Can someone illustrate the difference of the Fourier, Z ,Laplace and phasor transform methods in regards to why a certain one may be specifically used versus another or what the difference is between them all.

 

Can someone illustrate the difference of the Fourier, Z ,Laplace and phasor transform methods in regards to why a certain one may be specifically used versus another or what the difference is between them all.

I'm sure you can look up the formal definitions, so here are some informal comments.

First, they are all used with linear systems.

Second, Fourier, Laplace and Phasors used for systems that are functions of a continuous variable (such as time, or distance), but Z is used for systems that are a function of a discrete variable (e.g. digitally sampled time).

Fourier transform describes how a system, responds to pure sinusoidal signals.

Laplace transforms describes how a system responds to exponentially decaying/increasing or constant sinusoids. So, Fourier is a special case of Laplace.

Z transform is the discrete version of the Laplace transform.

Phasors are intimately related to Fourier Transforms, but provide a different notation and point of view.

EDIT: In thinking further, I don't see why the Phasor concept could not be used for discrete time systems just as well as continuous time systems. Although, I've never actually used Phasors in digital systems.

 

Thanks, I was not looking for formal def or anything along the line of proofs just a simple explanation... i.e. z transform deals with discrete time and so on