BIBO stability
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 formula_1 such that the signal magnitude never exceeds formula_2, that is Time-domain condition for linear time-invariant systems. Continuous-time necessary and sufficient condition. For a continuous time linear time-invariant (LTI) system, the condition for BIBO stability is that the impulse  response, formula_5 , be absolutely integrable, i.e., its L norm exists. Discrete-time sufficient condition. For a discrete time LTI system, the condition for BIBO stability is that the impulse response be absolutely summable, i.e., its formula_7 norm exists. Proof of sufficiency. Given a discrete time LTI system with impulse response formula_9 the relationship between the input formula_10 and the output formula_11 is where formula_13 denotes convolution. Then it follows by the definition of convolution Let formula_15 be the maximum value of formula_16, i.e., the formula_17-norm. If formula_21 is absolutely summable, then formula_22 and So if formula_21 is absolutely summable  and formula_25 is bounded, then formula_26 is bounded as well because formula_27 The proof for continuous-time follows the same arguments. Frequency-domain condition for linear time-invariant systems. Continuous-time signals. For a rational and continuous-time system, the condition for stability is that the region of convergence (ROC) of the Laplace transform includes the imaginary axis. When the system is causal, the ROC is the open region to the right of a vertical line whose abscissa is the real part of the "largest pole", or the pole that has the greatest real part of any pole in the  system. The real part of the largest pole defining the ROC is called the abscissa of convergence. Therefore, all poles of the system must be in the strict left half of the s-plane for BIBO stability. This stability condition can be derived from the above time-domain condition as follows: where formula_29 and formula_30 The region of convergence must therefore include the imaginary axis. Discrete-time signals. For a rational and discrete time system, the condition for stability is that the region of convergence (ROC) of the z-transform includes the unit circle. When the system is causal, the  ROC is the open region outside a circle whose radius is the magnitude of the pole with largest magnitude. Therefore, all poles of the system must be inside the unit circle in the z-plane for BIBO stability. This stability condition can be derived in a similar fashion to the continuous-time derivation: where formula_32 and formula_33. The region of convergence must therefore include the unit circle.