Orthogonal and Non-Orthogonal Signal Representations Using New Transformation Matrices Having NPM Structure

TL;DR

This paper introduces new non-orthogonal and orthogonal signal transforms based on novel transformation matrices with NPM structure, inspired by Ramanujan sums, enabling efficient period and frequency estimation.

Contribution

The paper proposes new CCPS-based NPMs, non-orthogonal and orthogonal transforms, and a fast algorithm for OCCPT, advancing signal analysis techniques inspired by Ramanujan sums.

Findings

OCCPT provides accurate period estimation.

FOCCPT reduces computational complexity compared to DFT.

Transforms effectively extract frequency information.

Abstract

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).