BIBO stability

In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for linear 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. In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for linear 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 B > 0 {displaystyle B>0} such that the signal magnitude never exceeds B {displaystyle B} , that is For a continuous time linear time-invariant (LTI) system, the condition for BIBO stability is that the impulse response 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 ℓ 1 {displaystyle ell ^{1}} norm exists. Given a discrete time LTI system with impulse response   h [ n ] {displaystyle h} the relationship between the input   x [ n ] {displaystyle x} and the output   y [ n ] {displaystyle y} is where ∗ {displaystyle *} denotes convolution. Then it follows by the definition of convolution Let ‖ x ‖ ∞ {displaystyle |x|_{infty }} be the maximum value of   | x [ n ] | {displaystyle |x|} , i.e., the L ∞ {displaystyle L_{infty }} -norm. If h [ n ] {displaystyle h} is absolutely summable, then ∑ k = − ∞ ∞ | h [ k ] | = ‖ h ‖ 1 < ∞ {displaystyle sum _{k=-infty }^{infty }{left|h ight|}=|h|_{1}<infty } and So if h [ n ] {displaystyle h} is absolutely summable and | x [ n ] | {displaystyle left|x ight|} is bounded, then | y [ n ] | {displaystyle left|y ight|} is bounded as well because ‖ x ‖ ∞ ‖ h ‖ 1 < ∞ . {displaystyle |x|_{infty }|h|_{1}<infty .}