The question is: how can I define $h_1[n]$ in such a way that $h [ n ] = \delta [n - 1 ] + 2 \delta [n -2 ] + h_ 1 [n]$ is a band-pass filter. My thought was the following.
Firstable, I wrote the $Z$-transform as $\displaystyle{H (z) = H_ 1 ( z) + z^ {-1} +2z^ {-2} = \frac{z^ 2 H_1 ( z) + z + 2 }{z ^ 2 }}$. Then I thought, since the sistem basically has two poles in 0, to add two poles, in $z = 1, -1 $ (in order to have the amplitude of $H( e^ {i\omega})$ vanishing for $\omega = 0, \pi $). I decided to take a signal of the form $h _1 [n] = A\delta [n + 1 ]+ B \delta [n+2 ] \implies H _1(z) = A z + B {z ^2 } $ and I got, by the previous conditions, a system of equations: \begin{equation*} \begin{cases} H ( -1 ) = H_1( -1 )+( -1)^ {-1} + 2 (-1 ) ^ 2 = -A+ B +1= 0 \\ H (1) = H _1(1) +(1)^ {-1} +2(1 ) ^ 2 = A + B +3 = 0 \end{cases} \end{equation*} so that $A= -1, B = -2 $. Therefore I concluded that $h[n] =- \delta [n+1 ] - 2\delta [n+2] + \delta [n-1] + 2 \delta[n-2]$ is a band-pass filter. I draw the graph of the amplitude and it seems to be a kind of absolute value of a damping sinusoidal function (with zeros for $\omega = 0, \omega _ 1, \pi $, where $\omega _ 1 $ is a value between $\frac \pi 2 $ and $ \frac 2 3 \pi $).
I was wondering if my argument is correct. Any help would be appreciated, thanks in advance.
