Step and delta.
Two additions to your mathematical modeling toolkit.
 Step functions [Heaviside]
 Delta functions [Dirac]
[1]
it is
Model of on/off process: a light turns on; first it is dark, then
light. The basic model is the Heaviside unit step function
u(t) = 0
for
t < 0
1
for
t > 0
Of course a light doesn't reach its steady state instantaneously; it
takes a
small amount of time. If we use a finer time scale, you can see what
happens.
It might move up smoothly; it might overshoot; it might move up in fits
and
starts as different elements come on line. At the longer time scale, we
don't care about these details. Modeling the process by
u(t)
lets us
just
ignore those details. One of the irrelevant details is the light output
at exactly
t = 0.
In fact as a matter of realism, you rarely care about the value of a
function at any single point. What you do care about is the average
value
nearby that point; or, more precisely, you care about
lim_{t>a} f(t)
The function is continuous if that limit IS the value at
t=a.
You will also often care about the values just to the left of
t=a,
or just to the right. These are captured by
f(a) = lim_{t>a
from below} f(t)
f(a+) = lim_{t>a
from above} f(t)
For example,
u(0) = 0 ,
u(0+) = 1. A function is continuous at t=a
if
f(a) = f(a) = f(a+) .
A good class of functions to work with is
the "piecewise continuous" functions, which are continuous except at
a scattering of points and such that all the onesided limits exist.
So
u(t)
is piecewise continuous but
1/t
is not.
The unit step function is a useful building block:
u(ta)
turns on at
t = a
Q1:
What is the equation for the function which agrees with
f(t) between
a
and
b
( a < b )
and is zero outside this window?
(1) (u(tb)  u(ta)) f(t)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Staff
 Addition, Derivative, Dirac delta function, Delta Functions

Click to edit the document details