Complex dual channel estimation: Cost effective widely linear adaptive filtering

Widely linear estimation for complex-valued data allows for a unified treatment of both second order circular (proper) and non-circular (improper) signals. We propose the complex dual channel (CDC) estimation technique as an alternative to widely linear estimation to both gain further insights into complex valued minimum mean square error (MMSE) estimation and to design computationally efficient adaptive filtering algorithms. This is achieved by finding two sets of optimal weights that minimize the mean square error (MSE) in estimating the real and imaginary parts of the signal independently. The concept is used in a stochastic gradient setting to design the dual channel complex least mean square (DC-CLMS). The analysis shows that any one of the sub-filters within the DC-CLMS can be used to estimate strictly linear models while the DC-CLMS is equivalent to widely linear estimation. This results in a reduction of computational complexity of complex-valued adaptive filters by a half, while providing enhanced physical insight and control over complex-valued estimation algorithms. & 2014 Elsevier B.V. All rights reserved. Notation: Lowercase letters are used to denote scalars, boldface letters for vectors and boldface uppercase letters for matrices. The symbol ð Þ denotes complex conjugation, ð Þ and ð Þ – transposition and conjugate transposition, ð Þ 1 – matrix inversions and Tr1⁄2 is the trace of a matrix. The operators R1⁄2 and I1⁄2 are used to extract respectively the real and imaginary parts of a complex variable and j1⁄4 ffiffiffiffiffiffiffiffi 1 p . The subscript k is used as a time index and E1⁄2 represents the statistical expectation operator.