🚀 go-pugleaf

|> 
|> 
|> 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