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Final Time Domain . . .
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Fundamentals ofCommunications
(XE37ZKT), Part I
Angle Modulation
Josef Dobes
3rd
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
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1. Outline
• Characterization of PM in time domain
– Initial formulae
– Formulation as a sum of harmonic components
– Graphical representation of frequency modulated carrier
– Necessity of Bessel functions
– Programming the Bessel functions
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
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1. Outline
• Characterization of PM in time domain
– Initial formulae
– Formulation as a sum of harmonic components
– Graphical representation of frequency modulated carrier
– Necessity of Bessel functions
– Programming the Bessel functions
• Characterization of PM in frequency domain
– Carson formula
– PM bandwidth
– PM energetic properties
– Components’s spectral diagram
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
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2. Initial Formulae
The following is the simplest formula for the phase-modulated signal(see a graphical representation):
vPM(t) = Vc sin[ωct + β sin (ωmt)
],
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Initial Formulae
Final Time Domain . . .
Frequency Domain
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2. Initial Formulae
The following is the simplest formula for the phase-modulated signal(see a graphical representation):
vPM(t) = Vc sin[ωct + β sin (ωmt)
],
where β is the modulation index, which is the peak phase deviation, inradians, of the carrier (β = ∆fmax
fm). The modulation index determines
the amplitudes and frequencies of the components of the modulated
wave.
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Initial Formulae
Final Time Domain . . .
Frequency Domain
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2. Initial Formulae
The following is the simplest formula for the phase-modulated signal(see a graphical representation):
vPM(t) = Vc sin[ωct + β sin (ωmt)
],
where β is the modulation index, which is the peak phase deviation, inradians, of the carrier (β = ∆fmax
fm). The modulation index determines
the amplitudes and frequencies of the components of the modulated
wave.Using the standard trigonometric formula, we obtain
vPM(t) = Vc
[sin (ωct) cos
(β sin (ωmt)
)+ cos (ωct) sin
(β sin (ωmt)
)]
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Initial Formulae
Final Time Domain . . .
Frequency Domain
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2. Initial Formulae
The following is the simplest formula for the phase-modulated signal(see a graphical representation):
vPM(t) = Vc sin[ωct + β sin (ωmt)
],
where β is the modulation index, which is the peak phase deviation, inradians, of the carrier (β = ∆fmax
fm). The modulation index determines
the amplitudes and frequencies of the components of the modulated
wave.Using the standard trigonometric formula, we obtain
vPM(t) = Vc
[sin (ωct) cos
(β sin (ωmt)
)+ cos (ωct) sin
(β sin (ωmt)
)]For the blue parts of the equation, the formulae that uses Besselfunctions of the first kind must be utilized:
cos(x sin α) = J0(x) + 2J2(x) cos(2α) + 2J4(x) cos(4α) + · · ·sin(x sin α) = 2J1(x) sin(α) + 2J3(x) sin(3α) + · · ·
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Final Time Domain . . .
Frequency Domain
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3. Final Time Domain Formula
Using the initial formulae of the above section, the final sequence canbe derived
vPM(t) = Vc
{J0(β) sin (ωct)
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3. Final Time Domain Formula
Using the initial formulae of the above section, the final sequence canbe derived
vPM(t) = Vc
{J0(β) sin (ωct)
+ J1(β)[sin
((ωc + ωm) t
)− sin
((ωc − ωm) t
)]
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Frequency Domain
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3. Final Time Domain Formula
Using the initial formulae of the above section, the final sequence canbe derived
vPM(t) = Vc
{J0(β) sin (ωct)
+ J1(β)[sin
((ωc + ωm) t
)− sin
((ωc − ωm) t
)]+ J2(β)
[sin
((ωc + 2ωm) t
)+ sin
((ωc − 2ωm) t
)]
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Initial Formulae
Final Time Domain . . .
Frequency Domain
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3. Final Time Domain Formula
Using the initial formulae of the above section, the final sequence canbe derived
vPM(t) = Vc
{J0(β) sin (ωct)
+ J1(β)[sin
((ωc + ωm) t
)− sin
((ωc − ωm) t
)]+ J2(β)
[sin
((ωc + 2ωm) t
)+ sin
((ωc − 2ωm) t
)]+ J3(β)
[sin
((ωc + 3ωm) t
)− sin
((ωc − 3ωm) t
)]
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Initial Formulae
Final Time Domain . . .
