Wide-sense stationary discrete-time random processes: Analyses in the discrete-Fresnel and discrete-frequency domains

• Published in: Journal of the Franklin Institute 2024-12, Vol.361 (18), p.107310, Article 107310
• Main Authors: de Lima Filomeno, Mateus, Giroto de Oliveira, Lucas, Camponogara, Ândrei, Zwick, Thomas, de Campos, Marcello Luiz Rodrigues, Ribeiro, Moisés Vidal
• Format: Article
• Language: English
• Subjects: Discrete-frequency domain; Discrete-Fresnel domain; Random process; Wide-sense stationarity
• ISSN: 0016-0032
• DOI: 10.1016/j.jfranklin.2024.107310

This article investigates the expectation and autocorrelation functions of random processes in the discrete-Fresnel and the discrete-frequency domains, originating from wide-sense stationary (WSS) random processes in the discrete-time domain. Through mathematical deduction, it is demonstrated that WSS random processes in the discrete-time domain exhibit the same expectation and autocorrelation functions as transformed to the discrete-Fresnel domain. In contrast, this property is not observed in the discrete-frequency domain. White, pink, and brown random processes are considered to illustrate these behaviors in the discrete-Fresnel and discrete-frequency domains. Furthermore, this study numerically compares data communication systems based on single-carrier with cyclic prefix, orthogonal chirp-division multiplexing, and orthogonal frequency-division multiplexing schemes, which are respectively designed in the discrete-time, discrete-Fresnel, and discrete-frequency domains. The numerical results show that orthogonal chirp-division multiplexing and single-carrier with cyclic prefix schemes achieve comparable communication performance under any additive noise modeled by a wide-sense stationary random process, regardless of the chosen linear equalization or the absence of it. •An expected value in the discrete-time domain is kept in the discrete-Fresnel domain.•The same autocorrelation function is seen in both time and Fresnel discrete domains.•The DFnT does not alter the WSS characteristic of random processes.•OCDM and SC-CP perform equally in terms of SNR for any kind of WSS additive noise.