The Importance of Continuity for Linear Time-Invariant Systems

• Date:2025-04-30
• Title of Paper:The Importance of Continuity for Linear Time-Invariant Systems
• Journal:IEEE Signal Processing Magazine
• Summary:It is widely known that an LTI system $T$ can be completely characterized by its impulse response, and the output signal of $T$ can be computed by the convolution of the input signal with the impulse response of $T$. However, in this note we will show that without the continuity of $T$, these results are not true in general. We construct two discontinuous LTI systems, which have the same impulse response but behave differently. Moreover, the convolution formula does not hold for these systems. We also prove that, for a linear system, the continuity implies the bounded input-bounded output (BIBO) stability, whereas the BIBO stability does not imply the continuity. Therefore, the continuity plays a very important role in LTI systems. However, this concept does not receive much attention as it should have in the literature of signal processing.
• Co-author:Ming Zhang and Xiaoming Chen
• Volume:37(2)
• Page Number:77-100
• Translation or Not:No
• Date of Publication:2020-03-01