In this article, 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).

中文翻译：

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使用具有 NPM 结构的新变换矩阵的正交和非正交信号表示

在本文中，我们介绍了两种类型的实值和，称为复共轭对和 (CCPS)，表示为 CCPS$^{(1)}$ 和CCPS$^{(2)}$，并讨论它们的一些特性。使用每种类型的 CCPS 及其循环位移，我们构建了两个非正交嵌套周期矩阵 (NPM)。由于 NPM 是非奇异的，这引入了两个非正交变换，称为复杂共轭周期变换 (CCPT)，表示为 CCPT$^{(1)}$ 和CCPT$^{(2)}$. 我们提出了另一种 NPM，它使用两种类型的 CCPS，使其列相互正交，这种变换被称为正交 CCPT（OCCPT）。在对周期性、循环移位等一些 OCCPT 特性进行简要研究后，我们对其提出了两种不同的解释。此外，我们提出了一种基于时间抽取 (DIT) 的快速计算算法 OCCPT（称为 FOCCPT），只要信号的长度等于$2^v,\ v \in \mathbb {N}$. 提议的总和和变换受到拉马努金求和和拉马努金周期变换 (RPT) 的启发。最后，我们表明可以使用所提出的变换来估计信号的周期（除数和非除数）和频率信息，与离散傅立叶变换 (DFT) 相比，计算复杂度显着降低。