🚀 go-pugleaf

In article <Bx3Io8.MGC@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>In article <1d3fv0INNfo3@usenet.INS.CWRU.Edu> at913@cleveland.Freenet.Edu (Mirko Vukovic) writes:

>>In a previous article, fang@phy.duke.edu (Fang Zhong) says:

>>>to have a Laplace Transform of our experimental data, so we can
>>>analyze all the exponential decaying modes.  Is there anything
>>>in numerical analysis that does the forward LT, just like FFT
>>>to help analyzing data in frequency demain?

>>OK, here is an extremely hazy suggestion  (I don't have any signal processing
>>books for firm definitions etc.)...

>This does not do the job, and in fact it is not clear how it can be done.

>The idea of the FFT is that one can use the n/k FTs for the points spaced
>k apart, and manipulate properties of the FT for k equally spaced points, to
>get the FFT for n equally spaced points.  One also increases the number of
>points at which it is calculated, which is the same as the number of points
>in the argument.  Nothing like this is possible for the Laplace transform
>UNLESS additonal sample points are interpolated.  

>Another indication of the problem is that inverting a Fourier series is 
>easy, but inverting a Laplace transform, while it can be done, in principle,
>without using complex arguments, is likely to be very difficult to do in
>practice that way.  The two transforms are not that similar

Note 1:  Fast Fourier Transform refers to a particular way to compute
the discrete fourier transform (DFT).  In turn, the DFT is commonly used
to approximate the Fourier integral transform.

Back when I was doing such things I had considerable success with the
following general procedure:

Suppose we are given m time points, not necessarily equispaced.  Suppose
that we have exponentially decaying data on these points and suppose that
we wish to determine the time constants spectrum (i.e. invert the Laplace
transform).  Pick n>m; n = 20*m is about right.  Pick n points spaced
logarithmically to approximately 10 times the time constant max.  This
gives us an n by m matrix E with entries e**(-s(i)*t(j)).  We have the
the under-determined equation E*S = T, where T is the time series data
and S is the desired spectrum of exponentials.  Now E is very
ill-conditioned.  The effective rank, given computational accuracy,
is r<m.  Determine E ~= L*D*R where L is n by r and R is r by m, D is
r by r, and L and R are column and row orthornomal.

Compute a solutions S' = T(R)*I(D)*T(L)*T, where T() is transpose
and I() is inverse, i.e. use the generalized inverse of E.  Now the
solutions S' actually has rather good properties.  What tends to happen
is that the conditioning noise and its disastrous effect on numerical
Laplace inversion is minimized.  The actual inverse (time constants
spectrum) is smeared and broadened, but the original general structure
is retained.

Warning:  This technique is computationally quite expensive and
requires high precision arithmetic in the generalized inverse
calcuations.
-- 
Richard Harter: SMDS Inc.  Net address: rh@smds.com Phone: 508-369-7398 
US Mail: SMDS Inc., PO Box 555, Concord MA 01742.    Fax: 508-369-8272
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