# Constraint Sparse Maximum Correntropy Beamforming Algorithm against Impulsive Noise to Improve the Performance of GNSS Signal Acquisition

Authors: Haichuan Zhang; Fangling Zeng
Journal: International Journal of Antennas and Propagation
 (2019)

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## Abstract

This paper presents the new constraint correntropy beamforming method to improve the performance of global navigation satellite system (GNSS) receivers. Although beamforming-based GNSS system possesses the ability of nulling interference sources, the distortions to satellite signal induced by impulsive noises are always neglected. This paper addresses the satellite navigation signal acquisition problem against impulsive noises using the constraint maximum correntropy criterion in framework of the GNSS system. In addition, in order to decrease the number of active elements for avoiding overmuch energy consumption, we introduce the L1-norm penalty equation to the list of constraints of the adaptive filter that forces the coefficients with small magnitudes to zero. From the above, we propose the norm-constraint beamforming method to satisfy the conflicting requirements between the coefficients sparsity and the performance of the satellite signal acquisition. The proposed CSMC maintains the robustness against impulsive outliers and achieve better performance in conjunction with less power consumption. An analysis of the mean square convergence properties of the proposed algorithms is presented, and the stability condition of the convergence is derived. Our simulation results demonstrate the superiority of the proposed methods.


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## Body

## 1. Introduction

Adaptive beamforming, as one of the most important techniques of array signal processing has been widely used in GNSS system as well as other applications because of its efficient ability to suppress interference and preserve desired signal [1, 2]. The GNSS signal is collected by receiver in open sky conditions, and the power of satellite signal on the ground is very weak [3]; thus, the GNSS receivers are susceptible to intentional or unintentional jamming. Moreover, the navigation performance of the receiver mostly depends on the process of satellite signal acquisition [4], and the theoretically predicted performance of GNSS system in practical application may not be achieved due to various factors including impulsive noises [5, 6]. Consequently, adaptive beamforming criterions which favor GNSS system and could also mitigate the effects of these imperfections are motivated into research including time-domain, transform-domain, and space-domain algorithms [7–9], while time-domain and transform-domain techniques can hardly be used to inhibit broadband interferences, and the spatial interference suppression based on the adaptive beamforming has been considered to be a promising solution to this problem. The algorithms proposed in [10, 11] have opened a way for the beamformer processing to play an important role in GNSS receivers; the effectiveness of these beamforming methods for suppressing interferences and enhancing satellite signals has been studied and verified in the papers. In the other open literature, Zhou designed a jointly adaptive beamforming method based on maximum signal-to-noise ratio (MSNR) criteria to suppress the jammers [12]; this technique is unable to track the interference rapidly for its relatively high computation complexity. Classical LMS (least mean square) family algorithms such as Normalized Least Mean Square (NLMS) [13, 14], Constrained LMS (CLMS) [15, 16], and modified robust variable step size (MRVSS) [17] are the popular beamforming methods since they can achieve fast convergence for real-time communication without causing too much performance degradation. However, the criterion of these algorithms utilized exhibits performance degradation in the presence of alpha noise, particularly on the process of satellite signal acquisition. Therefore, it is necessary to develop a robust beamforming method against impulsive noises with application to GNSS receiver.Maximum correntropy criterion (MCC) has gained a wide attention for its simplicity and insensitivity to outliers, MCC-based methods have been widely applied to principal component analysis [18], system identification [19], and other related fields in recent years [20]. However, this technique rarely has been applied to the beamforming and GNSS field. Besides, most beamforming algorithms suppress the interference effectively with high power consumption; as a result, the applicability of these techniques is restricted by the scenarios with limited power supply systems. With considerations on these issues, the main contributions of this paper are summarized as follows: we propose the novel adaptive beamforming algorithms in the presence of impulsive noise, referred to as constrained sparse maximum correntropy. The novel algorithm takes into account the possibility for provoking sparsity of L1-norm and adds a particular norm constraint into the optimization function. The results of this technique implementation can be regarded as producing the sparse solution of adaptive beamformer and achieving less waste of power at the GNSS receiver. Besides, we exploit the prior knowledge that MCC is robust against impulsive process of external perturbation; the optimization problem in mean square error (MSE) sense is transferred into the maximization of MCC function to attenuate the adverse effects of impulsive noises. Thus, the proposed CSMC beamforming technique can achieve large degrees of sparsity of the beamformer array at GNSS receiver while maintaining the robustness of MCC against impulsive noises.The structure of the paper is organized as follows: satellite navigation signal reception model of GNSS receiver is presented in Section2. Section 3 presents the proposed robust CSMC beamforming method and provides the mean square analysis of the algorithm. Section 4 illustrates and analyzes the simulation results. Section 5 summarizes the conclusions.


