How Do You Determine if a System is Causal and Distinguish Between FIR and IIR?

[cf9219]

I have a couple of questions about the theory about how to tell whether a system is casual and FIR or IIR.

First point is about a casual system. Is it true that all FIR systems are casual? How can you tell if an IIR system is casual?

I am I correct in thinking that a FIR system only has a numerator and IIR system has both a numerator and denominator?

For example: h(n) = u(n+1) -u(n-1) and h(n) = 5^n . U(-n)

Are both of these FIR systems?

 

[the_emi_guy]

cf9219
Not all FIR systems are causal. (If they operate in real time they must be causal).
Causal means that the present output depends only on the present and prior inputs and outputs. Non-causal means the output depends on future inputs. Your example h(n) = u(n+1) -u(n-1) is non-causal because the output at time n depends on the input at time n+1.

A FIR system's output depend only on the input (and its delayed copies). An IIRs output depends on prior outputs as well, in other words there is feedback. If the right hand side of your difference equation contains output terms, it is IIR. Your example h(n) = u(n+1) -u(n-1) is FIR since no "h" terms appear in right hand side.

You are right about the transfer functions. When we transform your difference equations into the z-domain, IIR filters will have a denominator polynomial (and potentially a numerator polynomial).

Finally, your second example: h(n) = 5^n . U(-n) seems strange.