5.Stability

Are our systems BIBO stable? i.e.: Will a bounded input necessarily give rise to a bounded output? No.

The integrals that describe the two systems need not converge for a bounded input signal. e.g.: they don't converge for a non-zero constant input signal.

Now that we have come to the issue of the Fourier transform and the Inverse Fourier transform not converging for a constant input signal, let us see what the Transform of the unit impulse is.

Note that the impulse, far from satisfying Dirichlet's conditions, is not even a function. It falls in the class of generalized functions. Thus what we are doing is extending our idea of the Fourier Transform. Why? Because we will find it useful.

That is, the Fourier transform of the unit impulse is the identity function. Thus, even though the inverse equation does not converge for the identity function, we say that that Fourier Transform of the unit impulse is the identity function.

Why stop here? Consistent with duality, we say that the Fourier Transform of the identity function is the unit impulse:

We will even apply the time-shift and frequency-shift properties we have just proved to make further generalizations: