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

Appendices
Contents I Representingacontinuousfunctionandasequenceasa
sumofweightedimpulses 979
II TheGreekalphabet 981
III SIunitsandprefixes 982
IV Thebinomialexpansionof ( )
n−N n
n
982
AppendixI REPRESENTINGACONTINUOUSFUNCTIONANDA
SEQUENCEASASUMOFWEIGHTEDIMPULSES
Inthisappendixweshowhowacontinuousfunction, f (t),oradiscretesequence, f[k],
obtained by sampling this function, can be represented as a sum of weighted impulses.
SucharepresentationisimportantinthestudyoftheztransformandthediscreteFourier
transform.
Thedeltafunction
ThedeltafunctionisintroducedinChapter2.Foreaseofreferencewerestateitsdevelopmenthere.Thedeltafunction
δ(t),orunitimpulsefunction,isthelimitofarectangle
function bounding an area of 1 unit, and located at the origin, as its width approaches
zero,anditsheightincreasesaccordinglytoensurethatthearearemains1.Theenclosed
area is known as the strength, or the weight, of the impulse. This is illustrated in Figure
AI.1. Note that the impulse is represented by an arrow and the height of the arrow
gives the strength of the impulse. If the impulse occurs at the pointt = d, then this is
written δ(t −d).Further,bymultiplyingthedeltafunctionbyanumberA,togiveAδ(t),
we obtain the limit of a rectangle function bounding an area ofA. This is an impulse of
strengthA.
Samplingacontinuousfunction
Now consider a function f (t) defined fort 0 as shown in Figure AI.2. Suppose this
function is sampled at timest = 0,T,2T,3T,...,kT,..., to generate a sequence of
values f(0), f(T), f(2T), f(3T),..., f(kT),..., which we shall write as f[0], f[1],
f[2],f[3],...,f[k],....
Note from the discussion above that the quantity f[0]δ(t) is the limit of a rectangle
functionlocated atthe originand boundinganarea equal to f[0].