In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for signals and systems that take inputs. If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded.

A signal is bounded if there is a finite value ${\displaystyle B>0}$ such that the signal magnitude never exceeds ${\displaystyle B}$, that is

For a continuous time linear time-invariant (LTI) system, the condition for BIBO stability is that the impulse response, ${\displaystyle h(t)}$ , be absolutely integrable, i.e., its L¹ norm exists.

For a discrete time LTI system, the condition for BIBO stability is that the impulse response be absolutely summable, i.e., its ${\displaystyle \ell ^{1}}$ norm exists.

Given a discrete time LTI system with impulse response ${\displaystyle \ h[n]}$ the relationship between the input ${\displaystyle \ x[n]}$ and the output ${\displaystyle \ y[n]}$ is

where ${\displaystyle *}$ denotes convolution. Then it follows by the definition of convolution

Let ${\displaystyle \|x\|_{\infty }}$ be the maximum value of ${\displaystyle \ |x[n]|}$, i.e., the ${\displaystyle L_{\infty }}$-norm.

If ${\displaystyle h[n]}$ is absolutely summable, then ${\displaystyle \sum _{k=-\infty }^{\infty }{\left|h[k]\right|}=\|h\|_{1}\in \mathbb {R} }$ and