Finding whether a filter is low/high/band pass

[t.kirschner99]

Homework Statement

Consider the IIR filter y_n = x_n - y_n-2

State whether the filter is low, high, or band pass.

Homework Equations

The Z-transform: $$H(z) = \frac {1} {1+z^2},$$

Subbing $z = e^{2πiw}$ : $$H(w) = \frac {1} {1+e^{4πiw}},$$

Amplitude response: $$|H(w)| = \sqrt{{(\frac {1} {1 + cos{(4πw)}})}^2 + {(\frac {1} {sin{(4πw)}})}^2},$$

The Attempt at a Solution

I set up the domain $w = [0,1]$

$w = 0$

Amplitude response would equal $\frac {1} {2}$

$w = \frac {1} {2}$

Amplitude response would equal $\frac {1} {2}$

$w = 1$

Amplitude response would equal $\frac {1} {2}$

Doesn't this not prove anything? I know the answer is band pass, but I would expect zeroes on both ends and a number greater than zero in the middle.

 

[FactChecker]

Try applying the initial and final value theorems (see http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html )