Laplace Transform

Abstract

With all its glory, power, and versatility the Fourier transform suffers from one ill condition and that is the starting signal has to be integrable over time, and more specifically from 0 to infinity. Things like the pulse function, negative exponential, or even the causal sine/cosine functions had Fourier transforms, albeit in some cases we picked up some singular delta functions in the frequency domain. For all other signals, which don’t necessarily have a finite integral over positive time, such as the ramp function t, the quadratic function t ² or even some positive exponential, we immediately ran into problems. The Laplace transforms come to the rescue! The trick is simple: multiply the signal upfront by a time-decaying function e ^−σt, ensure the resulting function has a finite time integral, now find the Fourier transform of the product, and then when finding inverse transform multiply back by e ^σt to take out the effect of the initial e ^−σt multiplication! So really the Laplace transform is a masquerading Fourier transform, or a Fourier transform in disguise. Rather than keeping continuous track of σ and jω why not lump them into the complex frequency s = σ + jω? With the origin and need of the Laplace transform covered we move to applying it to many examples, ranging from the unit step function, pulse one, causal sines/cosines, negative exponentials, and the list goes on. In the process we identify the impact of σ and the corresponding region of convergence. For each application case we plot the spectrum versus ω, for different values of σ. We observe that the impact of σ is to smear out the spectrum and eliminate what used to be delta functions in the frequency domain. We also practice in finding inverse transforms and monitor signal reconstruction as a function of harmonics. Finally we also touched briefly on the bilateral Laplace transform for reference.