Nonlinear Signal Estimation Using Semi-Blind Mutually Referenced Equalizers in Convolutive Mixture

Nonlinearity is a common source of signal and channel distortion in digital communication. In many circumstances, such distortions must be compensated by equalization algorithms or devices. This paper presents a semi-blind equalization method for nonlinear multichannel signal recovery in convolutive mixtures. The proposed work takes into account quadratic nonlinearities and applies the mutually referenced equalization (MRE) approach to estimate the equalizer as well as estimate the transmitted signal from the nonlinear convolutive mixture. The proposed semi-blind model combines both data and pilots in addition to the MRE technique to develop a cost function that provides an equalization solution. The proposed technique has various advantages, including robustness to channel order misspecification, implementational simplicity, and provides a single solution for several equalization delays. The simulation results show that the proposed technique has intriguing performance features with robustness to low SNR.


I. INTRODUCTION
In signal processing and communications, linear models have been extensively employed due to their versatility and simplicity. However, there are many practical applications involving nonlinearities in which a linear model cannot attain the optimum desired performance [1], [2]. Hence, a nonlinear model is essential to represent the behaviors and properties of such a system in order to effectively mitigate the nonlinear distortion. In system identification and equalization, an important class of nonlinear models is linear-in-parameter nonlinear models. These models are nonlinear in regards to the input-output relationship, however, the estimation problem is treated linearly. Some famous examples of this class are the Volterra filters [3] and polynomial filters [4].
The associate editor coordinating the review of this manuscript and approving it for publication was Easter Selvan Suviseshamuthu .
Similar to the Taylor series, the Volterra model has the additional advantage of capturing the memory effect. The authors in [5] present an adaptive least mean square using truncated Volterra series. In [6], the authors proposed a robust and stable sparse Volterra equalizer that addresses both linear and nonlinear distortions in optical interconnect while also attempting to reduce the Volterra equalizer's computational complexity. This equalizer uses l 0 regularization with a recursive least squares algorithm. In [7], the authors proposed a casecade neural network and maximum likelihood sequence estimation equalization scheme to mitigate nonlinearities and solve the over-equalization problem in a quasi-hard output sequence.
The authors in [8] investigate the performance of three different pre-equalization schemes, namely the Volterra series, memory polynomials, and artificial neural networks, in terms of mitigating linear and nonlinear impairments under various implementational conditions, such as received optical power, number of equalizer taps, and semiconductor optical amplifier current.
A variety of nonlinear system identification techniques have been developed; see, for example, [9], [10], and [11]. Some of these include training-sequence-based methods, which are often based on recursive least squares, least mean squares, and affine projection algorithms [12]. Other methods are fully blind, which means they only use observed (output) data to predict the input signal. Many of these techniques, including the standard subspace (SS) techniques [13], the structured subspace methods [14], [15], [16], the mutually referenced equalizers (MRE) techniques [17], the truncated transfer matrix (TTM) techniques [18], the minimum mean square error (MMSE) techniques [19], the maximum likelihood (ML) like techniques [20], have been employed for linear system identification.
Most semi-blind nonlinear techniques are extensions of blind linear methods, which integrate a few pilots with the cost function of the blind method to enhance performance and remove the scalar ambiguity present in the blind approach. The authors of [21], for instance, presented a semi-blind method for nonlinear multichannel systems based on ML processing that makes use of the Expectation Maximization (EM) methodology. The technique performs well with a rapid convergence speed but at the expense of a high computational, and also performs moderately at a low signal-to-noise ratio (SNR) due to initialization.
The proposed mutually referenced equalizer (MRE) method in [17] offers several benefits over the previously described blind approaches, including direct estimation of the signal equalizer, robustness to low SNR, and high estimation accuracy. Therefore, due to the appealing properties of the MRE approach, this work extends the method and introduces a semi-blind MRE technique for nonlinear multichannel systems. More precisely, the equalizer filter references and a few pilot symbols are jointly exploited to develop efficient and robust semi-blind criteria for the direct estimation of an equalizer for the input signal in a nonlinear SIMO system. An examination of the present literature indicates that there is no semi-blind MRE solution for nonlinear multichannel communication systems.
The goal of this study is to provide a semi-blind signal estimate method using MRE for nonlinear single input multiple output systems. Without channel pre-identification, the proposed semi-blind nonlinear SIMO MRE technique estimates the signal equalizer using a multidimensional mean square error cost function. In the setting of channel diversity, the nonlinear SIMO system is considered as a linear MIMO system. The semi-blind technique, unlike the entirely blind method, does not have the issue of scalar ambiguity, and it also does not require accurate knowledge of the channel order, which is impossible to know in practice.
The remainder of the manuscript is organized as follows. Section II presents the system model. Section III describes the blind MRE approach. Section IV describes the proposed semi-blind MRE method for nonlinear signals. Section V reports the various simulation results by the proposed approach. Section VI provides concluding remarks.
Notations: A scalar is written in lowercase x, a vector is written in bold lowercase x, and a matrix is written in bold upper case X. X T , X H , and X −1 denote respectively the transpose, conjugate transpose and the inverse of the matrix X. ∥ . ∥ 2 F and ∥ . ∥ 2 represent the Frobenius norm and Euclidean norm, respectively, vec() is a vectorization operator. I b and 0 a,b represent respectively a b × b identity matrix and an a×b matrix whose elements are all zeros, x(i, j) is the (i, j) of matrix X.

