tmkdt

tdtf fπθ

)()( tmkt p=θ

If the modulating signal is proportional to the angular frequency deviation, then we have frequency modulation (FM):

Hz/volt.in modulator,theofysensitivitdeviationtheiswhere frequencyk f

radiansinDeviationPhase)( =tθ

2

TEL312 Electronic Communications Fundamentals

Frequency Modulation

Frequency modulation implies that ddtθ is proportional to the modulating signal.

)(2 tmkdtd

fπθ =

Thus, in FM the instantaneous frequency varies linearly with the message signal.

( ))(

)(221

)(21)(

tmkf

tmkf

dttdftf

fc

fc

c

+=

+=

+=

ππ

θπ

per volt Hz of units has andmodulator FM the

ofysensitivitdeviationtheisfk

TEL312 Electronic Communications Fundamentals

The phase deviation θ(t) of FM signal is given by

Therefore, an FM signal can be expressed as:

in volts signal message theis Hz/voltin modulator FM theofy sensitivitdeviation theis

Hzin frequency carrier theis in volts, amplitude theis

m(t)k

fEwhere

f

cc

( ) ∫∫∫ ===t

f

t

f

t

dmkdmkdtdt

000

)(2)(2 ττπττπθθ

( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+=

∫ ττππ

θπ

dmktfV

ttfVtst

fcc

cc

0

22cos

)(2cos)(

3

TEL312 Electronic Communications FundamentalsFrequency deviation

Consider a sinusoidal modulating information signal given by

m t Am fmt( ) cos( )= 2π

The instantaneous frequency of the resulting FM signal equals

( ) )2cos()( tfAkftmkftf mmfcfc π+=+=

The maximum change in instantaneous frequency f(t) from the carrier frequency fc, is known as frequency deviation Δf. In the case of , thepeak frequency deviation is

mf Akf =Δ

The frequency deviation is a useful parameter for determining the bandwidth of the FM-signals

)2cos()( tfAtm mm π=

TEL312 Electronic Communications FundamentalsPhase deviation of FM signal

The ratio of the frequency deviation ∆f to the message frequency fm is called themodulation index of the FM signal. We denote it by:

mffΔ=β

β is unitless. For FM, it represents the depth of modulation achieved for a given modulating signal frequency.

( ) ( )

( ) ( )

( ) ( ) ( )tftffftf

fAk

ftfAkdfAk

dfAkdmkt

mmm

mm

mf

m

mmf

t

mmf

t

mmf

t

f

πβππ

πππττππ

ττππττπθ

2sin2sin2sin

22sin22cos2

2cos2)(2

0

00

=Δ

==

==

==

∫

∫∫In the case where the message signal is a sinusoid, the phase deviation is:

Hzin frequency message Hzin deviation frequency peak

==Δ

mff

4

TEL312 Electronic Communications Fundamentals

TEL312 Electronic Communications Fundamentals

The FM signal is given by

Depending on the value of the modulation index β, we may distinguish two cases of frequency modulation:

-Narrow-Band FM

-Wide-Band FM.

( ))(2cos)( ttfAts cc θπ +=

( ) ( )tft mπβθ 2sin=In the case where the message signal is a sinusoid, the phase deviation is:

The resulting FM signal is:

( )( )( )tftfA

ttfAts

mcc

cc

πβπθπ

2sin2cos)(2cos)(

+=+=

5

TEL312 Electronic Communications Fundamentals

Narrow-band Frequency ModulationFor small values of β,

cos(β sin(2π fm t)) ~ 1sin(β sin(2π fm t)) ~ β sin(2π fm t)

Thus the expression for FM signal can be expanded as:

x t Ac f ct Ac f ct fmt( ) cos( ) sin( ) sin( )= −2 2 2π π β πbecause ( ) BABABA sinsincoscoscos −=+

which may be written as follows

{ }x t Ac fct Ac fc fm t fc fm t( ) cos( ) cos[ ( ) ] cos[ ( ) ]= + + − −2 12 2 2π β π π

because

( ) ( )[ ]BABABA +−−= coscos21sinsin

TEL312 Electronic Communications Fundamentals

fc

Ac

f

fc +fmfc -fm

Bandwidth=2fm

cAβ21

cAβ21

Amplitude spectrum (single-sided plot)

6

TEL312 Electronic Communications Fundamentals

Wide-band Frequency ModulationThe general expression for FM signal can be analyzed to give the spectral components of wide-band FM signal. In order to compute the spectrum of an angle-modulated signal with a sinusoidal message signal, let

θ π( ) s i n ( )t ff m

f m t= Δ 2

The corresponding FM signal

))2sin(2cos()( tmftcfcAtx πβπ +=and may alternatively be written as

x t Ac e j cte j fmt( ) Re sin( )=ω β π2

where Re(x) denotes the real part of x.The parameter β is known as the modulation index and is the maximum value of phase deviation of the FM signal.