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:
Questions & Answers
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
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Samuel
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