Electronics & Communication Engg. Dept.
ANGLE MODULATION
Modulation
Amplitude Modulation
Double Sideband
Single Sideband
Vestigial sideband
Quadrature Amplitude Modulation
Angle
Modulation
Phase Modulation
Frequency Modulation
Angle Modulation
Let
i(t) be the instantaneous angle of a modulated sinusoidal carrier, i.e.,
where A
cis the constant amplitude.
The instantaneous frequency is
Observation: The signal s(t) can be thought of as a rotating phasor of length A
cand angle
i(t).
) (t m
), ( cos )
( t A t
s =
c
i) . ) (
( dt
t t d
ii
=
4
If s(t) were an unmodulated carrier signal, then the instantaneous angle would be
where c Constant angular velocity in rad/s
c Constant but arbitrary phase angle in radians
Let i(t) be varied linearly with the message signal m(t) and Фc = 0, then
where kp Phase sensitivity of the modulator in rad/volt In this case we say that the carrier has been phase modulated.
The phase modulated waveform is given by
Let the instantaneous frequency i(t) be varied linearly with the message signal m(t), i.e.,
c c
i
t t
( ) = +
), ( )
( t
ct k
pm t
i
= +
)) ( cos(
)
( t A t k m t
s
PM=
c
c+
p) ( )
( t
ck m t
i
= +
where k Frequency sensitivity of the modulator in rad/s/volt
In this case we say that the carrier has been frequency modulated and the instantaneous angle is obtained by integrating the instantaneous frequency, i.e.,
The modulated waveform is therefore described by
Observation: Both phase and frequency modulation are related to each other and one can be obtained from the other. Hence, we could deduce the properties of one of the two
modulation schemes once we know the properties of the other.
= + = +
=
t i t c c ti
t d k m d t k m d
0 0
0
) ( )
( )
( )
(
+
=
c c
tFM
t A t k m d
s
0
) ( cos
)
(
Relationship between Phase Modulation (PM) and Frequency Modulation (FM)
PM : FM :
such that,
8
PM modulation
The figure below shows a comparison between AM, FM and PM modulation of the same message waveform:
FM Generation
Frequency Modulation
Consider the frequency modulation of the message (tone)
The instantaneous frequency (in Hz) of the FM signal is
Define the maximum frequency deviation as
The instantaneous phase angle of the FM signal is
),
2 cos(
)
( t A f t
m =
m
m).
2 cos(
)
( t f k A f t
f
i=
c+
f m
mm
.
f
A k f =
) 2
sin(
2
) 2
sin(
2
) ( 2
) (
0
t f t
f
t f f
k A t
f
d f
t
m c
m m
m f
c t
i i
+
=
+
=
=
12
where is known as the FM modulation index (for a tone) or the maximum phase deviation (in rad) produced by the tone in question
The FM modulated tone is therefore given by:
The signal sFM(t) is nonperiodic unless fc = nfm, where n is a positive integer.
For the general case,
where the complex envelope of the FM signal is described by
Observation: Unlike sFM(t), se(t) is periodic with period 1/fm. ,
/ m
m
f A f
= k
( )
cos( cos 2 2 ) cos( sin( 2 2 ) sin( ) 2 ) sin( 2 ) .
) (
t f t
f t
f t
f A
t f t
f A
t s
m c
m c
c
m c
c FM
−
=
+
=
( ) ,
Re Re )
(
2
) 2 sin(
2 t f j e
t f t
f j c FM
c
m c
e t s
e A t
s
=
=
+) 2
)
sin((
c j f te
e
mA t
s =
Since se(t) meets the Dirichlet conditions, we can compute its Fourier series, i.e.,
where the Fourier series coefficients are given by
,
)
(
j2 nf tn
n e
e
mc t
s
−
=
=
. dt e
f A
dt e
e A f
dt e
) t ( s f
f / T
, dt e
) t ( T s
c
m
m m
m
m
m m
m
m
m m
f /
t f n ) t f sin(
j m
c
f /
f /
t f n j ) t f sin(
j c m
f /
f /
t f n j e
m
m /
T
/ T
t f n j e
n
−
−
−
−
−
−
−
=
=
=
=
=
2 1
2 2
2 1
2 1
2 2
2 1
2 1
2 2
2
2
1
1
14
Let Then,
and
where Is the Bessel function of the first kind of order n.
