# Determine filter type using recurrence relation

Given the recurrence relation: $$y[n] = x[n] + 0.5y[n-1]$$ I want to determine the filter type (i.e. LPF, HPF etc.)

I try to use Z transform, and get that the the transfer function is $$H(z) = \frac{2z}{2z-1}$$ So I derive that the filter is HPF (High pass). Is it correct?

• Hint : what is the gain at z = 1? I.e. DC. Is it 0? If it's not 0, it's not a high-pass filter. – Ben Feb 4 at 16:34
• H(1) = 2 and H(infinity) = 1, so it means it is LPF? – ron653 Feb 4 at 16:47
• Evaluating at z = infinity makes no sense. The frequency response is evaluated over the unit circle, with DC at z = 1, and freq pi at z = -1 – Juancho Feb 4 at 17:14
• So how I determine the type of the filter? – ron653 Feb 4 at 17:43
• Most filters don't fit an easy category like high/low pass. Why do you need to know the filter type and what is the range of "allowed" answers ? – Hilmar Feb 4 at 18:18