Narrow-band Frequency
Modulation
KEEE343 Communication Theory
Lecture #16, May 3, 2011
Prof.Young-Chai Ko
koyc@korea.ac.kr
Summary
·•Narrowband Frequency Modulation
·•Wideband Frequency Modulation
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:
m(t) = Am cos(2 fm t)
•
FM signal
•
Instantaneous frequency
fi (t) = fc + kf Am cos(2 fm t)
= fc +
•
Phase
i (t)
=
=
Z
t
f cos(2 fm t)

f = k f Am
f
2⇥
fi (⇤ ) d⇤ = 2⇥ fc t +
sin(2⇥fm t)
2⇥fm
0
f
2⇥fc t +
sin(2⇥fm t)
fm
•
•
Definitions
•
Phase deviation of the FM wave
•
Modulation index of the FM wave: fm
=
f
fm
Then, FM wave is
s(t) = Ac cos[2⇥fc t + sin(2⇥fm t)]
s(t) = Ac cos(2⇥fc t) cos( sin(2⇥fm t))
•
To be narrow-band FM,
•
For small
Ac sin(2⇥fc t) sin( sin(2⇥fm t))
should be small.
compared to 1 radian, we can rewrite
cos[ sin(2⇥fm t)] ⇡ 1,
s(t) ⇡ Ac cos(2⇥fc t)
sin[ sin(2⇥fm t)] ⇡
sin(2⇥fm t)
Ac sin(2⇥fc t) sin(2⇥fm t)
[Ref: Haykin & Moher, Textbook]
Polar Representation from Cartesian
•
Consider the modulated signal
s(t) = a(t) cos(2 fc t + ⇥(t))
which can be rewritten as
s(t) = sI (t) cos(2 fc t)
sQ (t) sin(2 fc t)
where
sI (t) = a(t) cos( (t)), and, sQ (t) = a(t) sin( (t))
and
a(t) =
⇥
s2I (t)
+
s2Q (t)
⇤ 12
,
and
(t) = tan
1

sQ (t)
sI (t)
•
Approximated narrow-band FM signal can be written as
s(t) ⇡ Ac cos(2⇥fc t)
•
Envelope
⇥
a(t) = Ac 1 +
•
2
⇤
sin2 (2⇥fm t) ⇡ Ac
Maximum value of the envelope
Amax = Ac
•
Ac sin(2⇥fc t) sin(2⇥fm t)
✓
1
1+
2
Minimum value of the envelope
Amin = Ac
2
◆
✓
1
1+
2
2
sin2 (2⇥fm t)
◆
for small
•
The ratio of the maximum to minimum value
Amax
=
Amin
•
2
◆
1
cos(2⇥(fc + fm )t)
2
Average power
Pav =
•
1
1+
2
Rewriting the approximated narrow-band FM wave is
s(t) ⇡ Ac cos(2⇥fc t) +
•
✓
1 2
A +
2 c
✓
1
Ac
2
◆2
+
✓
Average power of the unmodulated wave
Pc =
1 2
A
2 c
1
Ac
2
1
Ac cos(2⇥(fc
2
◆2
=
1 2
A (1 +
2 c
fm )t)
2
)
•
•
Ration of the average power to the power of the unmodulated wave
Pav
=1+
Pc
2
Comparison with the AM signal
•
Approximated Narrow-band FM signal
1
s(t) ⇡ Ac cos(2⇥fc t) +
cos(2⇥(fc + fm )t)
2
•
1
Ac cos(2⇥(fc
2
fm )t)
AM signal
1
sAM (t) = Ac cos(2 fc t) + µAc {cos[2 (fc + fm )t)] + cos[2 (fc
2
fm )t]}
difference between the narrow-band FM and AM waves
•
Bandwidth of the narrow-band FM: 2fm
•
Phasor interpretation
[Ref: Haykin & Moher, Textbook]
•
Angle
⇥(t) = 2⇤fc t + ⌅(t) = 2⇤fc t + tan
•
1
( sin(2⇤fm t))
Using the power series of the tangent function such as
tan
1
(x) ⇥ x
1 3
x + ···
3
•
Angle can be approximated as
⇥(t) ⇡ 2⇤fc t +
•
sin(2⇤fm t)
1
3
3
sin3 (2⇤fm t)
Ideally, we should have
⇥(t) ⇡ 2⇤fc t + sin(2⇤fm t)
•
The harmonic distortion value is
3
D(t) =
•
3
sin3 (2⇥fm t)
The maximum absolute value of D(t) is
3
Dmax =
3
•
For example for
Dmax
= 0.3,
0.33
=
= 0.009 ⇡ 1%
3
which is small enough for it to be ignored in practice.