Therefore,
and the FM tone waveform is described by .
t f x = 2
mf dx dt
m2
= 1
), (
J A
dx A e
c
n c
nx x sin c j
n
=
=
−
−
2
−
−
) e dx
(
J
n j sinx nx2 1
−
=
=
n
t nf j n
c e
e
mJ A
t
s ( ) ( )
2 2 ( ) .
cos ) ( )
(
−
=
+
=
n
m c
n c
FM
t A J f nf t
s
In the frequency domain,
Average power of the FM waveform:
Across a 1 ohm resistor, the power of the FM waveform is But,
−
=
−
=
−
=
−
− +
+ +
=
+
=
+
=
=
n
m c
m c
n c
n
m c
n c
n
m c
n c
FM FM
nf f
f nf
f J f
A
t nf f
F J
A
t nf f
J A
F
t s
F f
S
) (
) 2 (
) (
) (
2 cos )
(
) (
2 cos ) ( ) ( )
(
2
2 / Ac
P =
−
=
=
n
n
c
J
P A ( )
2
2
2
16
Let us now take a look at the properties of the Bessel function.
1.
2.
Hence, the average power of an FM tone is Suppose is small, i.e., 0 < ≤ 0.3, then
•
•
•
Under the assumption that is small, the Fourier series representation of the FM waveform can be simplified to three terms.
) ( )
1 ( )
(
n n
n J
J = − −
1 )
2( =
−
= n
Jn
. 2
2 / Ac
1 )
0(
J2 / )
1(
J2 ,
0 )
( n
Jn
18
Thus, for small, the FM tone may be described by
In the frequency domain,
) t ) f f
( cos(
A ) t ) f f
( cos(
A )
t f cos(
A
) t ) f f
( cos(
) t ) f f
( cos(
) t f cos(
A )
t ( s
m c
c m
c c
c c
m c
m c
c c
FM
−
− +
+
=
+ + − −
2 2 2 2
2
2 2 2 2
2
( ) ( )
( ) ( )
( ) ( )
c m c m
c
m c
m c
c c
c c
FM
f f
f f
f A f
f f
f f
f A f
f f
f A f
f S
−
− +
+ +
+
+
− +
− +
−
− +
+
4
4 ) 2
(
A plot of the magnitude spectrum of the FM tone with small is shown below
The time domain FM waveform can be represented in phasor form as follows:
For arbitrary t = t0, and small , we can illustrate graphically the phasor representation and arrive at some conclusion.
+ − +
+
= A A f t A f t
S
FM c c m c2
m2 2 1
2 0
1
20 The following figure shows an example of the phasor representation
Observation:
The resultant phasor , has magnitude and is out of phase with respect to the carrier phasor
Analytically,
SFM
c,
FM A
S .
0
c A
cos( 2 ) sin( 2 ) cos( 2 ) sin( 2 )
2
1 + + − + + − +
+
= A A f t j f t f t j f t
S
FM c c m m m mBut,
and
Consequently, the resultant phasor is given by
The magnitude of the resultant may be approximated by
)
2 cos(
) 2
sin(
sin cos
) 2
cos(
) 2
cos(
0
t f
t f t
f t
f
m
m m
m
−
=
+
= +
−
)
2 sin(
) 2
cos(
sin cos
) 2
sin(
) 2
sin(
0
t f
t f t
f t
f
m
m m
m
=
+
−
= +
−
) 2
sin( f t jA
A
S
FM=
c+
c
m, ) 2
( 2 sin
1 1
) 2
( sin
2 2
2 2
2 2
+
+
=
t f A
t f A
A S
m c
m c
c FM
22
Finally, the magnitude and phase of the resultant are found to be
Observation:
• For an FM tone, the spectral lines sufficiently away from the carrier may be ignored because their contribution (amplitude) is very small.
FM Transmission Bandwidth:
For an FM tone, as becomes large Jn() has significant lines only for
All significant lines are contained in the frequency range
where f is the peak frequency deviation.
