Approximation and design methods for efficient filters with less multiplication requirement

In consumer electronics, it is very important to design a filter with very less computation loading. In this paper, I try to approximate a filter by several geometric segments, geometric polynomial segments, or rectangular segments. With the approximation, the filter can be approximated by some IIR filter and implemented in a recursive way with very less multiplication requirement. I will introduce the way to design a filter whose number of multiplications is fixed to a small constant but the approximate error can be minimized. Several examples are given to show how to design the digital filter that can well approximate the original ideal filter but the number of real multiplications is much less.