Widely Linear Estimation Research Articles

Complementary Cost Functions for Complex and Quaternion Widely Linear Estimation

Widely linear (WL) models have been demonstrated to be superior to conventional strictly linear models for the estimation of noncircular complex and quaternion signals. Existing studies on their performance bounds focus on the analysis of mean square error (MSE). However, the single degree of freedom within standard MSE allows for only the minimization of error power, with no means to understand how the error contribution is distributed across the data channels. To this end, we introduce novel complex and quaternion valued complementary quadratic cost functions for complex and quaternion signal estimation, which are extensions of the recently proposed complementary MSE metric. It is shown that for WL minimum MSE estimation and least squares regression, the complementary cost function and the standard cost function attain the same stationary point. We also show that for the former, the stationary point is a saddle point. This novel finding provides insight into the performance of complex and quaternion WL estimators, and offers a rigorous foundation for further developments in this field.

A perspective on CLMS as a deficient length augmented CLMS: Dealing with second order noncircularity

Mean and mean square performance analyses of the strictly linear complex least mean square (CLMS) algorithm are addressed for widely linear estimation (WLE) of second order noncircular (improper) Gaussian inputs, for which both the covariance and pseudo-covariance matrices contain nonzero off-diagonal elements. A detailed performance analysis of standard CLMS in this ‘suboptimal’ context is not trivial but is important for practical applications. To this end, we here consider the strictly linear CLMS as a ‘deficient length’ version of the widely linear augmented CLMS (ACLMS) algorithm, which is second order optimal for improper inputs. Rigorous performance analysis is provided, which also statistically quantifies the suboptimality of CLMS in both the transient and steady state stages. The recently introduced approximate uncorrelating transform (AUT) is employed to derive closed-form expressions for the mean square stability and the steady-state performance of CLMS. In addition, since WLE with second order noncircular inputs is a general linear estimation problem in the complex domain C, these results provide a generalised framework from which current statistical descriptions of CLMS for strictly linear estimation (SLE) can be deduced as special cases. Simulations in system identification settings validate the findings.

Augmented Performance Bounds on Strictly Linear and Widely Linear Estimators With Complex Data

Novel physical insights are provided into the performances of strictly linear (SL) and widely linear (WL) estimators of the generality of complex-valued data, both proper (second order circular) and improper (second-order noncircular). This is achieved by first performing a novel complementary mean square error (CMSE) analysis, in order to quantify the degrees of improperness (second order noncircularity) of the SL and WL estimation errors. The exact bounds on the CMSE difference between the SL and WL estimators are investigated to show that only a joint consideration of the standard MSE analysis and the proposed CMSE analysis has enough degrees of freedom for a rigorous account of the performance of WL and SL estimators. This also makes it possible to rigourously quantify the contributions to the WL performance advantage from the individual real and imaginary channels, an important finding not possible to obtain by using the standard MSE analysis only.

Widely linear channel estimation for LD‐STBC MB‐OFDM UWB systems

SummaryThe combination of Linear Dispersion Space‐Time Block Codes (LD‐STBCs) and Multiband Orthogonal Frequency Division Multiplexing Ultra‐WideBand (MB‐OFDM UWB) technologies has been proposed in the literature to improve the error performance and data rate. Recently, it has been shown that the LD codes allow the repetitions of data symbols conjugates. They cause a transformation of the Multiple‐Input Multiple‐Output (MIMO) channel that requires a Widely Linear (WL) filtering. However, proposed LD‐STBC MB‐OFDM UWB system does not take this issue into account. In this paper, we propose a Widely Linear Minimum Mean Square Error (WLMMSE) channel estimator for LD‐STBC MB‐OFDM UWB systems. The proposed channel estimator proceeds to WL estimation instead of linear one. Via simulation results, we prove that the LD‐STBCs provide significant performance gain for MB‐OFDM UWB system compared to Single‐Input Single‐Output (SISO) configuration. We show also that the proposed WLMMSE channel estimator outperforms the LMMSE one. This performance improvement is observed for both proper and improper signal constellations. Additionally, the computational complexity of our proposal is comparable with that of LMMSE estimator. Copyright © 2016 John Wiley & Sons, Ltd.

Performance Bounds of Quaternion Estimators.

The quaternion widely linear (WL) estimator has been recently introduced for optimal second-order modeling of the generality of quaternion data, both second-order circular (proper) and second-order noncircular (improper). Experimental evidence exists of its performance advantage over the conventional strictly linear (SL) as well as the semi-WL (SWL) estimators for improper data. However, rigorous theoretical and practical performance bounds are still missing in the literature, yet this is crucial for the development of quaternion valued learning systems for 3-D and 4-D data. To this end, based on the orthogonality principle, we introduce a rigorous closed-form solution to quantify the degree of performance benefits, in terms of the mean square error, obtained when using the WL models. The cases when the optimal WL estimation can simplify into the SWL or the SL estimation are also discussed.