II. SYSTEM MODEL
The system model used in this work is described in this section. Consider a nonlinear SIMO system with a single transmitting antenna and multiple receiving antennas as shown in Fig. (1). The expression for the n th received signal at the i th antenna is expressed as where the elements of the linear and nonlinear i th received channel coefficient are represented by h i,L (k) and h i,NL (k) respectively, x(n) represents the transmitted input symbol that is assumed to be independent and identically distributed (i.i.d.) complex random variable.x(n) represents the nonlinear combination of the transmitted signal such thatx(n) = F(x(n), x(n − 1), . . .), F represents the nonlinear function that models the nonlinearity present in the system. v i (n) is the additive white Gaussian noise with variance σ 2 v . In particular, a quadratic nonlinearity is considered in this work that is, x(n) = x(n) 2 .This quadratic nonlinearity has been used extensively to model many real-world signals, notably in power amplifiers and optical devices [4], [21], [22]. Now, assuming an N r number of receiving antennas is considered, then the model in (1) can be written in a vector form as 56664 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. where

represents the additive white
Gaussian noise vector of similar dimension (N r × 1) with a covariance of σ 2 v I which is assumed to be independent of the transmitted input signal. The SIMO model proposed in this work can be thought of as a two-input linear MIMO system. Hence, the equivalent MIMO model is presented as; represents the k th channel matrix tap defined as and L = max{L L , L NL }. The channel length is assumed to be L + 1 implying that H(0) ̸ = 0 and H(L) ̸ = 0, respectively, although the channel is zero beyond the stated range. If N w samples are sequentially stacked into a single vector (N w being the selected processing window size), an N w N r dimensional vector is obtained and represented as follows: or in the following form: where T is the input signal vector of 2K × 1 dimension and H N w represents the block Sylvester channel matrix of dimension N w N r × 2K , and is given by: Finally, the data matrix may be constructed as follows: given as follows:

III. BLIND MRE METHOD
Before introducing the semi-blind MRE method for nonlinear systems. It is critical to present the blind MRE technique in [17]. Consider a multichannel equalizer vector w i and w i+1 , where i represents the equalizer delay. Hence, the following equation follows: where the i th -delay and (i+1) th -delay equalizer outputs can be considered to be ''referenced'' to each other up to a symbol duration delay. Similar relationships can be established for other equalizers and delays. In general, we have where k > i. Let w 0 , w 1 , . . . , w 2K −1 be respectively N r N w × 1 complex value vectors such that the relationship w H i y N w (n) = w H i+1 y N w (n + 1) for all n and i = 0, . . . , 2K −2 are satisfied. If we stack the vectors into a single matrix W = [w 0 , w 1 , . . . , w 2K −1 ] of dimension N r N w ×2K then the following points are valid: . . 2K − 1 Hence, the condition for estimating the equalizer in a noiseless case can be written as follows: where In the presence of noise, it is impossible to achieve perfect symbol recovery. Hence, a quadratic cost function is devised as follows: where Kronecker product property can be applied to (13) which leads to the following:

