HP 50g Graphing Calculator User Manual
Page 492
Page 16-15
Dirac’s delta function and Heaviside’s step function
In the analysis of control systems it is customary to utilize a type of functions that
represent certain physical occurrences such as the sudden activation of a switch
(Heaviside’s step function, H(t)) or a sudden, instantaneous, peak in an input to
the system (Dirac’s delta function,
δ(t)). These belong to a class of functions
known as generalized or symbolic functions [e.g., see Friedman, B., 1956,
Principles and Techniques of Applied Mathematics, Dover Publications Inc.,
New York (1990 reprint) ].
The formal definition of Dirac’s delta function,
δ(x), is δ(x) = 0, for x ≠0, and
Also, if f(x) is a continuous function, then
An interpretation for the integral above, paraphrased from Friedman (1990), is
that the
δ-function “picks out” the value of the function f(x) at x = x
0
. Dirac’s
delta function is typically represented by an upward arrow at the point x = x0,
indicating that the function has a non-zero value only at that particular value of
x
0
.
Heaviside’s step function, H(x), is defined as
Also, for a continuous function f(x),
Dirac’s delta function and Heaviside’s step function are related by dH/dx =
δ(x). The two functions are illustrated in the figure below.
)].
(
[
lim
)
(
lim
0
s
F
s
t
f
f
s
t
⋅
=
=
→
∞
→
∞
∫
∞
−∞
=
.
0
.
1
)
( dx
x
δ
∫
∞
−∞
=
−
).
(
)
(
)
(
0
0
x
f
dx
x
x
x
f
δ
⎩
⎨
⎧
<
>
=
0
,
0
0
,
1
)
(
x
x
x
H
∫
∫
∞
−∞
∞
=
−
0
.
)
(
)
(
)
(
0
x
dx
x
f
dx
x
x
H
x
f