Linear and nonlinear filters based on the improper Karhunen–Loève expansion

Performance analysis of the deficient length augmented CLMS algorithm for second order noncircular complex signals

“…For second order circular signals, the pseudo-covariance matrix P x = 0, and the standard adaptive filtering algorithms (based on the covariance only) are adequate. For noncircular signals, P x ̸ = 0, and complexvalued adaptive filtering algorithms should be designed based on the 2M × 1 so-called augmented input vector z(k) = [x T (k), x H (k)] T , for which the 2M × 2M augmented covariance matrix is given by [4,5,6,8,7,9] The augmented complex statistics have opened the possibility to design LMS-type adaptive filtering algorithms based on the widely linear model [4,6,7,10], leading to the so called augmented CLMS (ACLMS) [11,12], which is suitable for processing both second order circular and noncircular signals. This advantage over the conventional CLMS algorithm has led to its applications in signal processing [9,13,14], communications [15], and power systems [16,17].…”