Does a causal system have to be non-recursive?

[Bipolarity]

I have been trying to search for a clear definition of "discrete causal systems". The thing I want to know, which I've not been able to find, is whether non-recursiveness of the difference equation is part of the requirement for a causal discrete system?

Thanks!

BiP

 

[AlephZero]

A causal system is just one where the output at time $T$ only depends on the input at times $t \le T$. In other words, the filter doesn't know anything about the future. That is a requirement for any filter that operates in real time (unless you want to change the laws of physics!) . If you are filtering data that has been acquired in the past, the filter doesn't have to be causal.

Recursiveness is irrelevant. That just says the output at time $T$ also depends on the output at times $t \le T$, but those outputs ultimately depend on the inputs.

Recursiveness is useful in practice because you can design a filter that does a finite number of "operations" (either analog or digital) but the output depends on all the input from times $-\infty < t \le T$. A simple example would be an digital "exponential smoothing" filter where the output at time $T$ is a weighted average of the input at time $T$, and the output at the previous time $T-\delta t$.