## 2. Satellite Navigation Signal Reception Model

Consider a uniform linear antenna array ofM omnidirectional array elements employed in GNSS receiver, one satellite signal, and Q uncorrelated interference which impinge on the array at the kth snapshot as expressed by [21](1)xk=skaθd+∑i=1Qiikaθi+nkwhere sk and iik are the desired GNSS signal and the ith interference and nk is the noise vector. aθd and aθi represent M×1 steering vectors of sk and iik as given by(2)aθd=1,e-j2πdc/λcos⁡θd,…,e-j2πdc/λM-1cos⁡θdTaθi=1,e-j2πdc/λcos⁡θi,…,e-j2πdc/λM-1cos⁡θiTwhere θd and θi denote the direction of GNSS signal and the ith interference, dc is the interelement spacing, λ is the wavelength of GNSS carrier, and T is the transpose operation. The GNSS signal sk at L1 band can be written as(3)sk=EcC/AkdN/Aksin⁡2πfL1k+θL1where E is power of the receiving GNSS signal, dN/Ak represents the data code carrying navigational message, cC/Ak is spreading coarse acquisition (C/A) code sequence with a period of 1023 chips, and fL1 and θL1 are the carrier frequency and the carrier phase of L1.The beamformer output of GNSS receiver can be written as(4)yk=wHxkwhere w is the complex beamforming weight vector of the antenna array and H stands for the Hermitian transpose.The schematic of a linear adaptive antenna array processor on the front end of the GNSS receiver is shown in Figure1.Figure 1
The schematic of a linear antenna array processor in GNSS system.


## 3. Proposed Adaptive Beamforming Scheme for GNSS Receiver

To attain the aim of low used number sensors for the tight energy control GNSS system and reduce the performance misadjustment of satellite signal acquisition for impulsive noises simultaneously, we detail the proposed constraint sparse maximum correntropy (CSMC) beamforming method for GNSS receiver in this section. In addition, the mean square analysis of CSMC is carried out, and the convergence stability condition of CSMC algorithm is derived using the generalization of approximation methods.


### 3.1. CSMC Iteration Algorithm

In information learning theory, correntropy is proposed to obtain robustness against outliers. It is a local similarity measure between two arbitrary random variables A and B and can be expressed as(5)VσA,B=EkσA-Bwhere kσ is a kernel that satisfies the Mercer theory. One of the common choices of kσ adopted in this paper is the Gaussian kernel given as(6)kσA-B=1σ2πexp⁡-A-B22σ2where σ is the kernel size. In the field of GNSS array system discussed in this paper, the cost function of the MCC-based algorithm is considered to be the correntropy between the local satellite signal replica and outputs of the beamformer, as given by(7)Jk=E1σ2πexp⁡-dk-wHxk22σ2Based on the constrained equations in CLMS [15] and adding a specific L1-norm to the optimization function in (7), we can obtain the following constrained optimization of CSMC:(8)maxwJks.t.CHw=zw1=tIn order to solve the constrained optimization problem in (8), the Lagrange multiplier and related mathematical skills are used. The object function can be rewritten as(9)ζw=E1σ2πexp⁡-dk-wHxk22σ2-λ1HCHw-z-λ2w1-tUsing the instantaneous approximation, we can eliminate the expectation operator from (9) as follows:(10)ζw=1σ2πexp⁡-dk-wHxk22σ2-λ1HCHw-z-λ2w1-tCalculate the instantaneous gradient of the Lagrange function (10) relative to vector w concludes to the following relation:(11)∇wζw=12πσ3ekexp⁡-ek22σ2xk-Cλ1-λ2sign⁡wApplying the steepest ascent method, we can obtain the coefficient vector updated at each iteration as(12)wk+1=wk+μ∇wζwwhere μ is the step size. Inserting the gradient vector from (11) into (12), we can get(13)wk+1=wk+μBk-Cλ1-λ2sign⁡wkwhere Bk=1/2πσ3ekexp⁡-ek2/2σ2xk.It is possible to solve forλ1 premultiplying (13) by CH and using the constraint relation CHwk+1=z in [22], such that(14)λ1=Hwkμ+HBk-λ2Hsign⁡wk-Gzμwhere H=CHC-1CH and G=CHC-1.Due to the constraints in the maximization problem with L1-norm, the approximationsignH⁡wkwk+1≈t is utilized. Using the definition tk=signH⁡wkwk and the approximation signH⁡wksign⁡wk=M, it is possible to premultiply (13) by signH⁡wk and solve for λ2.(15)λ2=signH⁡wkBkM-signH⁡wkCλ1M-eL1kμMwhere eL1k=t-tk.It is necessary to solve the system of equations stated by (14) and (15) to obtain the Lagrange multipliers λ1 and λ2. Defining P=I-CH, the resulting update expression of GNSS array vector presented here can be written as(16)wk+1=PSkSHkMQkCH+Iwk+PI-SkSHkMQkPμBk+I-PSkSHkMQkCGz+PSkMQkeL1kwith(17)Qk=1-SHkCHSkM,Sk=sign⁡w,Fk=SkMQkImplementation of the CSMC algorithm is summarized in Table1.Table 1
Summary of the proposed CSMC algorithm.
Initialization k = 1, H=CHC-1C, G=CHC-1, wk=CGz, P=I-CH, S k = sign ⁡ w k For each snapshotk=1,2,…,Kmax While k < K m a x do e k = d k - w H x k, Bk=12πσ3ekexp⁡-ek22σ2xk e L 1 k = t - w k 1, Qk=1-SHkCHSkM, Fk=SkMQk w k + 1 = P F k S H k C H + I w k + P I - F k S H k P μ B k + I - P F k S H k C G z + P F k e L 1 k k = k + 1 End snapshot


