angle modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)pm.pp4.pdf · pm(t) = v...
TRANSCRIPT
![Page 1: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/1.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 1 of 11
Go Back
Full Screen
Close
Quit
Fundamentals ofCommunications
(XE37ZKT), Part I
Angle Modulation
Josef Dobes
3rd
![Page 2: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/2.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 2 of 11
Go Back
Full Screen
Close
Quit
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
![Page 3: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/3.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 2 of 11
Go Back
Full Screen
Close
Quit
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
![Page 4: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/4.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 3 of 11
Go Back
Full Screen
Close
Quit
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)
],
![Page 5: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/5.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 3 of 11
Go Back
Full Screen
Close
Quit
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.
![Page 6: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/6.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 3 of 11
Go Back
Full Screen
Close
Quit
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)
)]
![Page 7: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/7.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 3 of 11
Go Back
Full Screen
Close
Quit
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) + · · ·
![Page 8: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/8.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 4 of 11
Go Back
Full Screen
Close
Quit
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)
![Page 9: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/9.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 4 of 11
Go Back
Full Screen
Close
Quit
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
)]
![Page 10: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/10.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 4 of 11
Go Back
Full Screen
Close
Quit
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
)]
![Page 11: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/11.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 4 of 11
Go Back
Full Screen
Close
Quit
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
)]
![Page 12: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/12.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 4 of 11
Go Back
Full Screen
Close
Quit
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.
![Page 13: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/13.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 5 of 11
Go Back
Full Screen
Close
Quit
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
![Page 14: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/14.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 6 of 11
Go Back
Full Screen
Close
Quit
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;
![Page 15: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/15.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 7 of 11
Go Back
Full Screen
Close
Quit
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
))
![Page 16: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/16.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 8 of 11
Go Back
Full Screen
Close
Quit
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’);
![Page 17: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/17.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 9 of 11
Go Back
Full Screen
Close
Quit
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’);
}
}
![Page 18: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/18.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 10 of 11
Go Back
Full Screen
Close
Quit
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
![Page 19: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/19.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 10 of 11
Go Back
Full Screen
Close
Quit
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.
![Page 20: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/20.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 10 of 11
Go Back
Full Screen
Close
Quit
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.
![Page 21: Angle Modulation - radio.feld.cvut.czradio.feld.cvut.cz/personal/dobes2/(3)PM.pp4.pdf · PM(t) = V c sin ω ct+βsin(ω mt), where β is the modulation index, which is the peak phase](https://reader034.vdocuments.site/reader034/viewer/2022042410/5f27812c00edbf47a15b70cb/html5/thumbnails/21.jpg)
Outline
Initial Formulae
Final Time Domain . . .
Frequency Domain
Home Page
JJ II
J I
Page 11 of 11
Go Back
Full Screen
Close
Quit
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