Linear and Widely Linear Recursive Maximum Complex-Domain Loglikelihood Adaptive Filtering in Intricate Measurement Environments

In industrial measurements, when signal reception is affected by hardware defects or asymmetric interference, I/Q imbalance can occur, which may result in the noise exhibiting noncircular and non-Gaussian characteristics, along with significant heterogeneous probability distributions in real and imaginary parts. In this setting, previous adaptive filtering cost functions (implicitly assumed homogeneous distributions) may perform well on one part but suffer on the other heterogeneous part, leading to an overall deteriorating performance, sometimes even inferior to the mean square error (mse) criterion. This article presents an innovative criterion specifically designed for such intricate measurement environments. Leveraging the inherent diversity of the Gaussian mixture model (GMM) to accommodate any probability distribution, we model the additive noises in real and imaginary parts. Through a linear combination, the overall noise is modeled as a complex-domain noncircular GMM (CNGMM). Then, a new cost function found on the recursive maximum complex-domain loglikelihood (RMCL) of the CNGMM is derived, with its linear and widely linear (WL) algorithms. Due to the excellent adaptability of CNGMM to intricate measurement environments, the proposed cost function consistently maintains outstanding performance under noncircular, non-Gaussian, and heterogeneous noises. An analysis of mean value and mean square convergence is also conducted. Further investigation reveals that the proposed steady-state mean square deviation (SS-MSD) is always less than or equal to that of complex recursive least squares (CRLS)/WL-recursive least squares (WL-RLS), which strongly indicates that RMCL/WL-RMCL is a superior scheme in intricate measurement environments. All theoretical predictions align well with simulations.