### 3.2. Mean Square Analysis of CSMC

We perform the mean square analysis of the CSMC algorithm in this subsection; the following commonly used assumptions are given as in the derivations of the convergence behavior of the proposed algorithm:(i) Letw1o be the optimal weight vector that takes into account the L1-norm constraint. The weight coefficient deviation vector of the CSMC is defined as Δw1k=w1k-w1o.(ii) The desired response produced by the CSMC beamformer of GNSS system can be expressed asd1k=w1oHxk+vk; vk is the uncorrelated additive noise with zero mean and variance σv2 of filters and is independent of xk.(iii) The signalxk received by GNSS array is independent and identically distributed (i.i.d.) with zero mean and variance σx2. It is approximately independent of the a priori errors and uncorrelated to the noise samples.In the context of the previously defined CSMC update model (16), let us consider the constraint fact that CHw1k+1=CHw1k=z and simplify the iterative equation of CSMC; we obtain(18)w1k+1=Pw1k+μMkBk+F+FL1kwith (19)Mk=I-PSkSHkPSk2,F=CGz,FL1k=PSkPSk2eL1kWith the assumption Δw1k=w1k-w1o and the definitions Vk=PSk/PSk2 and Mok=PSkSHk/PSk2, the constraint vector F and the auxiliary vector FL1 can be, respectively, stated as(20)F=Ι-Pw1oand(21)FL1=t-SHkw1kVk=t-SHkΔw1k-SHkw1oVk=tVk-PMokΔw1k+w1oUsing the formulation from (20) and (21), the resulting update equation of weight deviation vector can be computed as(22)Δw1k+1=PI-MokΔw1k-PMokw1o+μPI-MokBk+tVkSubstituting the equations d1k=w1oHxk+vk and ek=d1k-w1Hkxk in (22), we obtain(23)Δw1k+1=AkHkΔw1k-PMokw1o+muvkfekAkxk+tVkwith(24)Ak=PI-Mok,Hk=I-mufekxkxHk,fek=exp⁡-ek22σ2,mu=μ2πσ3The error nonlinearity fek is independent of the input signal xk and weight w1k.Taking the expected value of (23) and using the independence assumption [23], we get(25)EΔw1k+1=Ak1-muEfekσx2EΔw1k-EPMokw1o+tEVkTo perform the mean square convergence of the CSMC algorithm, we define the autocovariance ofΔw1k as(26)Rk=EZkZTkwhere Zk denotes the zero-mean misalignment vector as(27)Zk=Δw1k-EΔw1k Now, subtracting (25) from (23) and substituting (27) give(28)Zk+1=AkHkZk+Dk+Wk+Lk+H1kEΔw1kwhere(29)H1k=AkHk+muEfekσx2-1(30)Dk=EPMokw1o-PMokw1o(31)Wk=muvkfekAkxk(32)Lk=tVk-EtVkIt is straightforward to verify that Dk, Lk, and H1k are of zero mean. Besides, with the independence assumption, i.e., Δw1k, xk, and vk are mutually independent [24], and substituting (28) into (26), we can derive the resulting Rk+1 after some tedious calculations(33)Rk+1=EAkHkZk+Dk+Wk+Lk+H1kEΔw1k×AkHkZk+Dk+Wk+Lk+H1kEΔw1kT=EAkHkZkZTkHTkATk+ELkLTk+EAkHkZkDTk+EAkHkZkLTk+EDkZTkHTkATk+EDkDTk+EWkWTk+ELkZTkHTkATk+EH1kEΔw1kETΔw1kH1TkTo evaluate (33), using the fact that the fourth-order moment of a Gaussian variable is three times the variance square [23], and by the independence assumption and symmetric behavior of the covariance matrix [24], we obtain(34)EAkHkZkZTkHTkATk=Ak1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMATkwhere tr∙ denotes the trace of a matrix. Also, with (23)-(28), we can get the other terms in (33)(35)EAkHkZkDTk=ETDkZTkHTkATk=1-muEf2ekσx2EAkwkDTk(36)EAkHkZkLTk=ETLkZTkHTkATk=1-muEf2ekσx2EAkwkLTk(37)EWkWTk=mu2σv2σx2Ef2ekEAkATk(38)EH1kEΔw1kETΔw1kH1Tk=mu2Ef2ekσx4AktrδkδTkIM+δkδTkATkwhere δk represents(39)δk=EΔw1kNow, to obtain an amenable formulation for the mean square convergence analysis, we find the trace of (33) using the results of (34)-(38), (40)trRk+1=trAk1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMATk+21-muEf2ekσx2EAkwkDTk+21-muEf2ekσx2EAkwkLTk+EDkDTk+ELkLTk+mu2σv2σx2Ef2ekAkATk+mu2Ef2ekσx4trAktrδkδTkIM+δkδTkATkUsing the properties of the trace of matrix, we can expand the first term of (40) through the inequality of traces products [25]; we have(41)trAk1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMATk≤tr1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMtrAkATk=1-2muEfekσx2+M+2mu2σx4Ef2ektrPPT-2trPMoTkPT-trPMokMoTkPTtrRkFrom (41), it can be obviously seen that δk,Ewk,Lk,Dk,EAk are all bounded terms [23, 24], and thus we can state that, in order to ensure convergence, it is necessary to maintain the trace of the second-order moment within the interval 0,1; therefore, the adaptive beamformer is stable if and only if(42)1-2muEfekσx2+M+2mu2σx4Ef2ektrPPT-2trPMoTkPT-trPMokMoTkPT≤1To further evaluate the mean square analysis, we apply the principle of statistical independence to (42) and expand the trace of matrix through an inequality of trace products for positive semidefinite matrices of the same order; we obtain(43)
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## 3.1. CSMC Iteration Algorithm

