Discrete Laplace transform

The relationship between the discrete Laplace transform and discrete Fourier transform is not quite the same as that between their continuous counterparts.

Continuous Fourier and Laplace transforms

The continuous versions of the Fourier and Laplace transforms are given as follows.

Fourier transform:

${\cal F}(f)(\omega) = \int_{-\infty}^\infty \exp(-i\omega x) f(x)\, dx$

Laplace transform:
${\cal L}(f)(s) = \int_0^\infty \exp(-s x) f(x)\, dx$

The Fourier transform is defined several ways, and I actually prefer the convention that puts a factor of 2π in the exponential, but the convention above makes the analogy with Laplace transform simpler. There are two differences between the Fourier and Laplace transforms. The Laplace transform integrates over only half the real line, compared to the entire real line for Fourier. But a variation on the Laplace transform, the Bilateral Laplace transform integrates over the entire real line. The Bilateral Laplace transform at s is simply the Fourier transform at x = is. And of course the same is true for the (one-sided) Laplace transform if the function f is only non-zero for positive values.

I’ve encountered the Fourier transform more in application, and the Laplace transform more in teaching. This is not to say the Laplace transform isn’t used in practice; it certainly is used in applications. But the two transforms serve similar purposes, and the Laplace transform is easier to teach. Because the factor exp(-sx) decays rapidly, the integral defining the Laplace transform converges for functions where the integral defining the Fourier transform would not. Such functions may still have Fourier transforms, but the transforms require distribution theory whereas the Laplace transforms can be computed using basic calculus.

Discrete Fourier and Laplace Transforms

There’s more difference between the discrete versions of the Fourier and Laplace transforms than between the continuous versions.

The discrete Fourier transform (DFT) approximates the integral defining the (continuous) Fourier transform with a finite sum. It discretizes the integral and truncates its domain. The discrete Laplace transform is an infinite sum. It discretizes the integral defining the Laplace transform, but it does not truncate the domain. Given a step size η > 0, the discrete Laplace transform of f is

${\cal L}_\eta(f)(s) = \eta \sum_{n=0}^\infty \exp(-sn\eta) f(n\eta)$

The discrete Laplace transform isn’t “as discrete” as the discrete Fourier transform. The latter takes a finite sequence and returns a finite sequence. The former evaluates a function at an infinite number of points and produces a continuous function.

The discrete Laplace transform is used in applications such as signal processing, as well as in the theory of analytic functions.

Connection with the z-transform and generating functions

If η = 1 and z = exp(-s), the discrete Laplace transform becomes the z-transform of the values of f at non-negative integers. And if we replace z with 1/z, or equivalently set z = exp(s) instead of z = exp(-s), we get the generating function of the values of f at non-negative integers.

z-transforms are common in digital signal processing, while generating functions are common in combinatorics. They are essentially the same thing.

Continuous Fourier and Laplace transforms

Discrete Fourier and Laplace Transforms

Connection with the z-transform and generating functions

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