I want to analyze a discrete signal in time, that is sum of exponentials distributed around two distinct values (T1, T2). My goal is to calculate these two different distributions, T1, T2, weights and standard deviations.


I have found two MATLAB scripts doing this using Inverse Laplace transform:


<http://www.wolfson.cam.ac.uk/~ssr24/invlap.html>


<http://www.mathworks.com/matlabcentral/fileexchange/6523-rilt/content/rilt.m>


I am trying to check the accuracy of the algorithm using as an input a sum of two exponentials with different time constants.


$S(t)=a\_1\exp(-t/T1)+a\_2\exp(-t/T2)$


Ideally the algorithm should return two peaks at T1 and T2 and weights a1 a2, but It returns two Gaussians with quite big dispersion around the T1 and T2 (see ex 2 on the second link).


Can you recommend any solution in order to get more specific results? even with another method (not inverse Laplace transform).


Thanks.



Inverse Laplace transform in a signal that is a sum of exponentials - MATLAB
The [Padé-Laplace method](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1280453/) works well in many contexts in which you want to recover the coefficients of a sum of exponentials from sampled values with some noise. The basic idea is that if you apply the Laplace transform again (not the inverse Laplace transform), you get a rational function because the Laplace transform of $c e^{-\alpha t}$ is $\frac{c}{s+\alpha}$. You can compute a rational (Padé) approximation to this Laplace transform via numerical integrals, and then use a partial fraction decomposition on the Padé approximation to identify the locations and coefficients of the poles. 


