A Unified Affine-Projection-Like Adaptive Algorithm for System Identification

A unified affine-projection-like adaptive (UAPLA) algorithm is devised and verified for system identification. The UAPLA algorithm uses a generalized cost function encompassing some data-reusing methods to cope with colored input signals. Furthermore, the UAPLA algorithm is derived based on the new cost function to approximate several affine-projection algorithms. As a result, the proposed UAPLA algorithm approximates some classical adaptive filters that can be considered as special cases of the UAPLA, allowing flexibility to achieve low estimation errors under impulsive noises. The obtained results conducted by simulations help to corroborate the superiority of the proposed UAPLA algorithm over other popular AP algorithms.


I. INTRODUCTION
I N COMPLEX noise environments [1], [2], the adaptive filters (AFs) commonly face Gaussian noises (GNs) and impulsive noises (INs) [3], which can be found in wireless communications, underwater acoustics, channel estimations and echo cancellations [4] applications.In particular, with high-speed vehicles crossing the mountains, cities, and deep seas, IN is ubiquitous, and its amplitudes might be higher than GNs [5].These intense noises can affect the quality of in-vehicle communication and system identification applications [6].Therefore, many AF algorithms were proposed and presented in detail, achieving good performance in GN and IN environments using distinct error criteria for constructing new cost functions (CFs) [7].However, these algorithms were independently devised with various CFs according to the applications.
Several error criteria addressed for IN mitigation in recent years have been well investigated, including maximum correntropy criterion (MCC) [8], minimum error entropy (MEE) [9], Cauchy [10], Versoria [11], and Lorentzian functions [12].Then, the MCC and MEE algorithms were developed and discussed within the AF framework, which can resist INs.Additionally, most AF algorithms have difficulty in dealing with colored input signals like the autoregressive (AR) model [13] and speech.As a result, the affine-projection (AP) or data-reusing strategy has gained importance by benefiting the use of second-order statistical error [14].As a result, the constructed AP-like AF algorithms can reduce the effects of colored inputs under GNs.However, they are still inefficient in combating INs, and their performance will be degraded when heavy-tailed noise contaminates the systems.Then, the proposed AP algorithm employs the sign scheme to improve its behavior when the system is corrupted by INs, where l 1norm is employed to mimic the posterior error vector to get the AP sign (APS) algorithm [15].As a result, the algorithm reduces the computations since the APS has no matrix inversions and converges faster than traditional AP methods.After that, several variant AP algorithms have been reported to get robust robustness and better performance, including robust APS (RAPS) and its shrinkage version (SRAPS) [16].Herein, l p -norm (p = 1, 2) was employed to improve the convergence rate and reduce estimation error.Thus, SRAPS and RAPS provide faster convergence and smaller misalignment than the APS.Still, they include matrix inversion operations resulting in a heavy computational burden.Then, the AP scheme is expanded to Versoria [11], Lorentzian [12], Ekblom norm [17], and MCC to combat not only the INs but also reduce the effects of colored inputs.After that, the Ekblom promoting adaptive algorithm (EPAA) [18] and AP Versoria (APV) [11] are obtained by considering the data-reusing scheme.However, each algorithm should be independently evaluated by taking different error criteria into consideration, which might reduce its versatility.This brief proposes a more general AF algorithm considering a simple error function.Here, a general error function is constructed to create a CF.Then the CF is driven using the gradient descent search method to get the proposed unified AP-like Adaptive (UAPLA) algorithm.By selecting the parameter ω, the proposed UAPLA can be used approximate to the popular AP-like algorithms without redesigning AF algorithms.Next, the characteristics of the UAPLA are analyzed and discussed.Finally, the proposed UAPLA is used for channel estimation and echo cancellation.The simulation results show that the proposed UAPLA outperforms recent popular AP-like algorithms in terms of the convergence rate and misalignment.Table I lists the mathematical operators used in the work.
II. PROPOSED UAPLA ALGORITHM Here, a system input signal x(l) and an unknown system g with a size of K × 1 are considered for channel estimation and 1549-7747 c 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
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TABLE I MATHEMATICAL OPERATOR NOTATIONS
echo cancellation.Then, the desired signal at the output is where n(l) is system noise.For AF applications, the system instantaneous estimated error is obtained by e(l) = d(l) − y(l), where y(l) is estimated output.It is easy to say that e(l) is to discuss the approximation between y(l) and d(l).Considering the data-reusing scheme, ] with recent Q inputs is input into the system, and then, the desired signal D(l), and error signal e(l) are obtained, respectively, giving by and where g(l) is estimation of g.To effectively use the advantages of the data-reusing and provide a more general algorithm to implement AF, a new CF is created by a quadratic bowl loss, which is presented as where ω is a shaping factor to provide robustness, and δ is a scaling factor.It is noticed that when the shaping factor ω tends to 2, the proposed CF approaches to second-order statistical error function, giving by which can be used for developing AP and least-mean square (LMS) algorithms.If ω equals 1, the proposed CF is an l 1norm like loss function, presenting as which can be used to approximate the APS algorithm.It is understood that ω of equation ( 5) cannot take the value of 0 since it is a denominator.Thus, we can take the limit of J(e(l), ω, δ) with respect to ω → 0 to get which can be used to get an approximation of Cauchy or Lorentzian functions.Thus, the proposed CF can also be used for getting an approximated performance of the AP Lorentzian (APL) algorithm.Moreover, when ω is equal to -2 or ω approaches negative infinity, the proposed CF changes to be the Geman-McClure function or Welsch function, respectively, given by and From the aforementioned analysis and discussion, we notice that the proposed CF can be used to construct a more general AF algorithm by selecting ω.Based on the AF theory, the partial derivative of ( 5) with respect to g(l) is where where k has the values of k = 1, 2, . . ., Q.By employing the gradient descent method, the weightupdating equation of the UAPLA algorithm is obtained where μ is a step size of the UAPLA algorithm, which is to trade-off the convergence rate and estimation error.Substituting equation (11) into equation ( 13), yields Here, e(l) is weighted by H(l) in ( 14) to achieve a constraint to improve the robustness of the UAPLA algorithm against impulse noise.

