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4.5 Impulse response and lti system causality

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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.

The impulse response and causality

There is a straightforward relationship between an LTI system's impulse response and whether or not the system is causal: An LTI system is causal if and only if its impulse response is 0 for all $n\lt 0$ (i.e., the impulse response is a causal signal). This follows naturally from the convolution sum. The system's output $y[n]=\sum_{m=-\infty}^\infty h[n-m]x[m]$ Note that if and only if $h[m]=0 \forall m\lt 0$, no $x[m]$ for $m\gt n$ contribute to the sum, which is to say that no future values of the input factor in to the output at time $n$.Below is an example of the impulse response for a causal system. Note that it is $0$ for $n\lt 0$. This also corresponds to the system matrix being lower triangular:
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An example of an impulse response for a causal system. The impulse response is zero for $n\lt 0$.
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A graphical representation of the system matrix of a causal system; its values are zero above the main diagonal.
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Source:  OpenStax, Discrete-time signals and systems. OpenStax CNX. Oct 07, 2015 Download for free at https://legacy.cnx.org/content/col11868/1.2
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