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In article <6576@news.duke.edu> fang@phy.duke.edu (Fang Zhong) writes:
>
>	We are studying a decaying time series.  We would like
>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?

Hi.  That's sounds cool.  Can someone send me the same info. please.

-- 
Ranjan Rai                             <rairan@ecf.toronto.edu> 
Engineering Science 9T3                <rairan@eecg.toronto.edu>
University of Toronto                      

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

>
>	We are studying a decaying time series.  We would like
>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?
>
>-- 
>	Fang Zhong				1-919-660-2524
>	Duke University Dept. of Physics	fang@phy.duke.edu
>	Durham, N.C.      27706			
>

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

LT is a generalization of FT to the whole complex plane, where FT is the LT on
the unit circle.

If you multiply your time series by alpha^t (I think), the LT of the new
series expands or contracts by the factor alpha (lookup Oppenheim's book
on descretem time signal analysis):

	f(t) <--> F(s)

f(t)*alpha^t <--> F(alpha*s)

So, you can modify your time series in the above way, to bring the appropriate
circles of the complex plane to the unit circle, and then you use the FFT.

Mind you, this is my (highly original :-)) contribution to science, meaning
that I've never seen it anywhere else, nor have seen (or looked) for an 
alternative way of doing a FLT.  FUrther more, I've never tried it, only 
thought about it.  Enough disclamers.

-- 
Mirko Vukovic 
University of Wisconsin -- Madison
VUKOVIC@UWMFE.NEEP.WISC.EDU 

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:


>>	We are studying a decaying time series.  We would like
>>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.)

>LT is a generalization of FT to the whole complex plane, where FT is the LT on
>the unit circle.

>If you multiply your time series by alpha^t (I think), the LT of the new
>series expands or contracts by the factor alpha (lookup Oppenheim's book
>on descretem time signal analysis):

			....................

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
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)

|> 
|> 
|> In a previous article, fang@phy.duke.edu (Fang Zhong) says:
|> 
|> >
|> >	We are studying a decaying time series.  We would like
|> >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?
|> >
|> >-- 
|> >	Fang Zhong				1-919-660-2524
|> >	Duke University Dept. of Physics	fang@phy.duke.edu
|> >	Durham, N.C.      27706			
|> >
|> 
|> OK, here is an extremely hazy suggestion  (I don't have any signal processing
|> books for firm definitions etc.)
|> 
|> LT is a generalization of FT to the whole complex plane, where FT is the LT on
|> the unit circle.
|> 
|> If you multiply your time series by alpha^t (I think), the LT of the new
|> series expands or contracts by the factor alpha (lookup Oppenheim's book
|> on descretem time signal analysis):
|> 
|> 	f(t) <--> F(s)
|> 
|> f(t)*alpha^t <--> F(alpha*s)
|> 
|> So, you can modify your time series in the above way, to bring the appropriate
|> circles of the complex plane to the unit circle, and then you use the FFT.
|> 
|> Mind you, this is my (highly original :-)) contribution to science, meaning
|> that I've never seen it anywhere else, nor have seen (or looked) for an 
|> alternative way of doing a FLT.  FUrther more, I've never tried it, only 
|> thought about it.  Enough disclamers.
|> 
|> -- 
|> Mirko Vukovic 
|> University of Wisconsin -- Madison
|> VUKOVIC@UWMFE.NEEP.WISC.EDU 
|> 

Strictly speaking, though, the Laplace transform (one-sided) is obtained by
taking the _continuous_ time Fourier transform and evaluating it at
s=j*omega.  Unit circles arise from conformal mappings and the like.  I've
never seen a definition for the Laplace transform on a unit circle - I'd be
curious to know where that is discussed.  As far as I know, the LT is
defined as

F(s) = Int(0..Inf) [f(t) e^(-s*t) dt]

George Woyansky
ECE Dept, Carnegie Mellon University

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
In the fields of Hell where the grass grows high
Are the graves of dreams allowed to die.

fang@phy.duke.edu (Fang Zhong) writes:


>	We are studying a decaying time series.  We would like
>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?

First note that the FFT is an approximation to a Fourier Transform
only when you interpret your data as being periodical and
discrete-time, the periodical interpretation being the harsher constriction.

Then note that a Laplace transform maps a realvalued function into
a complex-valued one. This complex-valued function has a region of
convergence somwhere for Re s > some constant. In that region you
can develop your data into a power series.

With Fourier Transforms, you have set Re s=0. Consequently, you can
represent the (Complex) Fourier transform by the values on the imaginary
axis. You can change this to a Laplace transform by multiplying your
time data with a decaying exponential factor, but, alas, then the
circularity theorem fails entirely.

One thing that might be interesting to you is the Chirp-Z-Transform,
which could possibly be a help for your problem. But other than that,
the Laplace transform is more likely to be useful in symbolical
form, just as the continuos Fourier Transform is.