Intuitive Analyis at High Frequencies

Consider input at $\displaystyle{\frac{f_s}{2}}$ Hz (highest possible frequency):

$\displaystyle x_2(n) = [A, -A, A, -A, ...].$

(maximum change from sample to sample).

The output of the filter is

$\displaystyle y(n)$	$\displaystyle =$	$\displaystyle x_2(n) + x_2(n-1)$
	$\displaystyle =$	$\displaystyle \quad [A, -A, A, -A, ...]$
		$\displaystyle +$ $\displaystyle [0, A, -A, A, ...]$
	$\displaystyle =$	$\displaystyle \quad [A, 0, 0, 0, ...]$
	$\displaystyle \approx$	$\displaystyle 0x_2(n)$ (except 1st sample) $\displaystyle .$