III. CONVERGENCE ANALYSIS OF THE UAPLA
According to the proposed UAPLA algorithm, a stability analysis of the weight update equation given in ( 14) is performed to obtain the bound of μ.
We define g(l) = g − g(l), and then, we get Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
= g(l) − μX(l)H(l) X T (l)g + n(l) − X T (l) g(l) = I − μX(l)H(l)X T (l) g(l) − μX(l)H(l)n(l), (15) where n(l) = [n(l), n(l − 1), . . ., n(l − Q + 1)] T is independent of the other signals and has an independent, identical distribution with zero mean.I is an identity matrix.Then, taking the expectation values on both sides of (15) and letting g(l) is independent of any other signals, we have Therefore, to ensure the convergence of the UAPLA algorithm, equation ( 16) should satisfy where . Also assuming that H(l) and X(l)X T (l) are statistically independent, we can obtain where R is covariance matrix of x(l) defined as R = E[x(l)x T (l)].Therefore, we take the derivative of ( 12) by e(l − k + 1), and the result is Letting equation ( 19) be equal to zero, we can conclude that e(l − k + 1) = 0 is the maximum point when ω < 2.Then, we have and According to equations ( 17) and ( 21), the sufficient condition for convergence of the UAPLA is From the discussions of the proposed UAPLA algorithm above, two remarks are concluded as follows.
Remark 1: The UAPLA algorithm can approximate to be AP, LMS, APS and EPAA algorithms by choosing different ω.
Remark 2: The convergence of the UAPLA has been presented and analyzed in detail.

IV. COMPLEXITY ANALYSIS
This section describes the implementation complexity of the UAPLA algorithm, given in Table II, where we consider the total number of multiplications, divisions, and additions required for each iteration.Steps 1 and 4 contribute to the main complexity of the UAPLA algorithm.Here, K represents the length of the filter, and Q is the order of the projection of the UAPLA algorithm.Table III compares the EPAA, AP, APS, APV, and UAPLA for each iteration's additions, multiplications, and divisions, where the matrix inversion lemma [19] is used for the AP algorithm to convert the matrix inversion operations into multiplication and addition operations.

