InverseLaplaceTransform problems

• To: mathgroup at smc.vnet.net
• Subject: [mg19325] InverseLaplaceTransform problems
• From: techie at mit.edu
• Date: Mon, 16 Aug 1999 02:14:56 -0400
• Organization: Massachvsetts Institvte of Technology
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,
I am having several difficulties with (Inverse)LaplaceTransform under
Mathematica 3.0.1 for Linux, with the SignalProcessing 3.0 package.
I have tried both sets of Laplace transforms located in the calculus
package (part of Mathematica) and the SignalProcessing 3.0 package;
neither one is sufficient for my needs, and I need help getting either
(or both)
to work properly.
Here is the description of the problems:

1. I have found that the (Inverse)LaplaceTransform functions in the
Signals package are less useful than the ones found in
Calculus`LaplaceTransform.
A simple example would be (for the Signals package):
InverseLaplaceTransform[LaplaceTransform[v[t], t, s], s, t]
yields:
(1/362880)(362880DiracDelta[0]DiracDelta[t]v[0] + 362880 v[0]
DiracDelta'[0]DiracDelta'[t] - .....

instead of the desired result: v[t]!
This is such a simple rule, you'd think that it would be standard.

For the standard LaplaceTransform:
<<Calculus`LaplaceTransform`

InverseLaplaceTransform[LaplaceTransform[v[t], t, s], s, t]
yields ->
v[t]

as expected.
Also,
InverseLaplaceTransform[s LaplaceTransform[v[t], t, s], s, t]
yields ->
DiracDelta[t] v[0] + v'[t]

as expected.
2. However, if the expression to be InverseLaplaceTransform'ed instead
contains a polynomial fraction function of s (for example: s / (s + 1) )

times the LaplaceTransform of some variable v[t], for example:
InverseLaplaceTransform[(s / (s+1)) LaplaceTransform[v[t], t, s],s, t]
yields ->
\!\(InverseLaplaceTransform[\(s\ LaplaceTransform[v[t], t, s]\)\/\(1 +
s\),
s, t]\)
which is the same!

It instead should be: v[t] * (-E^(-t) + DiracDelta[t]), where "*"
denotes the convolution integral from 0 to t.
This can be simplified further to: v[t] \!\(\(-\(\[Integral]\_0\%t\(
E\^\(\(-t\) + \[Tau]\)\ v[\[Tau]]\)
\[DifferentialD]\[Tau]\)\)\)

I would really like to find a way to get this to work since I am trying
to solve systems of circuit equations in the Laplace domain, then
inverse transform to get the
e.g.
Y[s] = H[s] X[s]
Or, equivalently in the time domain:
y[t] = h[t] * x[t]

h[t] is the InverseLaplaceTransform of H[s].
(where H[s] is the system function having a polynomial numerator and
denominator
in s; h[t] is the impulse response of the system; x[t] is the input to
the system; y[t] is the output).

I just wanted to check to see if anyone out there has already solved
this problem
/ deficiency in the LaplaceTransform functions.
thanks a lot for any suggestions / helpful hints, etc.
Ed Ouellette
techie at mit.edu

```

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