In information learning theory, correntropy is proposed to obtain robustness against outliers. It is a local similarity measure between two arbitrary random variables A and B and can be expressed as(5)VσA,B=EkσA-Bwhere kσ is a kernel that satisfies the Mercer theory. One of the common choices of kσ adopted in this paper is the Gaussian kernel given as(6)kσA-B=1σ2πexp⁡-A-B22σ2where σ is the kernel size. In the field of GNSS array system discussed in this paper, the cost function of the MCC-based algorithm is considered to be the correntropy between the local satellite signal replica and outputs of the beamformer, as given by(7)Jk=E1σ2πexp⁡-dk-wHxk22σ2Based on the constrained equations in CLMS [15] and adding a specific L1-norm to the optimization function in (7), we can obtain the following constrained optimization of CSMC:(8)maxwJks.t.CHw=zw1=tIn order to solve the constrained optimization problem in (8), the Lagrange multiplier and related mathematical skills are used. The object function can be rewritten as(9)ζw=E1σ2πexp⁡-dk-wHxk22σ2-λ1HCHw-z-λ2w1-tUsing the instantaneous approximation, we can eliminate the expectation operator from (9) as follows:(10)ζw=1σ2πexp⁡-dk-wHxk22σ2-λ1HCHw-z-λ2w1-tCalculate the instantaneous gradient of the Lagrange function (10) relative to vector w concludes to the following relation:(11)∇wζw=12πσ3ekexp⁡-ek22σ2xk-Cλ1-λ2sign⁡wApplying the steepest ascent method, we can obtain the coefficient vector updated at each iteration as(12)wk+1=wk+μ∇wζwwhere μ is the step size. Inserting the gradient vector from (11) into (12), we can get(13)wk+1=wk+μBk-Cλ1-λ2sign⁡wkwhere Bk=1/2πσ3ekexp⁡-ek2/2σ2xk.It is possible to solve forλ1 premultiplying (13) by CH and using the constraint relation CHwk+1=z in [22], such that(14)λ1=Hwkμ+HBk-λ2Hsign⁡wk-Gzμwhere H=CHC-1CH and G=CHC-1.Due to the constraints in the maximization problem with L1-norm, the approximationsignH⁡wkwk+1≈t is utilized. Using the definition tk=signH⁡wkwk and the approximation signH⁡wksign⁡wk=M, it is possible to premultiply (13) by signH⁡wk and solve for λ2.(15)λ2=signH⁡wkBkM-signH⁡wkCλ1M-eL1kμMwhere eL1k=t-tk.It is necessary to solve the system of equations stated by (14) and (15) to obtain the Lagrange multipliers λ1 and λ2. Defining P=I-CH, the resulting update expression of GNSS array vector presented here can be written as(16)wk+1=PSkSHkMQkCH+Iwk+PI-SkSHkMQkPμBk+I-PSkSHkMQkCGz+PSkMQkeL1kwith(17)Qk=1-SHkCHSkM,Sk=sign⁡w,Fk=SkMQkImplementation of the CSMC algorithm is summarized in Table1.Table 1
Summary of the proposed CSMC algorithm.
Initialization k = 1, H=CHC-1C, G=CHC-1, wk=CGz, P=I-CH, S k = sign ⁡ w k For each snapshotk=1,2,…,Kmax While k < K m a x do e k = d k - w H x k, Bk=12πσ3ekexp⁡-ek22σ2xk e L 1 k = t - w k 1, Qk=1-SHkCHSkM, Fk=SkMQk w k + 1 = P F k S H k C H + I w k + P I - F k S H k P μ B k + I - P F k S H k C G z + P F k e L 1 k k = k + 1 End snapshot


