HP 50G, User Manual

Page: 492 / 887

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