On Complex Conjugate Pair Sums and Complex Conjugate Subspaces

TL;DR

This paper explores properties of Complex Conjugate Pair Sums and Subspaces, demonstrating their applications in derivative computation, edge detection, and efficient projection calculations in signal processing.

Contribution

It introduces new insights into CCPS and CCS properties, including their use in derivative approximation, edge detection, and the structure of projection matrices.

Findings

CCPS-based systems can compute first and second derivatives of signals.

CCS projections have a circulant structure simplifying calculations.

CCS are shift-invariant and closed under circular cross-correlation.

Abstract

In this letter, we study a few properties of Complex Conjugate Pair Sums (CCPSs) and Complex Conjugate Subspaces (CCSs). Initially, we consider an LTI system whose impulse response is one period data of CCPS. For a given input x(n), we prove that the output of this system is equivalent to computing the first order derivative of x(n). Further, with some constraints on the impulse response, the system output is also equivalent to the second order derivative. With this, we show that a fine edge detection in an image can be achieved using CCPSs as impulse response over Ramanujan Sums (RSs). Later computation of projection for CCS is studied. Here the projection matrix has a circulant structure, which makes the computation of projections easier. Finally, we prove that CCS is shift-invariant and closed under the operation of circular cross-correlation.