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Given a general filter equation, how can one tell the type of filter that the same equation represents? Meaning how can I know if the filter is Low/High/Band Pass etc...?

For example, the following equation:

$ y_{n} = x_{n} + x_{n-2} $

Represents a "Band Stop" filter, but why?

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    $\begingroup$ gonna have to apply the Z transform. and set $z = e^{j\omega}$ to find out how this filter behaves. $\endgroup$ Commented Jan 12, 2016 at 20:00
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    $\begingroup$ @robertbristow-johnson , i see a down vote ,whats wrong with my solution if you can point out $\endgroup$ Commented Jan 13, 2016 at 7:28

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The $\mathcal Z$-transform of the given difference equation is :

$$H(z) = 1 + z^{-2}$$

Put $z= \exp(j\omega)$ and remember that in discrete time systems the low frequencies are at $2n\pi$ $(n=0,1,2, \ldots)$ and the high frequencies occur at $(2n+1)\pi$; this is just a consequence of the periodic behavior of the discrete time complex exponentials.

So, $H(z) = 1 + \exp(-2j\omega)$ at $z = \exp(j\omega)$.

When $\omega=0; H(z) = 2$ and $w = \pi$ gives $H(z)= 2$.

Thus, both at high and low frequencies the the system function provides same gain and hence the filter with the given $H(z)$ is a BAND REJECT/ NOTCH FILTER with $H(z) = 0$ at $\omega = \pi/2$.

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