Dirac's Delta Function And Heaviside's Step Function - HP 50g User Manual

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, (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, (x), is (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 -function "picks out" the value of the function f(x) at x = x
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 =
(x). The two functions are illustrated in the figure below.
f lim f ) (
t
∫
( dx
H ( x )
∫
( ) (
f x H x
t lim [ s F ( s
s 0
) 1 . 0 .
x
∫
( )
f x
, 1 x 0
⎧
⎨
, 0 x 0
⎩
∫
) (
x dx f
0
x
0
)].
( )
x x dx f
0
. Dirac's
0
) .
x dx
( ).
x
0
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