Maximum likelihood based identification for nonlinear multichannel communications systems☆
Introduction
Nonlinear behaviors can be encountered in many practical situations, in which case appropriate (nonlinear) processing is needed, when such nonlinearities are too important to be disregarded [1], [2]. Indeed, because most of real-life systems are inherently nonlinear in nature, nonlinear problems have drawn important interest and extensive attention from engineers, physicists, mathematicians and many other scientists [2]. In communications systems, and due to the presence of nonlinear devices such as optical equipments [3], [4] and power amplifiers, even in some MIMO-OFDM [5] and massive MIMO scenarios [6], [7] or millimeter wave based systems [8], communication channels are sometimes corrupted by nonlinear distortions such as nonlinear inter-symbol interference, nonlinear multiple access interference and nonlinear inter-carrier interference. These nonlinear distortions can significantly deteriorate the signal reception, leading to poor system performance. In order to overcome such an issue, nonlinear models are adopted to provide an accurate channel representation and to allow the development of efficient signal processing techniques capable of mitigating these nonlinear distortions. In the case of system identification, a widely used class of nonlinear models is the class of linear-in-the-parameters models. The input-output relation is essentially nonlinear but the estimation problem is linear with respect to the channel coefficients. Popular examples are polynomial filters, and more particularly Volterra filters [9]. They have been applied in many fields such as, electronic and electrical engineering, mechanical engineering, aeroelasticity problems and control engineering [10]. Indeed, the motivation for adopting these filters is that, they have the ability of modeling the behavior of nonlinear real-life phenomena, especially the ability to capture their “memory” effects [2]; and have mathematical relationship with other nonlinear system models namely the Wiener series, Hammerstein model, Wiener model, Wiener-Hammerstein model (block-oriented nonlinear systems), Taylor series or NARMAX model [10].
For nonlinear system identification, several approaches, most of them based on Volterra filters, have been proposed in the literature. Some works exploited training sequences and are essentially based on Least-Mean-Squares adaptive filters [11], Recursive Least-Squares algorithms [12], [13], and Affine Projection algorithms [14]. Other approaches are fully blind, thus, they seek to determine the system’s kernel using the output data only. One could cite the higher order output cumulant-based approach [15], the subspace-based approach [16], the genetic programming-based method using Volterra filter [17], the tensor-based frameworks in [18], or the Reversible Jump Markov Chain Monte Carlo approach [19]. One can notice that, these methods have been adopted and adapted to nonlinear systems mainly due to their efficiency for the linear case. Consequently, and due to the attractive advantages of the Maximum-Likelihood (ML) approaches (which is used in the current work), namely the consistency, and the asymptotic efficiency of the estimates, some works proposed ML-based identification techniques of certain nonlinear systems [20], [21]. In these works, an approximation of the complex likelihood function is minimized via modified Gauss-Newton methods assuming the input data to be white Gaussian and a block-structured system model. However, a review of the current literature reveals that a ML solution for the case of nonlinear, finite alphabet, multi-channel communications systems does not exist.
Other works, like those in [22], [23], have considered a Hammerstein model with cascaded nonlinear and linear blocks, where the initialization and the system identification (channel estimation) are performed by firstly estimating the impulse response of the linear filter, which is then used to estimate the nonlinear function parameters. By contrast, in the proposed work, both linear and nonlinear parameters are estimated simultaneously through solutions that fit into the framework of joint channel estimation and data detection.
Also, in [24] a blind nonlinear system identification is proposed based on the parallel factors (PARAFAC) tensor decomposition. However, it is shown that the input signals must satisfy some orthogonality constraints associated with the channel nonlinearities in order to allow the desired PARAFAC decomposition. Hence, a precoding scheme is introduced using temporal redundancy on the signals, which is carried out by imposing some constraints on the symbol transitions.
The aim of the current paper is to present ML-like blind and semi-blind channel estimators for Volterra-like nonlinear Single-Input Multiple-Output (SIMO) systems, that can be easily extended to MIMO scenarios. The proposed blind channel estimator combines a subspace-based estimation and an EM-based one. More precisely, firstly we exploit the received signal Second Order Statistics (SOS) using a subspace approach for channel estimation where the nonlinear SIMO system is treated as a linear Multiple-Input Multiple-Output (MIMO) system. A straightforward motivation is that, the use of SOS-based estimators avoids the need of high number of data symbols often required for High Order Statistics based methods, e.g. [15]. Then, unlike many blind-based works (e.g. [16]), we propose also an original method to remove the ambiguity inherent to such a blind approach. Finally, a second estimation is performed based on a maximum likelihood approach, where an iterative optimization is performed using the Expectation-Maximization (EM) algorithm. Indeed, due to its sensitivity to initialization, the EM-based estimator is initialized using the subspace-based one. Note that, efficient and practical initialization for blind EM-based techniques is often missing in the literature. Moreover, within the proposed EM-based framework, one could perform a joint channel estimation and data detection as will be highlighted later in the paper. The global scheme of the proposed blind EM-based estimator is given in Fig. 1, in which the signal at the received antennas is the input for the subspace-based estimator, whereas the estimates of linear and nonlinear channel coefficients represent the output of the EM algorithm after convergence.
