Orthogonal and Non-Orthogonal Signal Representations Using New Transformation Matrices Having NPM Structure
share
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∈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).
READ FULL TEXT