STABILITY OF RATIONAL SYSTEMS:

A continous LSI system is stable if and only if its impulse response is absolutely integrable, i.e.

.

Exploring the convergence of Laplace transform of impulse response of a stable LSI system, we find that

as

(Stability))

Thus H(s) converges on imaginary axis ( Re(s)=0 )

So, Re(s)=0 or imaginary axis is contained in ROC of system function for all stable LSI systems.

We can also look at this from a different point of view.Impulse response being absolutely integrable implies Fourier transform converges as is nothing but Fourier transform it is also bound to converge for Re{s} = 0 Re{s} = 0 is included in its ROC.

In general, Re{s} = 0 lies in ROC is not sufficent condition to imply stability. But for rational systems Re{s} = 0 lies in ROC system is stable.

Now, we will prove the above result .