What kind of filter is that? Is it IIR?

I am trying to answer the following question:

Is the system described by equation:

$$y[n]=0.5y[n-1]+x[n]-0.5x[n-1]$$

an IIR filter? My answer is yes.

Thank you

Answer

This is the FIR filter, although it looks like an IIR. If you calculate the coefficients you get finite impulse response:

$h=[1]$

This happens due to zero-pole cancellation:

$Y(z)-0.5Y(z)z^{-1}=X(z)-0.5X(z)z^{-1}$

$H(z)=\dfrac{Y(z)}{X(z)}=\dfrac{1-0.5z^{-1}}{1-0.5z^{-1}}=1$

Yes, it can be tricky. Seeing $y[n-k]$ coefficients in LCCDE (Linear Constant Coefficients Difference Equation) doesn't necessarily mean it's an IIR filter. It might be just a recursive FIR filter.