IV. SEMI-BLIND MRE METHOD
The goal of developing a semi-blind equalizer is to leverage both the information provided by blind techniques and the one provided by a known set of pilots. In addition to outperforming the blind and training sequence approaches, the semi-blind method provides a robust estimation approach. The training sequence approach has low bandwidth efficiency because a considerable portion of the bandwidth is consumed for training. Furthermore, training is impractical in some situations, particularly where explicit synchronization between receiver and transmitter is not achievable. Blind approaches, on the other hand, have a number of issues, including constant phase ambiguity, and channel order overestimation problems. The semi-blind offers remedies to all of these issues. Hence, the motivation for developing the semi-blind MRE technique.
In comparison to other approaches like the subspace method, the MRE method offers a wide range of benefits including global convergence and implementational flexibility. This approach directly provides all of the possible equalizers, thereby reducing noise amplification issues that are likely experienced due to channel inversion. Hence, as a result of the interesting benefits of the blind MRE method, a semi-blind MRE (SB MRE) method is proposed. The SB MRE method estimates the equalizers based on the full received pilots (Y p ) and data sequence ( Without loss of generality, to achieve this aim, the blind cost function is modified to accommodate the pilots in order to form a global cost function for a semi-blind scheme. where J 3 = I 2K an identity matrix of 2K × 2K dimension, Y p is the matrix of the received pilots of dimension N r N w × p which correspond to the first p columns of Y and Y n(d) = [y N w (p + 1) y N w (p + 2), · · · , y N w (N − 2)], and Y n+1(d) = [y N w (p + 2) y N w (p + 3), · · · , y N w (N − 1)]. X K (p) represents the first p columns of the matrix X K which corresponds to the transmitted pilots. In a similar way, the Kronecker product can be used to vectorize the cost function in (15) as follows: where Q 1 and Q 2 are defined, respectively, as and and x p = vec(X K (p) ). The optimization problem may be thought of as a least squares problem, 1 and the optimum 1 The detailed proof is illustrated in Appendix A.
solutionw is achieved as follows: Finally, the estimated vectorw in (19) is reshaped into the N r N w × 2K matrixŴ , and the transmitted signal is estimated by choosing a column i vectorŵ i fromŴ , i.e the equalizer of delay i and multiplying it with the received signal matrix Y . Hence, the estimated signal is given asx = w H i Y . The summary of the proposed algorithm is detailed in Algorithm 1. Note that each column ofŴ can provide an admissible equalization result with a delay corresponding to its index. However, for optimum performance, the columnŵ i is chosen as the one with index equal to the channel length.

Algorithm 1 Summary of the Proposed Semi-Blind Method
Form the matrix Y and X K (p) Form the selection matrices J 1 , J 2 , and J 3 Calculate Q 1 by (17) Calculate Q 2 by (18) Calculatew by (19) Reshapew into a N r N w × 2K matrixŴ , Choose a columnŵ i from the matrixŴ Estimate the transmitted signalx =ŵ H i Y

