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I am trying to figure out if $$y[n]-2.56y[n-1]+2.22y[n-2]-0.65y[n-3]=x[n]+x[n-3]$$ is either IIR or FIR, causal or non-causal, stable or unstable (BIBO).
I got the transfer function from the coefficient $$H(z)=\frac{1+z^{-3}}{1-2.56z^{-1}+2.22z^{-2}-0.65z^{-3}}$$ I saw this post and tried to do pole/zero cancelation but failed. How can I solve this?

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  • $\begingroup$ Why $0.5$ in the numerator of $H(z)$ if the right-hand side of your first equation has $x[n]$ instead of $\frac12 x[n]$? $\endgroup$
    – Matt L.
    Apr 7, 2023 at 16:29
  • $\begingroup$ @MattL. my bad, it should be 1. Edited $\endgroup$
    – Andrew
    Apr 7, 2023 at 17:35
  • $\begingroup$ That's kind of three questions. Which ones can you tackle ? Just apply the criteria for IIR/FIR, causality and stability to H(z) and see what you get and let us know where you stuck. $\endgroup$
    – Hilmar
    Apr 7, 2023 at 18:38

1 Answer 1

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This is a homework style question, so I won't give you a complete solution but instead I'll give you some hints to help you solve the problem yourself:

  1. the transfer function of an FIR system is a polynomial. Can the transfer function you came up with be written as a polynomial (of finite order)?
  2. given only the difference equation, there's no way to tell if the system is causal and/or stable or not. The given equation doesn't describe just one system, it describes three different systems. One of them is causal, the other two aren't. One of those three systems is stable, the others aren't. You need more information to answer the question of causality and stability.

To make my point clear, look at the following simple first-order system:

$$y[n]+ay[n-1]=x[n],\qquad a\neq 0\tag{1}$$

Eq. $(1)$ describes a causal system which can be implemented as

$$y[n]=-ay[n-1]+x[n]\tag{2}$$

But Eq. $(1)$ also describes a non-causal system:

$$y[n-1]=\frac{-y[n]+x[n]}{a}\tag{3}$$

Note that in Eq. $(3)$ in order to compute the output at time $n-1$ you need the output and the input at time $n$, i.e., you have to look one step into the future. Consequently, we have to make clear which of the two systems we want to talk about when we're given the difference equation $(1)$. For higher order difference equations, there are even more systems that are described by the same equation.

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  • $\begingroup$ Thank you for the reply. Yes, this is a homework question that I got wrong but a question that the answer was not provided afterward. For FIR or IIR, I thought of it as IIR because the transfer function cannot be simplified to a form that does not have a pole. Am I thinking correctly? For causality and stability, why are there four systems? By looking at your example, I'm thinking of making four systems by keeping one y term and transferring all others to the other side. when y[n] is on the left, it's causal and all the others are non-casual. Can you elaborate a bit more? $\endgroup$
    – Andrew
    Apr 8, 2023 at 19:30
  • $\begingroup$ @Andrew: You're right about the system being IIR. Same about one system being causal and the other being non-causal. So what would you like me to elaborate? $\endgroup$
    – Matt L.
    Apr 8, 2023 at 20:34

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