is stable. Causality A discrete-time LTI system is causal if the current value of the output depends on only the current value and past values of the input. A necessary and sufficient condition for causality is where is the impulse response. It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique. When a region of convergence is specified, then causality can be determined. Stability A system is bounded input, bounded output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if implies that (that is, if bounded input implies bounded output, in the sense that the maximum absolute values of and are finite), then the system is stable. A necessary and sufficient condition is that , the impulse response, satisfies In the frequency domain, the region of convergence must contain the unit circle (i.e., the locus satisfying for complex z). Notes See also Circulant matrix Frequency response Impulse response System analysis Green function Signal-flow graph References Further reading External links ECE 209: Review of Circuits as LTI Systems – Short primer on the mathematical analysis of (electrical) LTI systems. ECE 209: Sources of Phase Shift – Gives an intuitive explanation of the source of phase shift in two common electrical LTI systems. JHU 520.214 Signals and Systems course notes. An encapsulated course on LTI system theory. Adequate for self teaching. LTI system example: RC low-pass filter. Amplitude and phase response. Digital signal processing Electrical engineering Classical control theory Signal processing Frequency-domain analysis Time domain analysis