Henry writes: << The sinusoids are orthogonal to one another, allowing for the construction of excellent filters to detect a strong signal. The real exponentials aren't orthogonal, at least under the traditional defn of inner product. Perhaps the cleverness of Laplace was in coming up with a different inner product?
Does this have a direct bearing on what I wrote below? [edited: c_n -> c_(n_0)] --Dan At 04:49 AM 4/28/2007, Daniel Asimov wrote:
Henry writes:
<< Fourier analysis can easily separate out the various sinusoidal frequencies in a waveform.
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?
I recall studying Laplace transforms, but can't recall whether they solve this particular problem.
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_0) (modulo technical details).
--Dan