082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

600 Chapter 19 Ordinary differential equations I
and thisoutput waveform isshown inFigure 19.22.
VCO output y(t)
A
–A
Time, t
Figure19.22
Theoutput signal.
We may rewrite thisequation using phasor notation as:
v(t) =Re
[Ae j[2πf c t+βsin(2πf t)]] m
= Re [ Ae j2πf c t e jβsin(2πf m t)]
where Re means the real part of what follows. Notice that the second exponential
∞∑
term is in the same form as the Jacobi-Anger identity, e jxsinθ = J n
(x)e jnθ , and
n=−∞
can be expanded outinterms ofBessel functions of the first kind.Thus,
[ ∞
]
∑
v(t) =Re Ae j2πf c t J n
(β)e jn2πf m t
n=−∞
This can be rearranged by combining the exponentials:
[
]
∞∑
v(t) =Re A J n
(β)e jn2πf m t e j2πf c t
v(t) =Re
[
A
n=−∞
∞∑
n=−∞
J n
(β)e j2π(f c +nf m )t ]
Convertingbackfromthephasornotation,andforsimplicitytakingA = 1,weobtain:
v(t) =J 0
(β)cos(2πf c
t)
+J −1
(β)cos[2π(f c
− f m
)t] +J 1
(β)cos[2π(f c
+ f m
)t]
+J −2
(β)cos[2π(f c
−2f m
)t] +J 2
(β)cos[2π(f c
+2f m
)t]
+J −3
(β)cos[2π(f c
−3f m
)t] +J 3
(β)cos[2π(f c
+3f m
)t]
+ ...
We sawinSection 19.8.3 thatforinteger values ofn,J −n
(x) = (−1) n J n
(x), hence,
v(t) =J 0
(β)cos(2πf c
t)
−J 1
(β)cos[2π(f c
− f m
)t] +J 1
(β)cos[2π(f c
+ f m
)t]
+J 2
(β)cos[2π(f c
−2f m
)t] +J 2
(β)cos[2π(f c
+2f m
)t]
−J 3
(β)cos[2π(f c
−3f m
)t] +J 3
(β)cos[2π(f c
+3f m
)t]
+ ...
WecanseethattheoutputfromtheVCOisactuallymadeupfromasetofsinusoidal
waveformswithfrequenciesf c
−nf m
,...,f c
−3f m
,f c
−2f m
,f c
−f m
,f c
,f c
+f m
,f c
+
2f m
, f c
+ 3f m
,..., f c
+nf m
. This is known as the frequency content of the signal
andwillbediscussedinmoredetailinChapters23&24.Theequationsuggeststhat
a broadcast station with a carrier at f c
would have an infinite number of other fre-