Intuitive Analyis at High Frequencies

The running average of an input signal with significant variation from sample to sample will be very different than the input signal.

• consider a second input $x_2(n)$ at the highest possible frequency $f_s/2$ (the Nyquist limit), which swings more significantly from sample to sample:
  $$x_2(n) = [A, -A, A, ...].$$

• the output of the lowpass filter is
  $$y(n) = x_2(n) + x_2(n-1) = [A, 0, 0, ...]$$

• ignoring the first sample, it attenuates the Nyquist limit.