To make our method more flexible, and to consider the case where training sequences (pilots) are available, this work is extended to the semi-blind framework where data and pilot symbols are jointly exploited to improve the estimation accuracy and overcome certain limitations of the blind processing. In this case, the initialization of the EM algorithm is performed by exploiting the available pilots. The proposed blind and semi-blind approaches are supported by some identifiability results and performance bounds related to our context, that allow the reader getting more insights on the problem’s identifiability and its inherent performance limits.
The rest of the paper is organized as follows. The adopted system model is presented in Section 2. Section 3 describes the proposed blind channel estimation scheme. It starts by presenting the derivation of the subspace-based estimation, used for the initialization of our ML solution. Since this method suffers from inherent matrix indeterminacy, it is followed by the introduction of an original ambiguity removal technique. Then, the ML-based method which exploits the EM algorithm is presented in details at the end of this section. Motivated by the widespread use of pilots in communications systems, we then derive, in Section 4, an extension to the aforementioned ML method in a semi-blind context. In this section, we first detail the proposed semi-blind EM algorithm, then we provide a discussion on extending it to the MIMO case. In Section 5, some identifiability results and performance bounds are given to corroborate the proposed solutions. Section 6 is dedicated to certain comments related to the computational complexity of our algorithms and a discussion on their potential extension to other nonlinear models. Section 7 provides comparative simulation result analysis of the proposed algorithms. Finally, the last section contains concluding remarks
In the sections below, the following notations have been adopted. Lower-case letters (e.g., ) denote scalars; lower-case boldface letters (e.g., ) denote (column) vectors, and upper-case boldface letters (e.g., ) denote matrices. Operators , and stand respectively for complex-conjugation, transposition, Hermitian transposition, matrix pseudo-inverse, and the trace of a matrix. Operator denotes a diagonal matrix with entries of as diagonal elements. stands for an identity matrix; is the all-zeros matrix of size ; denotes the matrix (column) vectorization operator; and denotes the convolution operator.
Section snippets
System model
This section details the data model adopted in this paper. A nonlinear SIMO system is considered as illustrated in Fig. 2. It is composed of one single-antenna transmitter and a receiver equipped with antennas. The -th received signal at the -th receive antenna, denoted with , is given by:where (resp. ) refers to the elements of the linear (resp. nonlinear), -th receiver, channel’s finite impulse
Blind EM-based estimation
This section details the proposed blind channel estimation approach. A subspace-based estimation, for the nonlinear SIMO system, is firstly considered. Then, solutions for ambiguity removal are proposed to get rid of the inherent ambiguity of the blind processing. After that, an EM-based channel estimation is detailed, which helps refining the subspace-based estimation already performed, as illustrated in Fig. 1. Besides, a data estimation scheme, within the EM framework, is provided.
Semi-blind EM-based estimation
In most communications systems, some training symbols (pilots) are usually sent periodically within the wireless network frames besides the unknown data. Hence, a Semi-Blind (SB) approach, exploiting pilots, can be adopted in order to take advantage of this available data and reduce the different difficulties and issues related to the blind processing. To do so, and without loss of generality, the transmitted sequences and the received observations are assumed to be composed of pilots and
Identifiability results and performance bounds
The aim of this section is to corroborate the proposed blind and semi-blind approaches by providing some results related to the channel identifiability conditions. Also, in order to get more insight on the proposed solutions performance limits, we perform the derivation of the deterministic Cramér-Rao Bound (CRB), corresponding to the adopted nonlinear system model.
Generalization and numerical complexity
This section is dedicated to the extension of the proposed methods to other nonlinear models. It provides also details about the computational complexity of the proposed algorithms.
