Changing Filter Coefficients

Consider the following variation on the two-point averager (lowpass filter):

$\displaystyle y(n) = x(n) - x(n-1).$

How does changing the addition to a subtraction change the filter?

$\fbox{\parbox{5in}{Changing the addition to a subtraction changes the \emph{coefficients} of the filter.}}$

At DC the output becomes

$\displaystyle y(n)$	$\displaystyle =$	$\displaystyle x_1(n) - x_1(n-1)$
	$\displaystyle =$	$\displaystyle \quad [A, A, A, ....]$
$\displaystyle$		$\displaystyle -$ $\displaystyle [0, A, A, A, ...]$
	$\displaystyle =$	$\displaystyle \quad [A, 0, 0, 0, ...]$

At the Nyquist limit the output becomes

$\displaystyle y(n)$	$\displaystyle =$	$\displaystyle x_2(n) - x_2(n-1)$
	$\displaystyle =$	$\displaystyle \quad [A, -A, A, ....]$
$\displaystyle$		$\displaystyle -$ $\displaystyle [0, A, -A, A, ...]$
	$\displaystyle =$	$\displaystyle \quad [A, -2A, 2A, -2A, ...]$

``Music 206: Delay and Digital Filters I'' by Tamara Smyth, Computing Science, Simon Fraser University.