Im thinking whether any convolutional operation can output a unit impulse, an example to further explain: where a convolution between system $h[n]$ and unknown system $g[n]$ would output $\delta[n]$.
conv(h[n], g[n]) = δ[n]
where
$$h[n] = \delta[n] + \delta[n-1]/2 + \delta[n-2]/4 + \delta[n-3]/8 + \delta[n-4]/16$$
In regards to Z-transform - it can be concluded that
$$H[z] = 1 + 1/2z^{-1} + 1/4z^{-2} + 1/8z^{-3} + 1/16z^{-4}$$
Moreover, it is know that in z domain conv becomes multiplication therefore:
$$H[z] G[z] = 1$$
And in conclusion $G[z] = 1/H[z]$
$$G[z] = \frac{1}{1 + 1/2z^{-1} + 1/4z^{-2} + 1/8z^{-3} + 1/16z^{-4}}$$
If the above is somehow correct, then when trying to achieve the inverse z-transform we will get:
$$g[n] = \delta[n] + 2/\delta[n-1] + 4/\delta[n-2] + 8/\delta[n-3] + 16/\delta[n-4]$$
We get $1/\delta[n]$, which does not bode well for anybody.
Im I in the wrong? Can it be possible?