Cyclic Correntropy: Foundations and Theories

Over the past several decades, cyclostationarity has been regarded as one of the most significant theories in the research of non-stationary signal processing; therefore, it has been widely used to solve a large variety of scientific problems, such as weak signal detection, parameter estimation, pattern recognition, and mechanical signature analysis. Despite offering a feasible solution, cyclostationarity-based methods suffer from performance degradation in the presence of impulsive noise, so the methods are less adaptable and practicable. To improve the effectiveness of these algorithms, a nonlinear similarity measurement, referred to as cyclic correntropy or cyclostationary correntropy, was recently proposed that innovatively combines the cyclostationarity technology and the concept of correntropy and successfully changes the signal analysis from the finite dimensional space (Euclidean space) to the infinite dimensional space (Hilbert space). However, to date, the study of cyclic correntropy has been limited, and it needs to be explored further. In this paper, the foundations and theories of cyclic correntropy are elucidated rigorously to complete and develop the methodology, including basic definitions, statistical formalisms, mathematical derivations, convergence theorem, spectrum analysis, and kernel length estimation. It is believed that the cyclic correntropy, a novel methodology equipped with the precise framework of cyclostationarity, can address the problem of impulsive noise in mechanical and communication signals and that its algorithmic idea of crossing spaces will have a far-reaching impact on the development of signal processing.