V. SIMULATION
In this section, extensive simulations are performed and the results are presented. The performance metric adopted in this work is the symbol error rate (SER), which is the ratio of the number of incorrectly estimated symbols to the total number of transmitted symbols. The channel taps under consideration are generated as zero-mean, unit variance Gaussian random variables. Unless otherwise specified, the following simulation parameters will be used throughout: the window size is N w = 5, the equalizer delay is τ = 3, the channel order is L = 3, data size N = 1000, the number of pilots p = 30, the number of receiving antennas is N r = 4, and a Monte Carlo run of 1000. The first simulation investigates the proposed method's required number of pilots by varying the number of pilots against the SER for different SNR levels as illustrated in Fig. 2. One can notice that as the number of pilots increases the SER performance of the proposed SB MRE method improves significantly. As a result, increasing the number of pilots translates to an increase in the performance of the proposed approach.
In the second experiment, the effect of increasing the equalizer delay is investigated. As shown in Fig. 3, an increase in the equalizer delay corresponds to a significant increase in the SER performance which reaches an optimum value when the delay is τ = 4, after which the performance gradually decreases and performs worst when the equalizer delay is 8. This signifies that optimal performance is attained when the equalizer delay is equal to the channel length. It is important  to note that the estimates equalizer matrix in this case has 16 columns. However, the ones needed for estimating the desired signal from the nonlinear convolutive mixture are 8 in numbers.
Next, the effect of varying the data size is investigated using fixed pilots of size p = 30. As can be deduced from Fig. 4, an increase in SNR significantly improves the SER performance of the data size. However, for a given SNR level, the performance of the proposed SB MRE method slightly decreases with an increase in data size.
The effect of channel length is investigated in the following experiment at various SNR levels. Figure 5 depicts the relationship between channel length and SER performance. The plot shows that the SER performance decreases slightly with increasing channel length for all SNR levels considered. This implies that the proposed SB MRE performs better with shorter channel lengths. Figure 6 shows the effect of varying the window size N w against the SER performance at different SNR levels. In general, One can notice that increasing the SNR level from 0dB to 6dB significantly improves the performance of the   proposed method. However, an increase in window size for a particular SNR level result in a slight decrease in the SER performance.
A pilot's cost is often expressed as the percentage of the time slots it occupies during a particular transmission VOLUME 11, 2023 56667 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   frame. The algorithm's performance in relation to this pilot cost percentage is shown in Figure 7. This simulation demonstrates how, as compared to the pilot-based method, a semi-blind method can decrease the pilot size required and the amount of SNR needed while maintaining the target estimation quality. For example, the pilot-based method requires 10 dB SNR to achieve an SER performance of 7 × 10 −2 , whereas the proposed SB MRE method only requires 6dB to achieve the same SER performance. Furthermore, for a 10 dB SNR, the proposed SB MRE requires 2% of pilots to achieve 2 × 10 −2 SER performance, whereas the pilot-based method requires 7% of pilots to achieve similar performance. Figure 8 depicts the consequences of channel order misspecification. This incorrect specification reduces performance, as shown in the figure, but the proposed approach still maintains a reasonable estimation quality in such an event. As a result, the relative robustness of the proposed method against channel order misspecification is demonstrated. Furthermore, as the channel order mis- specification goes from 1 to 3, there is no substantial influence on the algorithm's performance. Hence, this demonstrates the proposed method's resistance to order misspecification.
In the final experiment, the proposed SB MRE method is compared to the nonlinear version of the semi-blind MMSE (SB MMSE) method, semi-blind SSS (SB SSS) method, and the pilot-based approach as illustrated in Fig. 9. The proposed SB MRE method outperformed the other method. Here, the proposed SB-MRE used an equalizer delay of τ = 4. It is important to note that the pilot-based approach is a least square estimation where some pilots are used to estimate the equalizer taps. The estimated equalizer taps are then used to filter out the interference and detect the signal.

VI. CONCLUSION
In conclusion, we proposed a semi-blind MRE method that takes advantage of some pilot signals. For different SNR levels, the proposed method is thoroughly investigated in various aspects such as equalizer delay, pilot size, channel length, window size, data size, and percentage of transmitted pilots. By increasing the number of pilots available, the proposed algorithm's performance improves significantly. However, increasing the channel length, processing window size, and data size results in a slight decrease in performance. When compared to the fully pilot-based method, a significant gain in SNR and percentage of pilots transmitted is achieved. In the future, a good trajectory is to explore the advantages of the proposed method in massive MIMO and reflective intelligent surface estimation problems. The possibility of exploring the time scales approach [23] is another direction.