Bounded input bounded output stability

Definition 

A sequence x(n) is bounded if there is a finite M so that |x(n)| < M for all n. (Note that, as expressed here, M is a bound for the negative values of x(.) as well. The other way of writing this is -M < x(n) < M.)

As an example, the sequence x(n) = [1+cos 5Πn] u(n) is bounded with |x(n)| ≤ 2. The sequence  is unbounded.

Definition
A discrete-time system is BIBO which is bounded input-bounded output stable if every bounded input sequence x(n) generates a bounded output sequence. That is, if |x(n)| ≤ M < ∞, then y(n) ≤ L ≤ ∞

BIBO stability theorem 

A linear shift invariant system with the impulse response of h(n) is bounded input-bounded output stable if and only if S, defined which is below, is finite.

that is, the unit sample response is absolutely summable.

Proof of the "If" part 

The system is given with impulse response h(n), let x(n) be such that |x(n)| £

M. Then output y(n) can be given by the convolution sum:

so that

By using the triangular inequality that sum of magnitudes ≥ magnitude of sum, we get

By using the fact that the magnitude of a product is product of the magnitudes,

Thus, a sufficient condition for the system to be stable is that the unit sample response should be absolutely summable; that is,

Proof of the "Only If" part
That it is also a necessary condition can be seen by considering as input, the following bounded signal (this is the signum function),

Or, equivalently,

1     where h(k) > 0

x(n-k) = sgn [h(k)] = 0      where h(k) = 0

     -1     where h(k) < 0

In the above we have implied that M = 1 (since M is some arbitrary finite number), and that |x(n)| ≤ 1. If, however, |x(n)|≤ M where M is finite but not equal to 1 we then will multiply the signum function by M. In either case x(n) is a bounded input. Thus

Clearly, if h(n) is not absolutely summable, y(n) will be unbounded. For a causal system the BIBO stability condition becomes

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