## 3.2. Mean Square Analysis of CSMC

We perform the mean square analysis of the CSMC algorithm in this subsection; the following commonly used assumptions are given as in the derivations of the convergence behavior of the proposed algorithm:(i) Letw1o be the optimal weight vector that takes into account the L1-norm constraint. The weight coefficient deviation vector of the CSMC is defined as Δw1k=w1k-w1o.(ii) The desired response produced by the CSMC beamformer of GNSS system can be expressed asd1k=w1oHxk+vk; vk is the uncorrelated additive noise with zero mean and variance σv2 of filters and is independent of xk.(iii) The signalxk received by GNSS array is independent and identically distributed (i.i.d.) with zero mean and variance σx2. It is approximately independent of the a priori errors and uncorrelated to the noise samples.In the context of the previously defined CSMC update model (16), let us consider the constraint fact that CHw1k+1=CHw1k=z and simplify the iterative equation of CSMC; we obtain(18)w1k+1=Pw1k+μMkBk+F+FL1kwith (19)Mk=I-PSkSHkPSk2,F=CGz,FL1k=PSkPSk2eL1kWith the assumption Δw1k=w1k-w1o and the definitions Vk=PSk/PSk2 and Mok=PSkSHk/PSk2, the constraint vector F and the auxiliary vector FL1 can be, respectively, stated as(20)F=Ι-Pw1oand(21)FL1=t-SHkw1kVk=t-SHkΔw1k-SHkw1oVk=tVk-PMokΔw1k+w1oUsing the formulation from (20) and (21), the resulting update equation of weight deviation vector can be computed as(22)Δw1k+1=PI-MokΔw1k-PMokw1o+μPI-MokBk+tVkSubstituting the equations d1k=w1oHxk+vk and ek=d1k-w1Hkxk in (22), we obtain(23)Δw1k+1=AkHkΔw1k-PMokw1o+muvkfekAkxk+tVkwith(24)Ak=PI-Mok,Hk=I-mufekxkxHk,fek=exp⁡-ek22σ2,mu=μ2πσ3The error nonlinearity fek is independent of the input signal xk and weight w1k.Taking the expected value of (23) and using the independence assumption [23], we get(25)EΔw1k+1=Ak1-muEfekσx2EΔw1k-EPMokw1o+tEVkTo perform the mean square convergence of the CSMC algorithm, we define the autocovariance ofΔw1k as(26)Rk=EZkZTkwhere Zk denotes the zero-mean misalignment vector as(27)Zk=Δw1k-EΔw1k Now, subtracting (25) from (23) and substituting (27) give(28)Zk+1=AkHkZk+Dk+Wk+Lk+H1kEΔw1kwhere(29)H1k=AkHk+muEfekσx2-1(30)Dk=EPMokw1o-PMokw1o(31)Wk=muvkfekAkxk(32)Lk=tVk-EtVkIt is straightforward to verify that Dk, Lk, and H1k are of zero mean. Besides, with the independence assumption, i.e., Δw1k, xk, and vk are mutually independent [24], and substituting (28) into (26), we can derive the resulting Rk+1 after some tedious calculations(33)Rk+1=EAkHkZk+Dk+Wk+Lk+H1kEΔw1k×AkHkZk+Dk+Wk+Lk+H1kEΔw1kT=EAkHkZkZTkHTkATk+ELkLTk+EAkHkZkDTk+EAkHkZkLTk+EDkZTkHTkATk+EDkDTk+EWkWTk+ELkZTkHTkATk+EH1kEΔw1kETΔw1kH1TkTo evaluate (33), using the fact that the fourth-order moment of a Gaussian variable is three times the variance square [23], and by the independence assumption and symmetric behavior of the covariance matrix [24], we obtain(34)EAkHkZkZTkHTkATk=Ak1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMATkwhere tr∙ denotes the trace of a matrix. Also, with (23)-(28), we can get the other terms in (33)(35)EAkHkZkDTk=ETDkZTkHTkATk=1-muEf2ekσx2EAkwkDTk(36)EAkHkZkLTk=ETLkZTkHTkATk=1-muEf2ekσx2EAkwkLTk(37)EWkWTk=mu2σv2σx2Ef2ekEAkATk(38)EH1kEΔw1kETΔw1kH1Tk=mu2Ef2ekσx4AktrδkδTkIM+δkδTkATkwhere δk represents(39)δk=EΔw1kNow, to obtain an amenable formulation for the mean square convergence analysis, we find the trace of (33) using the results of (34)-(38), (40)trRk+1=trAk1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMATk+21-muEf2ekσx2EAkwkDTk+21-muEf2ekσx2EAkwkLTk+EDkDTk+ELkLTk+mu2σv2σx2Ef2ekAkATk+mu2Ef2ekσx4trAktrδkδTkIM+δkδTkATkUsing the properties of the trace of matrix, we can expand the first term of (40) through the inequality of traces products [25]; we have(41)trAk1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMATk≤tr1-2muEfekσx2+2mu2σx4Ef2ekRk+mu2σx4Ef2ektrRkIMtrAkATk=1-2muEfekσx2+M+2mu2σx4Ef2ektrPPT-2trPMoTkPT-trPMokMoTkPTtrRkFrom (41), it can be obviously seen that δk,Ewk,Lk,Dk,EAk are all bounded terms [23, 24], and thus we can state that, in order to ensure convergence, it is necessary to maintain the trace of the second-order moment within the interval 0,1; therefore, the adaptive beamformer is stable if and only if(42)1-2muEfekσx2+M+2mu2σx4Ef2ektrPPT-2trPMoTkPT-trPMokMoTkPT≤1To further evaluate the mean square analysis, we apply the principle of statistical independence to (42) and expand the trace of matrix through an inequality of trace products for positive semidefinite matrices of the same order; we obtain(43)
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TUsing the properties that the trace of vector is greater than or equal to one and the number of elements in array is more than one by far, we have(44)trPPT-2trPMoTkPT-trPMokMoTkPT>>1Therefore, using (44), inequality (42) reduces to(45)1-2muEfekσx2+M+2mu2σx4Ef2ek<1Hence, combining (45) and the relation between mu and step size μ, we can obtain the range of μ to meet the mean square stability(46)0<μ<2M+2Efekσx2/2πσ3The deriving result in (46) satisfies the theoretical requirements for convergence step size of adaptive algorithm [26] and confirms the validity of proposed CSMC beamforming method.


