Module 4 : Laplace and Z Transform
Lecture 30 : Laplace Transform

Example of Laplace Transform

Consider that the impulse response h(t) = et u(t).

 Thus we notice that by multiplying by the term u(t) we are effectively considering the unilateral Laplace Transform whereby the limits tend from 0 to +∞

Also we notice that h(t) is not Fourier transformable as  it is not absolutely integrable.

Consider the Laplace transform of h(t) as shown below:

As stated earlier the symbol s is a complex number and is defined as s = σ + jΩ..
Substituting s in the above equation we get:

           

Observing the above equation closely, we realize that firstly H(s) converges if and only if

We analyze the part (1-σ) as follows:

  For a decaying e(1-σ)λ  it is essential that  (1-)< 0 . This implies that (> 1) which means that the Real part of 's' is greater than '1'  which is also denoted as  Re(s) > 1.

This is what defines the  " Region of Convergence " in an S-Complex Plane. The ROC of the Laplace Transform is always determined by the Re(s). The ROC in general gives us an idea of the stability of a system and is also a representation of  the poles-zero plot of a system. It is essential to note that the ROC never includes poles.
 

 Evaluation of the integral yields:
                                             H(s)= =1/(s-1)

We observe that there is a single pole at s=1. Since the Region of Convergence cannot contain poles therefore ROC start from '1' and tends outwards to infinity.

 est in physical systems:

We consider the real part of  est, where s = σ + jΩ .
                              
                                     

Such a response is visible in RLC (Resistance-Inductance and Capacitance)  systems. It is not only visible in the electrical field but also in other disciplines like mechanical field. In such cases the above expression is multiplied by a polynomial or a combination of such expressions.

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