# Recursive, non-recursive systems; FIR, IIR systems

I am confused with the classifications of an LTI system as recursive or non-recursive systems and FIR or IIR systems.

I understood what the FIR and an IIR systems are, but is it correct to say that FIR system is always non-recursive?
We could express an finite accumulator up to past N inputs (FIR system) in both non-recursive and recursive forms.

Also is it correct to say that a non-recursive system is always an IIR system or vice versa?

• Very related: this answer. – Matt L. Apr 5 at 19:59
• Here is another answer about the topic. There is a class of FIR filters called Truncated IIR or TIIR. They are recursive, approximate an IIR with the tail truncated. If the delay is long enough, a TIIR is a more computationally efficient than the transversal FIR structure that is $O\{L\}$. – robert bristow-johnson Apr 6 at 2:08

## 2 Answers

The logical implications are the following:

"non-recursive" $$\Longrightarrow$$ FIR

IIR $$\Longrightarrow$$ "recursive"

But the opposites are not necessarily true because a FIR system can be implemented recursively (transfer function poles can be cancelled by zeros).

Of course, when referring to "recursive" or "non-recursive" we always talk about implementations with finitely many operations per output sample. Clearly, any discrete-time LTI system can be described by a generally infinite convolution sum, but that is not what we mean by "non-recursive".
• Could you please clarify why is a non-recursive system always an FIR? Is it from a theoretical point of view or implementation point of view? Because theoretically an accumulator of past infinite number of inputs which is an IIR can be expressed non-recursively and also recursively. – Sai Krishna Garlapati Apr 6 at 4:37
• My understanding is that an FIR and IIR could both be theoretically expressed as a recursive or as a non-recursive system. But from an implementation point of view an FIR is mostly implemented as a non-recursive structure but could be implemented as a non-recursive structure also. But an IIR system is always implemented as arecursive structure. Am I right? – Sai Krishna Garlapati Apr 6 at 4:46
• @SaiKrishnaGarlapati How do you express an infinite accumulator non-recursively? – jpa Apr 6 at 5:39
• @SaiKrishnaGarlapati Ah, you mean with infinitely long expression. I wonder if there is a widely accepted definition whether non-recursive filters can have infinitely many coefficients. Infinity is always an annoying corner-case: for example all functions are polynomial, if the polynomial can have infinite coefficients. – jpa Apr 6 at 6:23
• @SaiKrishnaGarlapati: The terms "recursive" or "non-recursive" always refer to implementations, because it is a trivial fact that any discrete-time LTI system can be described by an (generally infinite) convolution sum. But it doesn't make much sense to say that any LTI system is (?) a non-recursive system ... – Matt L. Apr 6 at 6:57

is it correct to say that FIR system is always non-recursive?

You answer that yourself:

We could express an finite accumulator up to past N inputs (FIR system) in both non-recursive and recursive forms.

exactly. A common example of a recursive filter that's in fact an FIR is the CIC filter.

Also is it correct to say that a non-recursive system is always an IIR system

no,

or vice versa?

neither (see above)

• seriously, trivial to answer by finding the simplest IIR you can imagine, which is a single-pole IIR: y[t] = y[t-1]·a+x[t]·(1-a) . That's an IIR and it's recursive system. – Marcus Müller Apr 6 at 7:19