Angle Modulation and FM Intro
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Transcript of Angle Modulation and FM Intro
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
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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)(
+=+=
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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)
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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.