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

124 Chapter 3 The trigonometric functions
Sinc x
1
0.75
0.5
0.25
–4p –3p –2p –p p 2p 3p 4p
x
–0.25
Figure3.13
The standard sincfunction withoutnormalization.
Engineeringapplication3.1
Zerocrossingpointsofthenormalizedsinc(x)function
The sinc function often appears in communications engineering and signal processing.Ingeneral,asignalcanberegardedasbeingmadeofalargenumberofindividual
frequencycomponents.Inthecaseofasquarepulse,thegraphshowingtheamplitude
of these components against their frequencies takes the formof a sincfunction.
Find the zero crossing points of the normalized sincfunction,
⎧
⎨ sin(πx)
x≠0
sinc(x) = πx
⎩
1 x=0
Solution
It is evident from Figure 3.13 that the sinc function which has not been normalized
hasthesamezerocrossingpointsassinx,withtheexceptionofthecrossingatx = 0,
thatisatx=±π,±2π,±3π,...,±nπ,....
Thenormalizedsincfunctionis0whensin(πx)is0.Now,frompage121wesee
that
and so
sinx=0whenx=...−3π,−2π,−π,0,π,2π,3π,...
sinπx=0whenπx=...−3π,−2π,−π,0,π,2π,3π,...
and hence
sin(πx) =0whenx = −3,−2,−1,0,1,2,3...
Noting thatsinc(0) isdefined tohave a value of 1,then wesee that
sinc(x)=0whenx=...−3,−2,−1,1,2,3,...