Modulação FM

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'(t) = A ✓(t)
✓(t)
'(t) = A (!ct + ✓ )
✓(t) !c
�t '(t) !c
✓(t)
!i (t) =
d✓(t)
dt
✓(t) =
Z t
�1
!i (↵)d↵
✓(t)
✓(t) = !ct + ✓ + kpm(t)
✓ =
✓(t) = !ct + kpm(t)
'PM(t) = A [!ct + kpm(t)]
!i (t) =
d✓(t)
dt
= !c + kpm˙(t)
!i (t) = !c + kfm(t)
✓(t) =
Z t
�1
[!c + kfm(↵)]d↵ = !ct + kf
Z t
�1
m(↵)d↵
'FM(t) = A
h
!ct + kf
Z t
�1
m(↵)d↵
i
'EM(t) = A [!ct + (t)]
= A
h
!ct +
Z t
�1
m(↵)h(t � ↵)d↵
i
h(t) = kp�(t)
h(t) = kf u(t)
P =
A
'FM 'PM m(t)
kf = ⇡ ⇥ kp = ⇡ fc = MHz
fi = fc +
kf
⇡
m(t) = + m(t) (FM)
fi = fc +
kp
⇡
m˙(t) = + m˙(t) (PM)
'FM 'PM m(t)
kf = ⇡ ⇥ kp = ⇡ fc = MHz
fi = fc +
kf
⇡
m(t) = + m(t) (FM)
fi = fc +
kp
⇡
m˙(t) = + m˙(t) (PM)
'FM 'PM m(t)
kf = ⇡ ⇥ kp = ⇡/ fc = MHz
fi = fc +
kf
⇡
m(t) = + m(t) (FM)
'PM = A [!ct +
⇡
m(t)] (PM)
'FM 'PM m(t)
kf = ⇡ ⇥ kp = ⇡/ fc = MHz
fi = fc +
kf
⇡
m(t) = + m(t) (FM)
'PM = A [!ct +
⇡
m(t)] (PM)
a(t) =
Z t
�1
m(↵)d↵
'ˆFM(t) = Ae
j [!c t+kf a(t)] = Ae jkf a(t)e j!c t
'FM(t) = Re{'ˆFM(t)}
e jkf a(t)
'ˆFM(t) = A
h
+ jkf a(t)� kf! a (t) + · · ·+ j
n k
n
f
n!
an(t) + ...
i
e j!c t
'FM(t) = Re{'ˆFM(t)}
= A
h
!ct � kf a(t) !ct � kf! a (t) !ct
+
kf
!
a (t) !ct + · · ·
i
m(t) B a(t) B
a (t) a (t) an(t) B B
nB
|kf a(t)|⌧
'FM(t) ' A
h
!ct � kf a(t) !ct
i
B
'PM(t) ' A
h
!ct � kpm(t) !ct
i
m(t) = Amcos(wmt) = Amcos( ⇡fmt) fm = B
fi (t) = fc +
kFAm
⇡
cos( ⇡fmt) = fc +�f cos( ⇡fmt)
�f =
kFAm
⇡
�f = kFAm
�f
fc
�f
�f fm
� = �ffm =
�f
B
'FM(t) ' A
h
coswct � �sen(wmt) · sen(wct)
i
|kf a(t)|⌧
BP = BFM = �f + fm = �f ( + � )
W = ⇡fm
BP = WFM = (�! + wm)
� �f � fm = B
BFM ' �f
� �f ⌧ B
BFM ' fm = B
�f = kP
A0
⇡
A0 m(t)
BP = BPM = �f + fm = (kP
A0
⇡
+ fm)
m(t) = ↵ !mt
a(t) =
Z t
�1
m(↵)d↵ =
↵
!m
!mt
'ˆFM(t) = Ae
j [!c t+kf
↵
!m
!mt]
�w = kf ↵
� =
↵kf
!m
=
�!
!m
=
�!
