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

Review exercises 19 601
quencies, known as side bands associated with it. The amplitudes of the side bands
are given by the values of the Bessel functions. When these are evaluated, the values
ofJ n
(β) generally diminish with increasingnhence only the sidebands near to
the carrier frequency have significant amplitude. Therefore frequencies that are a
long way away from f c
can be safely ignored when designing the communications
system.
TabulatedvaluesofBesselfunctionsareavailablesimilartothoseshowninTable
19.3. It can be seen how the signal amplitudes generally decrease with increasing
order of the Bessel function. A larger value of β also implies a larger number of
side bands are significant and need to be considered. The use of Bessel functions
whendesigningradiocommunicationssystemsisveryimportantbecauseeachradio
channel has an allocated bandwidth outside of which it should not stray. To do so
would create the possibility of different radio channels interfering with each other.
Thismathematicalmethodallowstheamplitudesofthesidebandstobecalculatedto
ensuretheyaresmallenoughsoastonotinterferesignificantlywithotherchannels.
Table19.3
Table ofBesselfunctionsJ n (β)ofinteger order,n,fordifferent values of
modulation index, β.
β n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7
0.00 1.00 – – – – – – –
0.25 0.98 0.12 – – – – – –
0.50 0.94 0.24 0.03 – – – – –
0.75 0.86 0.35 0.07 – – – – –
1.00 0.77 0.44 0.11 0.02 – – – –
1.25 0.65 0.51 0.17 0.04 – – – –
1.50 0.51 0.56 0.23 0.06 0.01 – – –
1.75 0.37 0.58 0.29 0.09 0.02 – – –
2.00 0.22 0.58 0.35 0.13 0.03 – – –
2.25 0.08 0.55 0.40 0.17 0.05 0.01 – –
2.50 -0.05 0.50 0.45 0.22 0.07 0.02 – –
2.75 -0.16 0.43 0.47 0.26 0.10 0.03 – –
3.00 -0.26 0.34 0.49 0.31 0.13 0.04 0.01 –
3.25 -0.33 0.24 0.48 0.35 0.17 0.06 0.02 –
3.50 -0.38 0.14 0.46 0.39 0.20 0.08 0.03 –
3.75 -0.40 0.03 0.42 0.41 0.24 0.10 0.04 0.01
4.00 -0.40 -0.07 0.36 0.43 0.28 0.13 0.05 0.02
REVIEWEXERCISES19
1 Find the general solution ofthe followingequations:
(a)
(c)
dx
dt =2x (b) (1+t)dx
dt = 3
dy
dx =y2 cosx
3
2 Solve dy = 2,subjecttoy(0) = 3.
dx
Find the general solution oftẋ+x=2t.
4 Solve dx
dt +2x=e2t cost
(a)byusingan integrating factor
(b)byfinding itscomplementary function anda
particular integral.