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In addition to linearity and time-invariance, there are other significant classifications of discrete-time systems. One of these is causality. A system is causal if its output, for any $n$, depends only on inputs at or before time $n$. Causality is an important practical qualification on a system: it is not possible to implement a real-time system whose output depends on future values! While the qualification of causality is also relevant for non-LTI systems, it has a special relationship with LTI systems. Recall that for LTI systems, the impulse response can be used to find the system's output given some input (through convolution of the input and the impulse response). Likewise, for LTI systems, the impulse response can also tell us whether or not the system is causal.
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