[sdiy] Re: Walsh Generator Release!!!

[Ethan Duni]

> Both Laplace and Fourier transforms are complementary,
> no one is a generalization or simplification of the other.

-I think it's fair to say that the Fourier transform is a special case of
the Laplace transform. Mathematically, the Laplace transform is the Fourier
transform of the input signal scaled by an exponential. This allows you to
do the regular fourier transform (no scaling, or s = jw), and also to force
non square-integrable signals to be square-integrable and, hence, Fourier
transformable.

>The
> domain of Laplace transform is a complex halfplane, but it
> is limited to signals starting from zero, so you are basically
> limited to the analysis of transients.

-You're talking about the "one sided" Laplace transform, which is the one
commonly used in circuit analysis. The full-blown Laplace transform doesn't
have this restriction. The domain of the full Laplace transform is the
entire complex plane. The reason that the one-sided transform is used is
that, if you assume that your signal is right-sided (and reasonably
well-behaved), the Laplace transform will always exist. This isn't true for
some otherwise-well behaved two-sided signals. This makes the one-sided
Laplace transform ideal for transient circuit analysis, once you figure out
how to include initial conditions in your Laplace system model.

Ethan