Properties of Type-2 Complex Conjugate Pair Sums and Their Applications

Ramanujan Sum (RS) has recently been used in the Ramanujan Periodic Transform (RPT), which efficiently extracts period information with lower computational complexity. Building on RS and RPT, Complex Conjugate Pair Sums of type-1 (CCPS(1)) and type-2 (CCPS(2)) have been developed, forming the basis of the Orthogonal Complex Conjugate Periodic Transform (OCCPT), an alternative to the Discrete Fourier Transform (DFT) with reduced computational requirements. While RSs and CCPS(1) properties are well-studied, CCPS(2) characteristics remain underexplored. This paper investigates CCPS(2) properties and their potential applications. Specifically, it examines the behavior of a Linear Time-Invariant (LTI) system with a CCPS(2)-based impulse response, demonstrating that the system can approximate first- and second-order derivatives of the input signal. This property is applied to image edge detection and Electrocardiogram (ECG) preprocessing, comparing the performance with systems using RS and CCPS(1) impulse responses. Additionally, the paper shows that the DFT coefficients of any two distinct, CCPS(1) or CCPS(2), as well as CCPS(1) and CCPS(2), sequences are non-overlapping, ensuring orthogonality among the subspaces they span. Based on this, we propose a new modulation scheme, Orthogonal Complex Conjugate Periodic Subspace Division Multiplexing (OCCPSDM), which is compared with existing modulation techniques regarding Peak-to-Average Power Ratio (PAPR) and computational complexity.