Linear-phase lowpass filters: frequency responses alignment 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:


  $$\begin{align}
Y_1(z)&amp;=X(z)+2X(z)z^{-1}+X(z)z^{-2}\\
Y_2(z)&amp;=X(z)+X(z)z^{-1}
\end{align}$$
  
  if $Y_1(z)=Y_2(z)$
  
  $$
\begin{align}
\Longrightarrow 1+2z^{-1}+z^{-2}&amp;=1+z^{-1}\\
\Longrightarrow z^{-1}+z^{-2}&amp;=0\\
\Longrightarrow z\left(z+1\right)&amp;=0\\
\Longrightarrow z=e^{j\omega}=-1\\
\Longrightarrow \omega=\pi\\
\omega_0=\frac{f_s}2
\end{align}
$$


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.
Both methods are equivalent. Look at it like this: first write down the frequency responses (as already indicated in your question)

$$H_1(e^{j\omega})=1+2e^{-j\omega}+e^{-2j\omega}\\
H_2(e^{j\omega})=1+e^{-j\omega}$$

Now want to find $\omega_0$ such that

$$H_1(e^{j\omega_0})=H_2(e^{j\omega_0})$$

which gives

$$1+2e^{-j\omega_0}+e^{-2j\omega_0}=1+e^{-j\omega_0}$$

or, equivalently,

$$e^{-j\omega_0}(1+e^{-j\omega_0})=0\tag{1}$$

Equation (1) can only be satisfied if the term in parentheses vanishes. So you get

$$e^{-j\omega_0}=-1\Longrightarrow\omega_0=\pi$$

And since $\omega_0=2\pi f_0/f_s$ (where $f_s$ is the sampling frequency) you finally get

$$f_0=f_s/2$$