BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.

To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system. The boundedness of the input signal is represented by a constant, and the convolution integral helps determine whether the output remains bounded. Mathematically, this means that if the integrand of the convolution integral is finite, the output will also be finite. Specifically, a continuous-time system is BIBO stable if its impulse response is integrable, meaning the integral of the absolute value of the impulse response is finite and is represented as,

This condition ensures that the output remains within bounds for any bounded input signal, thus confirming the system's stability.

The same principle applies to discrete-time systems. BIBO stability in discrete-time systems is determined through the summation of the convolution series. For a discrete-time system, the output is finite if the summation term has a finite value, indicating that the system is BIBO stable if its impulse response is summable as given in the expression below.

In other words, if the sum of the absolute values of the impulse response is finite, the system will produce a bounded output for any bounded input, confirming BIBO stability.

The importance of BIBO stability lies in its application to real-world systems, where ensuring that outputs remain within acceptable limits in response to bounded inputs is crucial. Understanding and applying the concepts of convolution and impulse response integrals are vital for designing and analyzing stable systems in both continuous and discrete-time domains.