用具有NPM结构的新变换矩阵表示正交和非正交信号
Orthogonal and Non-Orthogonal Signal Representations Using New Transformation Matrices Having NPM Structure
摘要：
本文引入了两类实值和--复共轭对和（CCPS），记作CCPS $^{(1)}$和CCPS $^{(2)}$，并讨论了它们的一些性质.利用每种类型的CCPS及其循环移位，我们构造了两个非正交嵌套周期矩阵（NPM）。由于NPM是非奇异的，这引入了称为复共轭周期变换（CCPT）的两个非正交变换，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，使得它的列相互正交，这种变换被称为正交CCPT（OCCPT）。在简要研究了OCCPT的一些性质，如周期性、循环移位等之后，在此基础上，提出了一种基于时间抽取（DIT）的OCCPT快速计算算法（FOCCPT），当信号长度等于$2^v,\ v{\in} \mathbb{N}$时，该算法可以快速计算OCCPT。所提出的和和变换的灵感来自Ramanujan和和Ramanujan周期变换（RPT）。最后，我们表明，周期（除数和非除数）和频率信息的信号可以估计使用建议的变换与离散傅里叶变换（DFT）的计算复杂度显着降低。
英文摘要：
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