V. SIMULATION RESULTS AND NUMERICAL DISCUSSIONS
In this section, the performance of the proposed UAPLA algorithm is tested, investigated, and discussed through simulations for echo cancellation and system identification, where the EPAA, AP, APS, APV, APL, the Data-Reusing MCC (DRMCC) [20] and Random Data-Reusing GMCC (RDRGMCC) [21] are considered for the sake of performance comparison with the UAPLA algorithm under different ω. α and σ 2 β , respectively.All simulations for this experiment are obtained by averaging 100 trails, where the regularization parameters and weight vectors are initialized to zero for the above mentioned algorithms.
The unknown weight vector g has a length of K = 128 for system identification in the experiment.The normalized-mean-square-deviation (NMSD) defined as NMSD(l) = 10log 10 [ g(l) − g 2 / g 2 ] is adopted to evaluate the performance of the these mentioned algorithms.
In simulation 1, the convergence of the UAPLA is investigated and analyzed for different values of ω.The input sequence is a Gaussian signal, and the system being identified is contaminated by IN with P = 0.001 and P = 0.1, where the signal-to-noise ratio (SNR) is 30 dB.The signal-to-interference ratio (SIR) is -30 dB.The SNR is given by 10log 10 {E[y 2 (l)]/σ 2 α } and the SIR is given by 10log 10 {E[y 2 (l)]/σ 2 β }, and Q = 4. From the discussions above, we conclude that convergences for the newly created UAPLA Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.with different ω are given in Fig. 1, where ω is chosen as 0, 1, and -2.The UAPLA achieves a similar steady-state NMSD by choosing suitable parameters μ and δ to analyze the convergence rate.The UAPLA algorithm performs the best for the Gaussian input signal when ω = −2 for different P.
In simulation 2, the background noise is the same as that in simulation 1, and the input sequence is a first-order AR random process x(l) = 0.9x(l − 1) + z(l), where z(l) is zeromean Gaussian signal with unit variance.The step size μ is adjusted to let the UAPLA algorithm achieve approximately the same NMSD.Fig. 2 illustrates that the UAPLA algorithm performs very well for ω = −2 under different P.
In simulation 3, the input sequence is a random signal sequence, and the convergence rates and tracking performance of mentioned algorithms are compared with different P and ω in Fig. 3. Parameter μ and related parameters are selected to enable all mentioned algorithms to get the same convergence at their initial.The system noise are the same as in Simulation 1. From Fig. 3, the AP algorithm has the worst performance for dealing with large IN interference, and the proposed UAPLA algorithm can be used for approximating to the EPAA, APS and APV.Moreover, the UAPLA algorithm can achieve a small or even smaller NMSD when ω = −2.
In simulation 4, the AR process is used as the input sequence to discuss the behaviors of the different algorithms.The background noise and the other parameters are the same as in simulation 2 with P = 0.001 and P = 0.1.Fig. 4 shows that all the mentioned algorithms run well except for the AP algorithm.In addition, Fig. 4(a) shows that the UAPLA algorithm converges faster than the other algorithms for the same NMSD.
In Simulation 5, a speech signal obtained from ITU-T P.501 [22] with 8000 Hz sampling is considered as the system's input to analyze the UAPLA algorithm's performance, as shown in Fig. 5.The background noise is the same as simulation 1.It is found from Fig. 6 that the constructed UAPLA algorithm has a faster convergence and smaller steady-state misalignment (SSM) compared to other   algorithms under the effects of impulsive noise with P = 0.001 and P = 0.1.

VI. CONCLUSION
A unified affine-projection-like adaptive (UAPLA) algorithm has been proposed, analyzed, and tested for channel estimation and echo cancellation applications.The UAPLA algorithm is derived by a generalized cost function within the data-reusing scheme, which can be approximated to LMS, AP and APS by choosing the shaping factor ω. The parameter effects and the convergence analysis are presented to illustrate the performance of the UAPLA for channel estimation and echo cancellation.The results show that the UAPLA outperforms recent popular AP-like algorithms for applications in system identifications, reducing the repeated designs on AP-like algorithms.
Manuscript received 2 July 2023; revised 21 July 2023; accepted 31 August 2023.Date of publication 5 September 2023; date of current version 7 February 2024.This work was supported by the Natural Science Foundation of Anhui Province under Grant 2208085QF180.This brief was recommended by Associate Editor L. Chai.(Corresponding author: Yongchun Miao.)Yingsong Li, Yonglin Fu, Yongchun Miao, and Zhixiang Huang are with the Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University, Hefei 230601, Anhui, China (e-mail: liyingsong@ieee.org; sycmiao@gmail.com).Paulo S. R. Diniz is with the Program of Electrical Engineering, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-972, Brazil (e-mail: diniz@smt.ufrj.br).Color versions of one or more figures in this article are available at https://doi.org/10.1109/TCSII.2023.3312149.Digital Object Identifier 10.1109/TCSII.2023.3312149