Performance analysis and discussion
This section provides the performance analysis of the proposed blind and semi-blind channel estimators for the considered nonlinear systems. For benchmarking, we consider a ‘full’ training-based (fully-pilot) estimator, as done in many works (e.g., [19], [25]), where all transmitted symbols (pilots and data) are assumed known and used to estimate the channel parameters. The estimation performance is evaluated in terms of the Normalized -Mean-Square Error (NMSE) given by:
Conclusion
In this paper, blind and semi-blind Maximum Likelihood (ML) solutions are proposed for the identification of nonlinear multichannel communications systems. The ML criterion is maximized through the Expectation-Maximization (EM) algorithm. In the blind case, the EM algorithm is initialized by the subspace method followed by an original ambiguity removal technique. An identifiability study reveals that the success of the initialization step requires some stringent conditions that might not be
CRediT authorship contribution statement
Ouahbi Rekik: Writing – review & editing. Karim Abed-Meraim: Writing – review & editing. Mohamed Nait-Meziane: Writing – review & editing. Anissa Mokraoui: Writing – review & editing. Nguyen Linh Trung: Writing – review & editing.
Declaration of Competing Interest
The authors whose names are listed immediately below certify that they have NO affiliations with or involvement in anyorganization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus;membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the
References (42)
- et al.
Volterra-series-based nonlinear system modeling and its engineering applications: a state-of-the-art review
Mech Syst Signal Process
(2017) - et al.
Subspace-based identification algorithms for hammerstein and wiener models
European Journal of Control
(2005) - et al.
Tensor network alternating linear scheme for MIMO volterra system identification
Automatica
(2017) - et al.
Bayesian volterra system identification using reversible jump MCMC algorithm
Signal Processing
(2017) - et al.
Blind maximum likelihood identification of hammerstein systems
Automatica
(2008) - et al.
Volterra series based blind equalization for nonlinear distortions in short reach optical CAP system
Opt Commun
(2016) - et al.
A lookahead algorithm for the solution of block toeplitz systems
Linear Algebra Appl
(1997) Adaptive polynomial filters
IEEE Signal Process Mag
(1991)- et al.
Nonlinear system identification: a user-oriented road map
IEEE Control Syst. Mag.
(2019) - et al.
Statistical modeling and performance characterization of ultrashort light pulse communication system using power-cubic optical nonlinear preprocessor
IEEE Trans. Commun.
(2015)
A survey on fiber nonlinearity compensation for 400 gb/s and beyond optical communication systems
IEEE Communications Surveys & Tutorials
Analytical characterization of dual-band multi-user MIMO-OFDM system with nonlinear transmitter constraints
IEEE Trans. Commun.
Efficient nonlinear precoding for massive MIMO downlink systems with 1-bit DACs
IEEE Trans. Wireless Commun.
A digital predistortion scheme exploiting degrees-of-freedom for massive MIMO systems
IEEE International Conference on Communications (ICC)
Robust joint hybrid transceiver design for millimeter wave full-duplex MIMO relay systems
IEEE Trans. Wireless Commun.
Adaptive nonlinear system identification: The volterra and wiener model approaches
Second-order adaptive volterra system identification based on discrete nonlinear wiener model
IEE Proceedings-Vision, Image and Signal Processing
Low-complexity RLS algorithms using dichotomous coordinate descent iterations
IEEE Trans. Signal Process.
A low-complexity RLS-DCD algorithm for Volterra system identification
24th European Signal Processing Conference (EUSIPCO)
A multichannel hierarchical approach to adaptive volterra filters employing filtered-x affine projection algorithms
IEEE Trans. Signal Process.
Blind identification of volterra-hammerstein systems
IEEE Trans. Signal Process.
Cited by (8)
An efficient nonlinear adaptive filter algorithm based on the rectified linear unit
2024, Digital Signal Processing: A Review JournalSemi-Blind structured subspace method for signal estimation in nonlinear convoluted mixture
2023, Signal ProcessingAuxiliary model-based interval-varying maximum likelihood estimation for nonlinear systems with missing data
2024, International Journal of Robust and Nonlinear ControlMaximum likelihood based multi-innovation stochastic gradient identification algorithms for bilinear stochastic systems with ARMA noise
2023, International Journal of Adaptive Control and Signal ProcessingSemi-Blind Signal Estimation Using Toeplitz Structure-Based Subspace Method
2023, IEEE Wireless Communications LettersInteractive two-stage recursive least squares identification algorithms for controlled autoregressive systems
2023, Proceedings of the 35th Chinese Control and Decision Conference, CCDC 2023
- ☆
This work was supported by the National Foundation for Science and Technology Development of Vietnam under Grant No. 01/2019/TN.