## 4. Simulation Results

In this section, several groups of simulation experiments are conducted using Matlab R2015b to verify the proposed CSMC beamforming algorithm for GNSS receivers; a uniform linear array of 15 omnidirectional antenna elements with the interelement spacing of half wavelength is considered in the simulation; the desired satellite signal is the form of C/A code with 90° for incident angle. The carrier frequencies of GNSS signal are set at 1575.42 MHz of the L1 center frequency. The sampling frequency is 5 MHz and the acquisition time is set to 20 ms (K=100000). The code phase delay and the Doppler frequency shift of the received GNSS signal are set to 300 sampling points and -5kHz, respectively. The step size of Doppler frequency search is 500 Hz. We consider that four single frequency interferences with 45 interference-to-noise ratio (INR) are located at 20, 40, 110, and 150, respectively.Additive noise in array elements is modeled as Gaussian noise with zero mean and unit variance. The impulsive noises applied in this paper consider to be a four-parameter family of alpha noises and can be denoted asSα,βα,γ,δα, α∈0,2 is the characteristic exponent, βα∈-1,1 is the skewness, and the last two parameters γ>0, δα∈R denote the scale and the location of the model.The performance for GNSS system with the proposed method is evaluated from the following aspects: the beampattern performance of the array antenna for satellite system receiver, the steady state output SINR of the proposed CSMC beamformer, the acquisition results of the code phase delay and the Doppler frequency shift in time-frequency domain, and the probability of detection for the satellite signal. The sparsity performance of GNSS array induced by the proposed algorithm is also evaluated. All simulation results in the figures are obtained by averaging 200 trials. The proposed methods are compared with the four conventional beamforming methods. The Linear Constrained Minimum Variance (LCMV) solution is used as a closed-form expression to the uniformly optimum weight vectorwopt. All the other parameters used in the simulations are summarized in Table 2.Table 2
Parameter values for MSE.
Parameter ULA Kernel Sizeσ 3  Intermediate Frequency 1.25MHz  Skewnessβα 0  Parameterδα 0  Parameterγ 1


### 4.1. Beampattern Performance Comparison

In the first example, we compare the power patterns formed by the different types of beamformers using ULA. The signal-to-noise ratio (SNR) of GNSS signal is -5dB. All the beamformers are designed to operate for the impulsive noise scenario. Figures2(a)–2(d) show the beampatterns in presence of the impulsive noise with varied characteristic exponent representing different impulsive levels of noises. As can be seen, the proposed algorithms have suppressed the jammers in all cases while maintaining the beampattern gain in the direction of the desired signal. The other algorithms are able to achieve the similar beamforming performance for lowly impulsive noises, as impulsive degree of the noise interference increase, the four compared beamforming methods suffer from performance degradation of corresponding beampatterns, and the proposed algorithm has evident advantages over these algorithms, especially in the most severely impulsive noise situation. Hence, the proposed CSMC achieved improved robustness against impulsive noises than their competitors.Figure 2
Beampatterns of the proposed and the other existing beamformers with differentα: (a) α=1.65; (b) α=1.5; (c) α=1.35; (d) α=1.2.
(a) (b) (c) (d)


### 4.2. Output SINR Performance

For the simulation presented in this example, the output SINR performance for the CSMC beamformer is investigated in terms of SNR. We compare the proposed technique to the four aforementioned adaptive beamforming methods. The simulation scenario and parameters setup are the same as used in Section4.1. When the SNR varies from −10 to 30 dB, the resulting curves are shown in Figure 3. It can be seen from the figures that the SINR of all algorithms increase with higher SNR, but the increasing rates of the comparisons in the simulation show fluctuations gradually as α decreasing, which is due to the heavier disturbance of impulsive noises. The proposed beamforming algorithm leads to the improved SINR performance, especially in the situation of α=1.05, which means the most impulsive process of the noises. This fact is due to the MCC criterion which is much more robust against impulsive outliers than MSE estimation. Absolutely, the proposed methods outperform the other algorithms in all the cases and are close to the optimum SINR.Figure 3
SINR performance against SNR with differentα: (a) α=1.05; (b) α=1.25; (c) α=1.45.
(a) (b) (c)


