Question

Which of the following is true for discrete-time stable systems?

The question was asked by my school teacher while I was bunking the class.

This intriguing question originated from Properties of the Impulse Response Representation for LTI Systems in section Time Domain Representation for LTI Systems of Signals and Systems

Answer 1

For discrete-time stable systems, the key truth is that the system is BIBO (Bounded Input, Bounded Output) stable if and only if the impulse response \( h[n] \) is absolutely summable. This means that:

\[

\sum_{n=-\infty}^{\infty} |h[n]| < \infty

\]

In other words, the system is stable if the sum of the absolute values of the impulse response is finite. This condition ensures that for any bounded input, the output will also remain bounded.