These two filters have very similar frequency responses, but those responses are not identical except at a single frequency $\omega_0$. I want to find $\omega_0$.
One way I've seen that done is like this:
$$\begin{align} Y_1(z)&=X(z)+2X(z)z^{-1}+X(z)z^{-2}\\ Y_2(z)&=X(z)+X(z)z^{-1} \end{align}$$
if $Y_1(z)=Y_2(z)$
$$ \begin{align} \Longrightarrow 1+2z^{-1}+z^{-2}&=1+z^{-1}\\ \Longrightarrow z^{-1}+z^{-2}&=0\\ \Longrightarrow z\left(z+1\right)&=0\\ \Longrightarrow z=e^{j\omega}=-1\\ \Longrightarrow \omega=\pi\\ \omega_0=\frac{f_s}2 \end{align} $$
But, to be perfectly frank that is quite confusing to me I don't quite know what's going on there, would someone please break it down for me?
Another way I've seen it done is like this:
$$ H_1\left(\omega_0\right)= H_2\left(\omega_0\right)\rightarrow 1+2e^{-j\omega_0}+e^{-j2\omega_0}=1+e^{-j\omega_0}. $$
With algebraic simplification the above equality can be written as:
$$ e^{-j\omega_0}=-e^{-j2\omega_0}. $$ Here's the tricky part,...we multiply both sides of the above equation by $e^{j\omega_0}$ to give us: $$ 1=-e^{-j\omega_0}. $$
I guess they're equivalent, but, in what ways specifically are they identical and in what ways are they divergent?
If someone could help me to see I would be so grateful.