### 4.3. Adaptive Beamforming for GNSS Signal Acquisition

In this part, we investigate the GNSS signal acquisition performance using regular searching method [27] with the proposed beamforming method; four methods mentioned above are compared in terms of the code phase delay and the Doppler frequency shift acquisition. The input signal-to-noise ratio (SNR) is set as 8 dB. Figures 4(a)–4(d) show the three-dimension search results for the GNSS signal in impulsive noises with α=1.25 that was acquired with traditional beamformers. We can observe from the figures that hardly any dominant peak could be observed at the correct code phase-Doppler frequency shift in combination with the comparison beamforming algorithms. Peaks of indistinct and irregular magnitude exist due to noises and interferences disturbance, which indicates that CLMS-L1, MRVSS CLMS, and LMS are susceptible to impulsive noises and the acquisition of the GNSS signal with these techniques fails. The correlation peak with the proposed CSMC beamforming method is depicted in Figure 5(a), as evidenced by the figure, the impact of impulsive noises is drastically reduced, and a significant peak is present in time-frequency domain. The sharp peak occurs at correct code phase=300 sampling points and Doppler frequency shift=-5KHz which is shown in Figures 5(b) and 5(c). The success of the acquisition with CSMC is due to the novel beamforming criterion used in the GNSS receiver.Figure 4
Acquisition results in the impulsive noises with traditional adaptive beamformers in GNSS receiver: (a) CLMS-L1; (b) MRVSS; (c) CLMS; (d) LMS.
(a) (b) (c) (d)Figure 5
Acquisition results in the impulsive noises with the CSMC beamformer in GNSS receiver: (a) three-dimension domain; (b) the code phase; (c) the Doppler frequency.
(a) (b) (c)


### 4.4. Detection Probability for Acquisition Performance

In this subsection, the acquisition performance with CSMC beamformer is assessed in terms of the detection probability. To check the robustness validity of our approach, we consider three sets of parameters withα=1.05, α=1.45, and α=1.85 representing the alpha noises of different impulsive level. The acquisition test statistic and detection metric used in this paper is based on the test detector in [28]. Figure 6 shows the detection probability versus SNR in the presence of impulsive noises. It can be observed from the results shown in Figure 6(a) that the proposed beamforming algorithm enjoys a higher detection probability than most of the conventional beamformers for the value of α=1.85 which indicates lowly impulsive noises. As the value of α decreases in Figures 6(b) and 6(c), which means more highly impulsive noises exist, the proposed algorithms outperform the comparison methods significantly and achieve the highest rate of acquisition for satellite signal, whereas the detection probability of traditional algorithms is deteriorating and fluctuating in these two cases, which demonstrates that these methods are more sensitive to impulsive noises and fail to acquire GNSS signal in severely impulsive noises scenarios.Figure 6
Acquisition performance in terms of detection probability in presence of the alpha noises of different impulsive level: (a)α=1.05; (b) α=1.45; (c) α=1.85.
(a) (b) (c)


### 4.5. The Sparsity of the ULA with CMSC in GNSS Receiver

This section presents three sets of simulations carried out with a 30-element ULA (simulated large number of sensor array utilized in satellite communication system) to explore the sparsity of the proposed beamforming method for GNSS receiver. All the simulations were carried out using 3 distinct levels of norm constraint, according to Table3. The elements represented by white circles are those turned off by the algorithm. The results depicted in Figure 7 indicate that the acquisition of satellite signal with ULA favors the sparse solution produced by CSMC algorithm, the GNSS beamformer synthesized by the CSMC with 25%, 45%, and 65% norm constraints keeps only 16.67%, 26.67%, and 40% of the total number of elements online, and the increasing sparsity was achieved decreasing the norm constraint value. From the results of simulation, we can conclude that the efficient ability of the proposed algorithm to produce sparse solutions and shut down a number of sensors in adaptive antenna arrays is suitable for GNSS receiver system.Table 3
Parameter values for sparsity simulations.
The Number of Elements 30 L1 Norm Constraintt t=wopt1  Norm Constraint Ratio 25%, 45%, 65%Figure 7
The sparsity of GNSS array obtained from employing the CSMC beamformer with different norm constraints: (a)t=25%×wopt1; (b) t=45%×wopt1; (c) t=65%×wopt1.
(a) (b) (c)


