Cos sin sin, Grf-1300a user manual and teaching materials, Cos[ ( ) – GW Instek GRF-1300A User Manual
Page 74: Cos[ sin( ), Ut m t, Ω + ϕ, Cos( sin ) ( ) 2 ( ) cos 2 2 ( ) cos 4
GRF-1300A User Manual and Teaching Materials
is the offset of the angular frequency determined by the modulating
signal U
Ω
The general expression for the FM signal:
0
0
( )
cos[
( )
]
t
cm
c
f
u t
U
t k
u t dt
Ω
=
ω +
+
∫
ϕ
ϕ
0
0
cos[
cos
]
t
cm
c
f
m
U
t k U
tdt
Ω
=
ω +
Ω +
∫
0
cos[
sin(
)
]
f
m
cm
c
k U
U
t
t
Ω
=
ω +
Ω +
Ω
ϕ
Assume that,
f
m
m
f
k U
M
Ω
Δω
=
=
Ω
Ω
In this formula, M
f
is called the FM index, Δω
m
is called the
maximum angular frequency deviation, its value is proportional to
the amplitude of the modulating signal.
2.
Frequency domain analysis
Expressed by the time domain FM wave
( )
cos[
sin(
)
0]
f
m
cm
c
k U
u t
U
t
t
ω
Ω
=
+
Ω + ϕ
Ω
0
cos[
sin(
)
]
cm
c
f
U
t m
t
ω
=
+
Ω + ϕ
Let the initial phase angle be 0 and expand as follows:
( )
[cos
cos(
sin
) sin
sin(
sin
)]
cm
c
f
c
f
u t
U
t
m
t
t
m
t
ω
ω
=
Ω +
Ω
When m
f
<<
1, cos(m
f
sin Ωt) ≈1
sin(m
f
sin Ωt)≈ (m
f
sin Ωt)
Then we get,
( )
cos
sin
sin
cm
c
f
cm
c
u t
U
t m U
t
t
ω
ω
=
+
Ω
cos
cos(
cos(
)
2
2
f
cm
f
cm
cm
c
c
c
m U
m U
U
t
t
ω
ω
ω
=
+
+ Ω) +
−
t
Ω
+
We can see when m
f
<<
1, the FM wave spectrum is composed of the
carrier, (ω
c
+Ω) frequency component and (ω
c
- Ω) frequency
component.
When m
f
>>
1
0
2
4
cos(
sin
)
( ) 2 ( ) cos 2
2 ( ) cos 4
......
f
f
f
f
m
t
J m
J m
t
J m
t
Ω =
+
Ω +
Ω
1
3
5
sin( sin ) 2 ( )sin
2 ( )cos3
2 ( )sin5
......
f
f
f
f
m
t
J m
t
J m
t
J m
t
Ω =
Ω +
Ω +
Ω +
In this formula, J
n
(m
f
) is called an n-order Bessel function of the
first kind.
There are an infinite number of frequency components in FM
waves, and they are distributed symmetrically around the center of
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