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

980 Appendices
d(t)
as h 0
1
1–
h
0 t
0
h
t
FigureAI.1
The limitofthe rectangle function is the delta function.
f(t)
f(3T)
f(kT)
f(0)
f(2T)
f(T)
0 t 0 T 2T 3T....... kT t
FigureAI.2
Sampling a continuous function f (t)gives the sequence f[k].
Similarly, f[1]δ(t −T )isthelimitofarectanglefunctionboundinganareaequalto
f[1]and located att =T.
In general f[k]δ(t −kT) is the limit of a rectangle function bounding an area equal
to f[k]andlocatedatt =kT.
The quantity
∞∑
f[k]δ(t −kT)
k=0
isaseries of delta functions. The area bounded by these isgiven by
∞∑
area=f[0]+f[1]+f[2]+···+f[k]+···,
thatis
f[k]
(AI.1)
k=0
Approximatingtheareaunderacurve
Now consider approximating the area under the continuous function f (t) by a series of
rectangular areas as shown in Figure AI.3. The first rectangle has widthT and height
f (0) and so its area isTf (0). Using our sampling notation this may be writtenTf[0].
Similarly, the second rectangle has widthT and height f (T) and so its area isTf (T).
Using the sampling notation this is written Tf[1]. The third rectangle has area Tf[2]
and soon. The total area isthus given by
area=Tf[0]+Tf[1]+Tf[2]+···+Tf[k]+···=T
∞∑
f[k]
k=0
(AI.2)