# Complex filter factorizations – continued

Continuing from this rather silly trivial question factoring real valued filters into shorter complex ones, hoping this won’t be as trivial.

If we modify it a bit:
$$z_0 = e^{2\pi i / 8}$$
and
$$\left(z_0^{[3k,2k]} * z_0^{[3k,-2k]}\right)$$

will for $k \in \{2,3,4\}$:
$$k=2 \rightarrow \left[\begin{array}{ccc} -1&2i&1 \end{array}\right]$$
derivative in real part
$$k=3 \rightarrow \left[\begin{array}{ccc} i&0&1 \end{array}\right]$$
captures displacements by -1 and 1 position. E.g. both “lazy” filters.
$$k=4 \rightarrow \left[\begin{array}{ccc} 1&-2&1 \end{array}\right]$$
which is a second derivative approximation.

Theses ones I found by trial and error. I guess what I am asking for is systematic ways to find filters which are easy to build “on the fly”, cheap to calculate, and which capture previously known important features.