Till now we have been dealing with continuous and discrete domains . Then we studied the relationships involved using the transform domains.  A system actually operates in a natural domain but it can be well understood in transform domains . The advantage of transform domains is that a few of the properties which may not be observed in natural domains are clear in transform domains. Most of the LTI-Systems act in time domain but they are more clearly described in the frequency domain instead.

Till now ,we have seen the importance of Fourier analysis in solving many problems involving signals and LTI systems. Now, we shall deal with signals and systems which do not have a Fourier transform.
 

But what was so special about Fourier transform in case of LSI systems?

We found that continuous-time Fourier transform (F.T.) is a tool to represent signals as linear combinations of complex exponentials. The exponentials are of the form  e^st with s=j  and  e^jw  is an eigen function of the LSI system. Also , we note that the Fourier Transform only exists for signals which can absolutely integrated and have a finite energy.
 

This observation leads to generalization of continuous-time Fourier transform by considering a broader class of signals using the powerful tool of  "Laplace transform". It will be trivial to note that the L.T can be used to get the discrete-time representation using relevant substitutions. This leads to a link with the Z-Transform and is very handy for a digital filter realization/designing. Also it will be helpful to note that, the properties of Laplace Transform and Z-Transform are quite similar.

With this introduction let us go on to formally defining both Laplace and Z-transform.