[Henry Baker:]
Is there an analogous analysis that can separate out various real exponentials in a real waveform?
I.e., if a signal is the sum of various real exponentials (i.e., no sinusoidal components), is there a simple analysis that will pull out the coefficients & exponents? Is there a "fast" version analogous to the FFT for this procedure? ...
[Dan Asimov:]
It would indeed seem that the Laplace transform is what you're looking for; see < http://en.wikipedia.org/wiki/Laplace_transform >. Since the (bilateral) Laplace transform is equivalent to the Fourier transform under a change of variables (real <-> imaginary), there by rights should be a corresponding fast [or finite] Laplace transform. (Cf. V. Rokhlin, "A fast algorithm for the discrete Laplace transform", J. Complexity, v. 4, 1988.)
In simple discrete cases -- say where you already know
f(x) == sum_{n=-oo to oo} c_n exp(nx)
then integrating f(x)*exp(-n_0 x) over the imaginary interval [0, 2pi*i] will equal 2pi*i c_n (modulo technical details).
That's not going to help much if what Henry has is an empirically obtained signal for real values of x only... Parameter estimation for linear combinations of decaying exponentials is notoriously ill-conditioned, so I'd be a bit pessimistic about the chances of any algorithm doing what Henry wants at all well in the presence of noise. -- g