A rule of thumb for a FIR filter that would:
- preserve the amplitude of low (resp. high) frequency signals
- reduce the amplitude of high (resp. low) frequency signals
could be memorized as follows.
The prototype of a low-frequency discrete signal is the constant one $x_l=[\ldots,\,1,\,\,1,\,\,1,\,\,1,\,\ldots]$. The prototype of a high-frequency discrete signal is the alternating one $x_h=[\ldots,\,-1,\,\,1,\,\,-1,\,\,1,\,\ldots]$.
Therefore, by convoluting a low-pass filter $h_l$ by $x_l$ one may expect that:
- $\sum_k h_l[k] \simeq 1$ (so that the constant input signal $x_l$ yields a filtered output almost identical)
- $\pm1\sum_k (-1)^k h_l[k] \simeq 0$ (or small enough).
Respectively, by convoluting a high-pass filter $h_h$ by $x_h$ one may expect that:
- $\sum_k h_h[k] \simeq 0$ (or small enough); the amplitude of the constant DC $x_l$ signal is shrunk
- $\sum_k (-1)^k h_h[k] \simeq \pm1$ alternatively (so that the odd/even output samples are similar to $x_h$, up to a 1-sample shift)