A New Signal Representation Using Complex Conjugate Pair Sums

Abstract

This letter introduces a real valued summation known as Complex Conjugate Pair Sum (CCPS). The space spanned by CCPS and its one circular downshift is called {\em Complex Conjugate Subspace (CCS)}. For a given positive integer $N\geq3$, there exists $\frac{\varphi(N)}{2}$ CCPSs forming $\frac{\varphi(N)}{2}$ CCSs, where $\varphi(N)$ is the Euler's totient function. We prove that these CCSs are mutually orthogonal and their direct sum form a $\varphi(N)$ dimensional subspace $s_N$ of $\mathbb{C}^N$. We propose that any signal of finite length $N$ is represented as a linear combination of elements from a special basis of $s_d$, for each divisor $d$ of $N$. This defines a new transform named as Complex Conjugate Periodic Transform (CCPT). Later, we compared CCPT with DFT (Discrete Fourier Transform) and RPT (Ramanujan Periodic Transform). It is shown that, using CCPT we can estimate the period, hidden periods and frequency information of a signal. Whereas, RPT does not provide the frequency information. For a complex valued input signal, CCPT offers computational benefit over DFT. A CCPT dictionary based method is proposed to extract non-divisor period information.