Signal Approximations Based on Nonlinear and Optimal Piecewise Affine Functions

In this work, we address the problem of piecewise affine approximations, that is, to find piecewise affine functions that well-approximate a given signal. The proposed approach is optimal in the sense of L-2 norm and formulated in a compact and explicit way; no fitting stage is needed. Also, affine parameters are obtained as closed formulas, and affine approximation functions are optimal in their corresponding subdomains. In addition, we state and prove a recursive formula for approximation errors, which makes the approach optimal and nonlinear, links also the subdomains and helps derive an algorithm of complexity of order O (MN2), where M represents the number of piecewise affine approximants and N is the number of samples of the processed signal. Finally, obtained qualitative and quantitative results show that the presented method obtains good approximations and provides improvement over piecewise constant approximations.