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 i (t) contains harmonic distortion in the form of thirdand higher order harmonics of the modulation frequency fm
Wide-Band Frequency Modulation
•
Spectral analysis of the wide-band FM wave
s(t) = Ac cos[2⇥fc t +
sin(2⇥fm t)]
or
s(t) = < [Ac exp[j2⇥fc t + j sin(2⇥fm t)]] = <[s̃(t) exp(j2⇥fc t)]
where s̃(t) = Ac exp [j sin(2⇥fm t)] is called “complex envelope”.
Note that the complex envelope is a periodic function of time with a
fundamental frequency fm which means
s̃(t) = s̃(t + kTm ) = s̃(t +
where
Tm = 1/fm
k
)
fm
•
Then we can rewrite
s̃(t) = s̃(t + k/fm )
= Ac exp[j sin(2⇥fm (t + k/fm ))]
= Ac exp[j sin(2⇥fm t + 2k⇥)]
= Ac exp[j sin(2⇥fm t)]
•
Fourier series form
s̃(t) =
1
X
cn exp(j2 nfm t)
n= 1
where
cn
= fm
Z
= f m Ac
1/(2fm )
s̃(t) exp( j2⇥nfm t) dt
1/(2fm )
Z
1/(2fm )
exp[j sin(2⇥fm t)
1/(2fm )
j2⇥nfm t] dt
•
Define the new variable: x = 2 fm t
Then we can rewrite
Ac
cn =
2⇥
•
Z
nx)] dx
nth order Bessel function of the first kind and argument
1
Jn ( ) =
2⇥
•
exp[j( sin x
Z
exp[j( sin x
nx)] dx
Accordingly
c n = Ac J n ( )
which gives
s̃(t) = Ac
1
X
n= 1
Jn ( ) exp(j2⇥nfm t)
•
Then the FM wave can be written as
s(t)
=
=
=
<[s̃(t) exp(j2⇥fc t)]
"
#
1
X
Jn ( ) exp[j2⇥n(fc + fm )t]
< Ac
Ac
n= 1
1
X
Jn ( ) cos[2⇥(fc + nfm )t]
n= 1
•
Fourier transform
1
Ac X
S(f ) =
Jn ( ) [⇥(f
2 n= 1
fc
nfm ) + ⇥(f + fc + nfm )]
which shows that the spectrum consists of an infinite number of delta
functions spaced at f = fc ± nfm for n = 0, +1, +2, ...
Properties of Single-Tone FM for Arbitrary Modulation
Index
1.
For different values of n
Jn ( ) = J
Jn ( ) =
2.
6.
n(
J
),
n(
for n even
),
For small value of J0 ( )
⇡
J1 ( )
⇡
Jn ( )
⇡
1,
2
0,
n>2
The equality holds exactly for arbitrary 1
X
n= 1
Jn2 ( ) = 1
for n odd
[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 fm , 2fm , 3fm ...
2.
The FM wave is effectively composed of a carrier and a single pair of sidefrequencies at fc ± fm .
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 1ohm resistor is also constant:
Pav
1 2
= Ac
2
The average power of an FM wave may also be determined from
Pav
1
1 2 X 2
= Ac
J ( )
2 n= 1 n