An algebraic explanation for Maximum Decision

One of the aims of communication is to increase the data rate. However, frequency resources are scare and valuable for wireless communication systems. How to use frequency resources efficiently is a significant area which needs the attentions of researchers. Ultra Narrow Band (UNB) has attracted wide attentions since it was proposed. M-ary Position Phase Shift Keying (MPPSK) is one of the UNB modulation modes. The differences in phases of the MPPSK modulated waveform can be transformed into the peaks of the amplitudes through the Impacting Filter. Maximum Decision is selected as the decision module of the MPPSK demodulator which needs to search the position where the response of the Impacting Filter has a maximum value. As a novel viewing angle, this paper will provide a mathematic explanation for Maximum Decision and prove its validity.