Changing Filter Coefficients

The difference (instead of the sum) of adjacent samples:

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

is like changing the coefficient of $x(n-1)$ to -1.

At 0 Hz:

$$y(n) = x_1(n) - x_1(n-1)$$

$$= \quad [A, A, A, ....]$$

$$ - [0, A, A, A, ...]$$

$$= \quad [A, 0, 0, 0, ...]$$

$$\approx 0x_1(n).$$

At $f_s/2$ Hz:

$$y(n) = x_2(n) - x_2(n-1)$$

$$= \quad [A, -A, A, -A, ...]$$

$$ - [0, A, -A, A, ...]$$

$$= \quad [A, -2A, 2A, -2A, ...]$$

$$\approx 2x_2(n).$$