Fourier transform and Laplace transform are two mathematical tools frequently used in various fields of applied math and engineering sciences. If one is good at or very good at both transforms (e.g., awgn), then, there is a good chance that he/she might be majoring in electric engineering. Generally speaking, math is a common language to all science and engineering fields. Therefore, an effective communication between people from different fields can as well start from the common math tools they used in their researches.

On the surface or solely from their mathematical expressions, Fourier transform and Laplace transform look very similar:

Fourier{f(t)} = F(omega) ~ int.[f(t)*exp(-i*omega*t)dt]

Laplace{f(t)} = L(s) ~ int.[f(t)*exp(-s*t)dt]

Given a function F(omega) or L(s) one can also perform an inverse transform to derive the original function:

Fourier^-1{F(omega)} = f(t) ~ int.[F(omega)*exp(i*omega*t)(d omega)]

Laplace^-1{L(s)} = f(t) ~ int.[L(s)* exp(s*t)ds]

While solving an engineering or science problem one often has to apply both the forward and inverse transforms to get the complete solution. One first applies the forward transform to convert an equation into a different space/variable (say, s) that can be solved easily. The inverse transform gives the final solution to the problem in the original space/variable (say, t).

There is a “tiny” difference between two transforms: the kernel functions of the transforms are different. For the Fourier transform, the kernel functions for the forward and inverse transforms are exp(-i*x*t) and exp(i*x*t), respectively. For the Laplace transform, they are exp(-x*t) and exp(x*t).

Yet, such an apparent small difference actually makes these two transforms fundamentally different in their mathematical properties and applications. For one thing or for one very important thing in practical applications, we note that exp(-i*x*t) and exp(i*x*t) are orthogonal functions with respect to different values of x or t. On the other hand, exp(-x*t) and exp(x*t) are not orthogonal functions so they do not form a set of beautiful orthogonal bases. Mathematically, we have, for example,

int.[exp(-i*x_j*t)*exp(i*x_k*t)dt] = 0 if x_j != x_k
int.[exp(-x_j*t)*exp(-x_k*t)dt] != 0 even if x_j != x_k

Why do I say the orthogonal bases are beautiful? It is beautiful because one can accurately, very accurately or as accurately as one wishes, to expand any given function by a set of orthogonal bases, i.e., a set of complete orthogonal functions because the coefficients in those expansions can always be computed numerically. In fact, Fourier transform is much more widely used in various fields mainly because we have the so-called FFT (Fast Fourier Transform) technique. Here, the word “Fast” is of secondary importance. It is the accuracy of the discrete Fourier transform that matters. Increase of the resolution in numerical computation, i.e., increase of the number of points in numerical evaluation of the Fourier integral will not affect but actually will only increase the accuracy of the derived forward and inverse transforms.

We hardly see anyone performing forward or inverse Laplace transform numerically. Most people study and derive the forward and inverse Laplace transforms in analytic forms. Well, I tried numerical inversion of Laplace transform once while working on a project many years ago and I found the following two books on the numerical inversion of Laplace transform while starting working the project:

Bellman, R., R. Kalaba, and J. Lockett, 1966: The Numerical Inversion of the Laplace Transform. American Elsevier, New York, 249 pp.

Krylov, V. I., and N. S. Skoblya, 1969: Handbook of Numerical Inversion of Laplace Transform. Israel Program for Scientific Translation, Jerusalem.

I thought it was an easy and straightforward problem for the numerical inverse of Laplace transform by following a regular procedure of numerical quadratures at the beginning but it was not! Here was the problem, the very serious problem, I found out about the numerical inversion of Laplace transform: the non-orthogonal basis functions of the Laplace transform make the numerical inversion unstable because the ill-conditioned matrices will quickly enhance the error amplification as the resolution of the quadratures for the integration increases. Well, luckily, I found the solution to the problem related to my project:

................................
................................
................................

which led to the publication of one of my science papers.

(2007.08.10)