I have the following system: $$ y[n]=\frac{1}{3}(x[n+1]+x[n]+x[n-1]) $$ After the Z-Transform we get

$$ \frac{y[z]}{x[z]}=\frac{z^2+z+1}{3z} $$ which is of course the transfer function of the system. Now, I know that it would be a FIR if $\sum h(z) < \infty$ with $h(z)$ the impulse response. But As far as I know, it should be clear if it is FIR or IIR by just looking the transfer function. How can I know that?

  • $\begingroup$ Your system is not causal as your current output y[n] depends on a future input x[n+1]. If your system were causal then it would definitely be an FIR. $\endgroup$ – Ben Oct 12 '17 at 12:59
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    $\begingroup$ @Ben: An FIR filter need not be causal. $\endgroup$ – Jason R Oct 12 '17 at 13:03
  • $\begingroup$ Yeah, you're right. My DSP theory needs some brushing up ;) $\endgroup$ – Ben Oct 12 '17 at 13:05

The forward $z$-transform is written with negative powers on $z$. So, $z$ transform of your difference equation would be:

$$ Y(z) = \frac{1}{3}\left(zX(z) + X(z) + z^{-1}X(z)\right) $$

yielding a transfer function of:

$$ H(z) = \frac{Y(z)}{X(z)} = \frac{1}{3}(z + 1 + z^{-1}) $$

The easiest way to determine whether a filter is IIR or FIR is to identify its pole locations. For FIR filters, there is a rule for this that is based on the structure of the impulse response:

  • If the system is causal (i.e. it is zero for all $n < 0$), then it is FIR if all of its poles are located at the origin ($z=0$).
  • If the system is anticausal (i.e. it is zero for all $n \ge 0$), then it is FIR if all of its poles are located at $z=\infty$.

Your system is neither causal nor anticausal. Instead, it has a causal component ($\frac{1}{3}(x[n] + x[n-1])$) and an anticausal component ($\frac{1}{3}x[n+1]$). So, it can be helpful to look at each component separately:

  • $\frac{1}{3}(x[n] + x[n-1])$: The $z$-transform of the causal portion of the transfer function is $\frac{1}{3}(1 + z^{-1})$. This expression $\to \infty$ as $z \to 0$, so it has a pole at $z = 0$. This meets the criterion for an FIR response.

  • $\frac{1}{3}x[n+1]$: The $z$-transform of the anticausal portion of the transfer function is $z$. This expression $\to \infty$ as $z \to \infty$, so it has a pole at $z = \infty$. This meets the criterion for an FIR response.

This would imply that you gave a difference equation for a non-causal FIR filter.

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    $\begingroup$ An easier test might be to see if $H(z)$ is a polynomial in $z^{-1}$, or a polynomial in $z$ or the sum thereof (as in this case) and if so, the filter is an FIR filter (non causal in this case). In applying this test, one needs to distinguish between polynomials and series in order to decide whether we have an FIR filter or an IIR filter. This meets the desired criterion of "just looking at the transfer function" that the OP wants. $\endgroup$ – Dilip Sarwate Oct 13 '17 at 15:04

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