Special Functions
The Dirac
Delta Function
The
function d(t) is the Dirac delta
function. The function u(t) is the Heaviside step
function.
d(t) looks like the plot below, i.e. it has a value of ¥ at
time t=0, and a value of zero everywhere else.
It follows
that for any arbitrary constant, t,
The delta
function is also defined to have the
following property:
That is,
the area under the delta function is unity.
One way to
view the delta function is as the limit of the triangle function below.
The area is
½ bh, according to simple
geometry. Thus, regardless of the value
of H, the area is:
Now let H
go to ¥.
The triangle gets taller and narrower until it looks like the delta
function (infinitely tall and infinitely narrow).
An
important property of the delta function is that whenever any function is
integrated against it, the result is the value of the function at time t=0.
This
follows directly from the statement above that the integral of the delta
function is 1.
because
is zero at all values except 0.
(since
f(0) is continuous, its value at 0+ is the same as its value at 0-).
Similarly,
The Laplace Transform of the function f(t)
is defined as:
Thus, the Laplace
Transform of the delta function is:
The Heaviside
Step Function
The step function
is:
t
It is used
to specify functions which are “turned on” at time t=0.
The Laplace Transform of the step function is:
d’(t) t
Note that the
derivative of the step function is the delta function since the derivative is 0
everywhere and infinite at t=0. We can
also talk about the derivative of the delta function, which is called the unit
doublet. This is infinitely large and
positive at 0- and infinitely negative at 0+. It looks like the figure to the right. When any function is integrated against this
function, the derivative is obtained. To
see this, do a u substitution with u=f(t), du=f’(t)dt, v=d(t), dv=d’(t)dt
The
corresponding
Steven A. Jones
Biomedical Systems
(BIEN 225),