+ −
= cos( 4 )
4 1 4
) (
2 2
t f A
t
S
FM c
m sin( 2 )
tan )
( t
1f t
S
FM=
S=
−
m
. /
/
m mm
f
A f f f
k
n = =
, f f
f
f
c
m=
c
Let be small, i.e., 0 < ≤ 0.3, then
which means that only the first pair of spectral lines is significant, i.e., the significant lines are contained in the range fc ± fm
Observation: The previous analysis of an FM tone suggests that 1. For large the FM bandwidth is
2. For small the FM bandwidth is
In general, the FM transmission bandwidth may be approximated by
This is known as the Carson’s rule.
0 ),
( )
0
( J n
J
n
f B
FM= 2
. 2 m
FM f
B =
) / 1 1 ( 2
) /
1 ( 2
2 2
+
=
+
=
+
f
f f
f
f f
B
m m T
24
Alternative definition of FM tone transmission bandwidth:
A band of frequencies that keeps all spectral lines whose magnitudes are greater than 1% of the unmodulated carrier amplitude Ac, i.e.,
where
, 2
max mT
n f
B =
: ( ) 0 . 01 .
max
= max n J
n
n
General Case:
Let an arbitrary message signal m(t) have bandwidth Wm.
Define the peak frequency deviation and the deviation ratio by
and
Then Carson’s rule can be used to define the transmission bandwidth of an arbitrary FM signal, i.e., when m(t) is arbitrary.
Specifically, the FM transmission bandwidth can be defined by
) / 1 1 ( 2
) /
1 ( 2
2 2
D f
f W
f
W f
B
m m T
+
=
+
=
+
( )
ˆ k max m t f
f t
=
.
ˆ f / W
mD =
26
Example: In commercial FM in the US, f = 75 kHz, Wm = 15 kHz.
Therefore, the deviation ratio is D = 75 kHz/15 kHz = 5.
Using Carson’s rule, the transmission bandwidth is
Using the Universal curve, the transmission bandwidth is
In practice, FM radio in the US uses a transmission bandwidth of BT = 200 kHz.
, 180
) / 1 1 (
2 f D kHz
B
T= + =
. kHz f
.
B
T= 3 2 = 240
Generation of FM
The frequency of the carrier can be varied by the modulating signal m(t) directly or indirectly.
Direct generation of FM
If a very high degree of stability of the carrier frequency is not a concern, then we can generate FM directly using circuits without external crystal oscillators. Examples of this method are VCO’s, varactor diode modulators, reactance modulators, Crosby modulators (modulators that use automatic frequency control), etc..
+
R2
R1
C1
RFC1
C2
R5
RFC2
C5
L1 L2
+ +VCC
) (t s
28
Indirect generation of FM
Commercial applications of FM (as established by the FCC and other spectrum governing bodies) require a high degree of stability of the carrier frequency. Such restrictions can be satisfied by using external crystal oscillators, a narrowband phase modulator, several
stages of frequency multiplication and mixers.
Let us begin with the synthesis of narrow-band FM.
Narrowband frequency modulator The narrow band FM signal is given by
+
=
c c f
tNB
t A f t k m d
s
0
) ( 2
2 cos )
(
with kf (and thus fNB) small
Let us now consider a technique to increase the FM signal bandwidth.
Let sNB(t) be input to a nonlinear device with transfer characteristic y(t) = axn(t), where x(t) is its input, namely.
Nonlinear device.
Let , then at the output of the nonlinear device, we observe
Let us expand this last equation to infer the effect of this nonlinear device.
+
= c f t
i t f t k m d
0
) ( 2
2 )
(
) ( cos
)
( t aA t
y =
cn n
i30
cosni(t) can be expanded as follows:
Likewise,
Thus,
Expanding the last term of the previous equality, we get
) ( cos
) ( 2 2 cos ) 1
( 2 cos
1
) ( cos
) ( 2 cos 2 1
1
) ( cos
) ( cos
) ( cos
2 2
2 2
2
t t
t
t t
t t
t
i n i
i n
i n i
i n i
i n
−
−
−
−
+
=
+
=
=
) ( cos
) ( 2 2 cos ) 1
( 2 cos
) 1 (
cos
n−2
it =
n−4
it +
it
n−4
it
) ( cos
) ( 2 4 cos
) 1 ( cos
) ( 2 4 cos ) 1
( 2 cos
) 1 (
cos
n
it =
n−2
it +
it
n−4
it +
2
it
n−4
it
1 cos 4 ( ) cos ( ).
