Uma investigação sobre a representação matricial real para sistemas OCDM e suas limitações estruturais

Abstract

This work presents the development and analysis of an orthogonal chirp division multiplexing (OCDM)-based data communication system for baseband transmission. Stemming from the fundamentals of the discrete Fresnel transform (DFnT), which forms the mathematical core of OCDM, a novel architecture based on the real representation of complex unitary matrices is proposed. The methodology converts the N×N complex DFnT matrix into a 2N × 2N real matrix, enabling the transmission of purely real signals. A critical analysis reveals that this representation breaks the circulant structure of the DFnT matrix, increasing the demodulation computational complexity from O(N log N) to O(N2 ). Furthermore, a fundamental structural limitation is demonstrated: The impossibility of constructing a purely real transformation matrix of the form M = Λ1WN Λ2 for dimensions N > 2, where WN is the Discrete Fourier Transform (DFT) matrix. It is concluded that the proposed approach, though functional, introduces a significant computational penalty compared to existing alternatives in the literature, with its main contribution being the theoretical analysis of the limitations of real transforms based on the DFT.