18.03 Class 24, April
5
, 20
10
Unit impulse and step responses
1. Generalized derivative
2. Rest initial conditions
3. First order unit step/impulse response
4. Second order unit step/impulse response
[1] Generalized derivative
The unit step and delta functions help deal with events happening on
a time scale which is very short relative to our interest.
u(t) can be thought of as a function which, except for
t
very near zero,
is given by
u(t) = 0
for
t < 0
and
u(t) = 1
for
t > 0 .
delta(t) can be thought of as a function which is zero except for
t
very
near zero, and has area under its graph equal to
1 .
u'(t) = delta(t) .
A generalized function is by definition a sum
g(t) = g_r(t) + g_s(t) ,
where its *regular part*
g_r(t)
is piecewise smooth, and its
*singular part*
g_s(t)
is a linear combination of shifted delta functions.
Any regular
f(t)
function has a
"generalized derivative"
f'(t)
which is
a generalized function:
f'(t) = f'_r(t) + f'_s(t) .
The regular part
is the ordinary derivative of
f(t) (except at the break points, where it
is undefined).
The singular part is a sum of delta functions, one for each
break in the graph:
(f(a+)f(a)) delta(ta)
There's no separate notation for the generalized derivative to distinguish
it from the ordinary derivative, and we will just write
f'(t)
or
dotx (t).
For example, if
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 Spring '09
 vogan
 Derivative, Continuous function, Dirac delta function, rest initial conditions

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