4 ) 1 ( cos
) ( 2
cos
2
it
n−4
it = +
it
n−4
it
Rewriting the equation before the last one, we get
The last term in the expansion of cosni(t) is given by
) ( cos
) ( 6 32 cos
1
) ( cos
) ( 2 32 cos
) 1 ( cos
) ( 4 16 cos
1
) ( cos
) ( 2 4 cos ) 1
( 8 cos
) 1 ( 2 cos
1
) ( cos
) ( 4 8 cos ) 1
( 8 cos
1
) ( cos
) ( 2 4 cos ) 1
( 2 cos
) 1 ( cos
6
6 6
4 4
2
4 4
4 2
t t
t t
t t
t t
t t
t t
t
t t
t t
i n
i
i n
i i
n i
i n
i i
n i
n
i n
i i
n
i n
i i
n i
n
−
−
−
−
−
−
−
−
−
−
+
+ +
+ +
=
+ +
+
=
).
( cos
) ( 1 cos
1
k
it
n k it
k−
−
32
Let n be an even number, then, when k = n, the last term is
If, on the other hand, n is an odd number, then when k = n-1, the last term is
Therefore, the last term in the expansion of cosni(t) is
So, can be expanded as
) ( 2 cos
1
1
n
it
n−
cos ( )
2 ) 1 ( ) 2 2 cos(
) 1 ( cos ) ( ) 1 2 cos(
1
1 1
2
n
it
it
nn
it
nn
it
n−
− =
−− +
−
) ( 2 cos
1
1
n
it
n−
) ( 2 cos
) ( 2 cos )
( cos )
(
0 1 2A
1n t
a t
c t
c c
t
y
n in c i
i
+ + +
−+
=
Example: Consider the cases when n = 2 and n = 3.
Let n = 2, then
or
Let n = 3, then
) ( cos
)
( t aA
2 2t
y =
c
i) ( 2 2 cos
2 2
) ( 2 cos ) 1
(
2 2
2
t aA aA t
aA t
y
c
i c c
i+
=
= +
) ( 3 4 cos
) ( 4 cos
3
) ( 2 cos
) 1 ( 3 2 cos 1 2 ) 1 ( 2 cos
1
) ( cos ) ( 2 2 cos ) 1
( 2 2 cos ) 1
(
3 3
3 3
aA t aA t
t t
t aA
t t
t aA
t y
i c
i c
i i
i c
i i
i c
+
=
+
+
=
+
=
34
Finally,
Let y(t) be input to an ideal bandpass filter with unity gain, bandwidth wide enough to accommodate spectrum of wide band signal and center frequency nfc, i.e.,
Ideal bandpass filter Then,
+
+ +
+
+
+
+
=
−
t f n c
n c
t f c
t f c
d m
k n t
f A n
a
d m
k t
f c
d m
k t
f c
c t
y
0 1
0 2
0 1
0
) ( 2
2 2 cos
) ( 4
4 cos )
( 2
2 cos )
(
+
=
n−cn c f
tWB
aA n f t n k m d
t s
0
1
cos 2 2 ( )
) 2
(
The instantaneous frequency of sWB(t) is
Observations about sWB(t) :
1. The carrier frequency is nfc
2. The peak frequency deviation is nfNB
These are the desired properties of the WB FM signal.
The overall frequency multiplier device is shown below:
Complete frequency multiplier
)
( )
( t nf nk m t
f
i=
c+
f36
Example: Noncommercial FM broadcast in the US uses the 88-90 MHz band and
commercial FM broadcast uses the 90-108 MHz band (divided into 200 kHz channels). In either case f = 75 kHz. Suppose we target a station with fc = 90.1 MHz. Then the
indirect FM generation method suggested by Armstrong enables us to achieve our goals.
Armstrong indirect method of FM generation
