In this paper, we introduce two types of real-valued sums known as Complex Conjugate Pair Sums (CCPSs) denoted as CCPS$^{(1)}$ and CCPS$^{(2)}$, and discuss a few of their properties. Using each type of CCPSs and their circular shifts, we construct two non-orthogonal Nested Periodic Matrices (NPMs). As NPMs are non-singular, this introduces two non-orthogonal transforms known as Complex Conjugate Periodic Transforms (CCPTs) denoted as CCPT$^{(1)}$ and CCPT$^{(2)}$. We propose another NPM, which uses both types of CCPSs such that its columns are mutually orthogonal, this transform is known as Orthogonal CCPT (OCCPT). After a brief study of a few OCCPT properties like periodicity, circular shift, etc., we present two different interpretations of it. Further, we propose a Decimation-In-Time (DIT) based fast computation algorithm for OCCPT (termed as FOCCPT), whenever the length of the signal is equal to $2^v,\ v{\in} \mathbb{N}$. The proposed sums and transforms are inspired by Ramanujan sums and Ramanujan Period Transform (RPT). Finally, we show that the period (both divisor and non-divisor) and frequency information of a signal can be estimated using the proposed transforms with a significant reduction in the computational complexity over Discrete Fourier Transform (DFT).
翻译:在本文中,我们引入了两种实际价值的金额,即被称为Complicate Congate Pair Sumes(CCPS)的复杂组合组合(CCPS),称为CCPT$(1)美元(CCPS)和CCPT$(CCPS)美元(CCPS)美元(CCPS)美元(CCPS)美元(CCPS)和CCPP$(CCPS)美元(CCPS)美元(CCPS)),并讨论其中的一些属性。我们使用两种类型的CCPS(CCPS)和它们的循环转换(CCPS),我们建造了两种非横向的NEST定期(NPS)周期(CCPS),这两类非横向的变换价(CCPT),我们提出两种不同的解释(DIMT)基于OCPT(CCPT)的快速算法(CPT),只要提议的变价期为2美元(FCCPT),拟议变价期的变价(FNBT)的变价可以显示。