🚀 go-pugleaf

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.