A filter of impulse response *h*(*k*) is bounded-input, bounded-output (BIBO) stable when a bounded input signal *x*(*k*) produces a bounded output *y*(*k*).

Let us consider a bounded input signal *x*(*k*);

where *h**(− *k*) and |*h* (−*k*)| respectively designate the conjugate and the complex modulus of *h*(−*k*). With equation (10.1), the output *y*(*k*) is written as:

More specifically, equation (10.2) verifies for *k* = 0:

If the filter is BIBO stable, then it verifies the following condition:

Inversely, if the impulse response *h*(*k*) of the filter satisfies equation (10.4) then, for any input signal *x*(*k*) bounded by *M*, the output *y*(*k*) verifies:

Let us assume now that the filter admits a transfer function^{1}:

We write C the open convergence ring (i.e. circular band) represented as:

or it can be represented ...

Start Free Trial

No credit card required