⇡fm
=
�f
fm
'ˆFM(t)
'ˆFM(t) = Ae
j [!c t+� !mt] = Ae j!c t(e j� !mt)
⇡/!m
e� !mt =
1X
n=�1
Cne
j!mt
Cn =
!m
⇡
Z ⇡/!m
�⇡/!m
e j� !mte�jn!mtdt =
⇡
Z ⇡
�⇡
e j(� x�nx)dx
Jn(�)
Jn(�)
e j� !mt =
1X
n=�1
Jn(�)e
j!mt
'ˆFM(t) = A
1X
n=�1
Jn(�)e
j(!c t+n!mt)
'FM(t) = A
1X
n=�1
Jn(�) (!c + n!m)t
!c ± !m
!c ± !m !c ± n!m
Jn(�) n > � +
� +
BFM = nfm = (� + )fm = (�f + fm) = (�f + B)
BFM BPM m(t)
kf = ⇡ ⇥ kp = ⇡ B
m(t) =
X
n
Cn n! t, ! = ⇡
Cn =
⇢
⇡ n , n
, n
�
fm = B = kHz
BFM = (�f + B) = kHz(a)
BPM = (�f + B) = kHz(a)
BFM = (�f + B) = kHz(b)
BPM = (�f + B) = kHz(b)
wc = ⇡x
'FM(t) = cos(wct + t + ⇡t)
�
'FM(t)
'FM(t) ' A
h
!ct � kf a(t) !ct
i
'PM(t) ' A
h
!ct � kpm(t) !ct
i
A˙(t)
A
A
vi (t) = A(t) ✓(t), ✓(t) = !ct + kf
Z t
�1
m(↵)d↵
vo(✓) =
⇢
, ✓ >
� , ✓ <
�
vo(✓) ⇡
vo(✓) =
⇡
⇣
✓ � ✓ + ✓ + · · ·
⌘
vo [✓(t)] = vo [!ct + kf
Z t
�1
m(↵)d↵]
=
⇡
n
[!ct + kf
Z t
�1
m(↵)d↵]
� [!ct + kf
Z t
�1
m(↵)d↵] + · · ·
o
vo [✓(t)] !c
eo(t) =
⇡
[!ct + kf
Z t
�1
m(↵)d↵]
!c , !c , · · · , n!c
�f , �f , · · · , n�f
n!c
, MHz
�f = kHz
fc = kHz
|kf a(t)|⌧
� ⌧
�f = Hz
Hz < B < kHz , < � < ,
�f kHz
⇥ = �f = , kHz
fc = ⇥ kHz = , MHz
�f = ⇥ Hz = , kHz
fmix = fc � fI = , MHz � , MHz = , MHz
fc = , MHz �f = , kHz
fc = ⇥ , MHz = , MHz
�f = ⇥ , kHz = , kHz
m(t)
!i (t) = !c + kfm(t)
!o = p
LC
C = C � km(t)
!o = q
LC [ � km(t)C ]
= p
LC [ � km(t)C ] /
⇡ p
LC
[ +
km(t)
C
],
km(t)
C
⌧
!c = p
LC
, kf =
k!c
C
! = !c + kfm(t)
C = C � km(t)
�C = C � C = kmp = kf C mp
!c
�C
C
=
kfmp
!c
=
kfmp
⇡fc
=
�f
fc
!i = !c + kfm(t)
|H(!)| = a! + b
|H(!)| = !
'FM(t)
'˙FM(t) =
d
dt
{A [!ct + kf
Z t
�1
m(↵)d↵]}
= A[!c + kfm(t)] [!ct + kf
Z t
�1
m(↵)d↵]
!VCO = !c + ceo(t)
B [!ct + ✓o(t)]
!c + ✓˙o(t)
✓˙o(t) = ceo(t)
H(s)
eo(t) = h(t) ⇤ AB [✓i (t)� ✓o(t)]
= AB
Z t
h(t � ⌧) [✓i (⌧)� ✓o(⌧)]d⌧
✓˙o(t) = AK
Z t
h(t � ⌧) ✓e(⌧)d⌧
✓i (t)
✓i (t) = kf
Z t
�1
m(↵)d↵
✓o(t) = kf
Z t
�1
m(↵)d↵� ✓e
eo(t) =
c
✓˙o(t) ' kf
c
m(t)
�f
y(t) = a x(t) + a x (t)
x(t) = [!ct + (t)]
y(t) = a [!ct + (t)] + a [!ct + (t)]
=
a
+ a [!ct + (t)] +
a
[ !ct + (t)]
y(t) =
a
+ a
h
!ct + kf
Z t
�1
m(↵)d↵
i
+
a h
!ct + kf
Z t
�1
m(↵)d↵
i
!c !c
y(t) = a + a x(t) + a x (t) + · · ·+ anxn(t)
y(t) = c + c
h
!ct + kf
Z t
�1
m(↵)d↵
i
+c
h
!ct + kf
Z t
�1
m(↵)d↵
i
+ · · ·+ cn
h
n!ct + nkf
Z t
�1
m(↵)d↵
i
A !ct
I (!c + !)t
r(t)
r(t) = A !ct + I (!c + !)t
= (A+ I !t) !ct � I !t !ct
= Er (t) [!ct + d(t)]
 d(t) =
I !t
A+ I !t
⇡ I
A
!t, (I ⌧ A)
yd(t) =
I
A
!t (PM)
yd(t) =
I!
A
!t (FM)
!
Hp(j!)
Hd(j!)
Hp(!)
Hp(!) = K
j! + !
j! + !
K K = ! /!
Hp(!) =
⇣!
!
⌘ j! + !
j! + !
'
⇢
, ! ⌧ !
j!
! , ! ⌧ ! ⌧ !
�
f = , kHz f = kHz
Hd(!)
Hd(!) =
!
j! + !
�f
fI , MHz
L R
L+ R
	Conceitos Básicos
	Análise Espectral
	Geração de FM
	Demodulação de FM
	Extras