Frequency Domain
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3. Final Time Domain Formula
Using the initial formulae of the above section, the final sequence canbe derived
vPM(t) = Vc
{J0(β) sin (ωct)
+ J1(β)[sin
((ωc + ωm) t
)− sin
((ωc − ωm) t
)]+ J2(β)
[sin
((ωc + 2ωm) t
)+ sin
((ωc − 2ωm) t
)]+ J3(β)
[sin
((ωc + 3ωm) t
)− sin
((ωc − 3ωm) t
)]+ J4(β)
[sin
((ωc + 4ωm) t
)+ sin
((ωc − 4ωm) t
)]+ · · ·
},
which is infinite, of course. However, only a little group of members ofthe sequence is necessary to represent the frequency-modulated signal– see the Bessel functions.
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The modulating LF signal and frequency-modulated HF carrier canbe demonstrated using the following figure:
t
t
Modu
lating
Modu
late
d
fc
fm
= 24, β = 500
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The plot of the phase-modulated carrier was created using the follow-ing MetaPost-language code:
path p;
color c;
c:=(0.0,0.0,0.666);
p:=(0,0)
for ix=0 upto 1440/24:
...(4ix,40sind6ix)
endfor;
draw p withcolor c;
draw (0,-100)
for ix=0 upto 1440:
...(ix/6,-100+40sind(6ix+500sind0.25ix))
endfor;
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Frequency Domain
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Computed components’ amplitudes relative to the unmodified carrieramplitude (i.e., the Bessel functions of the first kind) can be demon-strated by the following standard plot:
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8
J0(β)
J1(β)J2(β)
J3(β) J4(β) J5(β) J6(β)
Modulation index (β)
Rel
ativ
eto
unm
odu
late
dca
rrie
r(J
n(β
))
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The graph of Bessel functions has been created using the followingC-language code (the Watcom compiler used):
#include <math.h>
#include <stdio.h>
const int bess_max = 8, beta_max = 8;
const int point_per_1 = 10;
void main () {
int np = point_per_1 * beta_max + 1, point, i_bess;
double beta, bess;
for (point = 0; point < np; point++) {
beta = (double)beta_max * point / (np-1);
bess = j0(beta);
printf("%lg %lg\n", beta, bess);
}
putchar(’\n’);
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for (i_bess = 1; i_bess <= bess_max; i_bess++) {
for (point = 0; point < np; point++) {
beta = (double)beta_max * point / (np-1);
bess = jn(i_bess, beta);
printf("%lg %lg\n", beta, bess);
}
putchar(’\n’);
}
}
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4. Frequency Domain
As shown, the theoretical infinite spectrum can be limited in an em-pirical way. For this purposes, the Carson formula can be used (here,the formula for FM is defined)
BFM = 2 (∆fmax + fm) = 2(1 + β)fm,
because the modulation index definition
β ,∆fmax
fm
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Frequency Domain
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4. Frequency Domain
As shown, the theoretical infinite spectrum can be limited in an em-pirical way. For this purposes, the Carson formula can be used (here,the formula for FM is defined)
BFM = 2 (∆fmax + fm) = 2(1 + β)fm,
because the modulation index definition
β ,∆fmax
fm
For the European standards, maximum LF frequency fm = 15 kHzand β = 5 are used. Therefore, the bandwidth for one transmitter isnecessary
2× (1 + 5)× 15 kHz = 180 kHz,
which is worse than that in AM.
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Final Time Domain . . .
Frequency Domain
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4. Frequency Domain
As shown, the theoretical infinite spectrum can be limited in an em-pirical way. For this purposes, the Carson formula can be used (here,the formula for FM is defined)
BFM = 2 (∆fmax + fm) = 2(1 + β)fm,
because the modulation index definition
β ,∆fmax
fm
For the European standards, maximum LF frequency fm = 15 kHzand β = 5 are used. Therefore, the bandwidth for one transmitter isnecessary
2× (1 + 5)× 15 kHz = 180 kHz,
which is worse than that in AM. However, the energetic properties ofFM are (much more) better than those in AM.
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The spectral and energetic properties can be demonstrated using thecomponents’ histogram – see Bessel functions (again, here for FM):
J 0(5
)
−J 1
(5)
J 1(5
)
J 2(5
)
J 2(5
)
−J 3
(5)
J 3(5
)
J 4(5
)
J 4(5
)
−J 5
(5)
J 5(5
)
J 6(5
)
J 6(5
)
−J 7
(5)
J 7(5
)
J 8(5
)
J 8(5
)
f c
f c−
f m
f c+
f m
f c−
2f m
f c+
2f m
f c−
3f m
f c+
3f m
f c−
4f m
f c+
4f m
f c−
5f m
f c+
5f m
f c−
6f m
f c+
6f m
f c−
7f m
f c+
7f m
f c−
8f m
f c+
8f m
BFM = 2fm(1 + β)∣∣β=5
= 12fm
f
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