, Apr 21, 2010
Laplace Transform V:
Poles and amplitude response
1. Recap on pole diagram
2. Stability
3. Exponential signals
4. Transfer and gain
5. Fourier and Laplace
[1] To recap: The kind of function
F(s)
that arises as a Laplace
transform can be understood, in broad terms, by giving the set of points
at which it becomes infinite. This is the "pole diagram."
Examples:
The following
F(s)'s
all have pole diagram
{2i,2i}
(a not zero) :
[Slide]
F(s)
f(t)
a/(s^2+4)
(a/2) sin(2t)
as/(s^2+4)
a cos(2t)
e^{bs}/(s^2+4)
cos_b(2t)
1 + a/(s^2+4)
delta(t) + (a/2) sin(2t)
4s/(s^2+4)^2
t cos(2t)
and many other examples. All these functions
f(t)
share some common
features, for sufficiently large
t :
 they oscillate with circular frequency 2 .
 they may grow or shink, but less than exponentially.
These features are common to all functions
f(t)
such that
F(s)
has
this pole diagram.
Pole diagram
{2i,2i,1} :
any of the above plus
(c not zero)
F(s)
f(t)
c/(s1)
c e^t + .
..
c/(s1)^2
c t e^t + .
..
and many other examples. All these functions
f(t)
share some common
features, for sufficiently large
t :
 they grow as
t > infinity
"like
e^t ." This means that they
grow faster than
e^{kt}
for any
k < 1,
and slower than
e^{lt}
for any
l > 1.
 they oscillate but the oscillations become insignificant relative to
the size of
f(t)
as
t > infinity.
The rightmost poles of
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 Spring '09
 vogan
 Complex number, imaginary axis, pole diagram

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