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

2.4 Review of some common engineering functions and techniques 111
f
6
4
2
u(t – 1)f
6
4
2
u(t – 1)f – u(t –2)f
6
4
2
(a) (b)
–1
0
1
2 3 t
(c)
–1
0
1
2 3 t
(d)
–1
0
1
2 3 t
FigureS.5
2.4.9 Thedeltafunctionorunitimpulsefunction, δ(t)
Consider the rectangle function,R(t), shown in Figure 2.56. The base of the rectangle
ish, the height is 1/h and so the area is 1. Fort >h/2 andt < −h/2, the function is 0.
Ashdecreases, the base diminishes and the height increases; the area remains constant
at1.
Ashapproaches 0, the base becomes infinitesimally small and the height infinitely
large. The area remains at unity. The rectangle function is then called a delta function
or unit impulse function. Ithas a valueof0everywhere except atthe origin.
δ(t) = rectanglefunction ashapproaches0
We write thisconcisely as
δ(t) =R(t) as h →0
The position of the delta function may be changed from the origin tot =d. Consider a
rectangle function,R(t −d), shown in Figure 2.57.R(t −d) is obtained by translating
R(t)anamountd totheright.Again,lettinghapproach0producesadeltafunction,this
time centredont =d.
δ(t−d)=R(t−d) as h→0
We have seen that the delta function can be regarded as bounding an area 1 between
itselfand the horizontalaxis.More generallythe area bounded by the function
f(t) =kδ(t)
isk.Wesaythatkδ(t)representsanimpulseofstrengthkattheorigin,andkδ(t −d)is
animpulseofstrengthkatt =d.Itisoftenusefultodepictsuchanimpulsebyanarrow
where the height of the arrow gives the strength of the impulse. A series of impulses is
oftentermedan impulse train.
R (t)
1–
h
R (t – d)
1–
h
– h–
2
h–
2
t
d – h– d d + h–
2 2
t
Figure2.56
The rectangle function,R(t).
Figure2.57
Thedelayed rectangle function,R(t −d).