Tessarine signal processing under the T-properness condition

• Published in: Journal of the Franklin Institute Vol. 357; no. 14; pp. 10100 - 10126
• Main Authors: Navarro-Moreno, Jesús, Fernández-Alcalá, Rosa María, Jiménez-López, José Domingo, Ruiz-Molina, Juan Carlos
• Format: Journal Article
• Language: English
• Published: Elmsford Elsevier Ltd 01.09.2020; Elsevier Science Ltd
• Subjects: Algebra; Estimating techniques; Hilbert space; Quaternions; Random signals; Signal processing; State space models; Stochastic models
• ISSN: 0016-0032; 1879-2693; 0016-0032
• DOI: 10.1016/j.jfranklin.2020.08.002

The paper analyzes the processing of 4D commutative hypercomplex or tessarine signals under properness conditions. Firstly, the concept of T-properness is introduced and a procedure to test experimentally whether a tessarine random signal is proper or not is proposed. Then, for the class of T-proper signals, the linear minimum mean square error estimation problem is addressed. In this regard, it should be highlighted that although the tessarine algebra is not a Hilbert space, a metric which guarantees the existence and unicity of the optimal estimator is defined. Moreover, the equivalence, under T-properness conditions, between the optimal estimator based on a tessarine widely linear processing and the one based on a tessarine strictly linear (TSL) processing is also shown, attaining thus a notable reduction in computational burden. Finally, two T-proper models, a TSL state-space model and a TSL stationary model, from which the optimal estimator can be recursively obtained are considered. In both cases, simulated examples are developed where the superiority of TSL processing over the counterparts in the quaternion domain is exhibited.