### 4.6. The Influence of the Kernel Size and Step Parameter on CSMC Algorithm Performance

To evaluate the effects of all the used parameters on the behavior of the proposed CSMC algorithm and fully justify these parameters and their influence on the results closely examined, we conduct a series of simulations in the context of adaptive beamforming, including simulations on kernel size and step parameter. The mean square error (MSE) measure between the local satellite signal replica and beamformer output is employed to quantify the performance in the literature. A total of10000 iterations were performed for the verification. The simulation experiment environment of this example is the same as Section 4.1. In the first case, for investigating the effect of the kernel size on MSE with different level, we consider five sets of parameters to justify this parameter on our approach. Figure 8(a) illustrates the resultant MSE performance versus the number of iterations for different kernel sizes. It can be observed from the results shown in the figure that choosing a proper value for the kernel size σ is a crucial issue in CSMC algorithm since it directly affects the adaptive learning speed and accuracy. σ=0.5 will lead to a very poor performance so that the kernel size cannot be chosen arbitrarily small. However, the large values of the kernel size σ will lead to a low speed performance; thus, in this paper, we fix σ=3 in the simulation which benefits from both the high speed and the high accuracy, simultaneously. In the next case, we investigate the effects for different values of the step parameter on the CSMC performance. The MSE curve of simulation results for different μ is shown in Figure 8(b). As can be seen, the large step size leads to fast convergence rate, but the steady state MSE of the proposed algorithms increases as the step sizes increase, achieving higher steady state error, while small step size has small misadjustment for adaptive beamforming, but suffers from slow convergence speed. From the above, we can conclude that, to obtain performance improvement and stability with balanced convergence rate, we select μ = 0.01 to guarantee the fast convergence and satisfied stable steady state MSE for the proposed CSMC algorithm.Figure 8
MSE learning curves of the proposed CSMC with different used parameters: (a) MSE for different kernel size; (b) MSE for different step parameter.
(a) (b)


## 4.1. Beampattern Performance Comparison

In the first example, we compare the power patterns formed by the different types of beamformers using ULA. The signal-to-noise ratio (SNR) of GNSS signal is -5dB. All the beamformers are designed to operate for the impulsive noise scenario. Figures2(a)–2(d) show the beampatterns in presence of the impulsive noise with varied characteristic exponent representing different impulsive levels of noises. As can be seen, the proposed algorithms have suppressed the jammers in all cases while maintaining the beampattern gain in the direction of the desired signal. The other algorithms are able to achieve the similar beamforming performance for lowly impulsive noises, as impulsive degree of the noise interference increase, the four compared beamforming methods suffer from performance degradation of corresponding beampatterns, and the proposed algorithm has evident advantages over these algorithms, especially in the most severely impulsive noise situation. Hence, the proposed CSMC achieved improved robustness against impulsive noises than their competitors.Figure 2
Beampatterns of the proposed and the other existing beamformers with differentα: (a) α=1.65; (b) α=1.5; (c) α=1.35; (d) α=1.2.
(a) (b) (c) (d)


## 4.2. Output SINR Performance

For the simulation presented in this example, the output SINR performance for the CSMC beamformer is investigated in terms of SNR. We compare the proposed technique to the four aforementioned adaptive beamforming methods. The simulation scenario and parameters setup are the same as used in Section4.1. When the SNR varies from −10 to 30 dB, the resulting curves are shown in Figure 3. It can be seen from the figures that the SINR of all algorithms increase with higher SNR, but the increasing rates of the comparisons in the simulation show fluctuations gradually as α decreasing, which is due to the heavier disturbance of impulsive noises. The proposed beamforming algorithm leads to the improved SINR performance, especially in the situation of α=1.05, which means the most impulsive process of the noises. This fact is due to the MCC criterion which is much more robust against impulsive outliers than MSE estimation. Absolutely, the proposed methods outperform the other algorithms in all the cases and are close to the optimum SINR.Figure 3
SINR performance against SNR with differentα: (a) α=1.05; (b) α=1.25; (c) α=1.45.
(a) (b) (c)


## 4.3. Adaptive Beamforming for GNSS Signal Acquisition

In this part, we investigate the GNSS signal acquisition performance using regular searching method [27] with the proposed beamforming method; four methods mentioned above are compared in terms of the code phase delay and the Doppler frequency shift acquisition. The input signal-to-noise ratio (SNR) is set as 8 dB. Figures 4(a)–4(d) show the three-dimension search results for the GNSS signal in impulsive noises with α=1.25 that was acquired with traditional beamformers. We can observe from the figures that hardly any dominant peak could be observed at the correct code phase-Doppler frequency shift in combination with the comparison beamforming algorithms. Peaks of indistinct and irregular magnitude exist due to noises and interferences disturbance, which indicates that CLMS-L1, MRVSS CLMS, and LMS are susceptible to impulsive noises and the acquisition of the GNSS signal with these techniques fails. The correlation peak with the proposed CSMC beamforming method is depicted in Figure 5(a), as evidenced by the figure, the impact of impulsive noises is drastically reduced, and a significant peak is present in time-frequency domain. The sharp peak occurs at correct code phase=300 sampling points and Doppler frequency shift=-5KHz which is shown in Figures 5(b) and 5(c). The success of the acquisition with CSMC is due to the novel beamforming criterion used in the GNSS receiver.Figure 4
Acquisition results in the impulsive noises with traditional adaptive beamformers in GNSS receiver: (a) CLMS-L1; (b) MRVSS; (c) CLMS; (d) LMS.
(a) (b) (c) (d)Figure 5
Acquisition results in the impulsive noises with the CSMC beamformer in GNSS receiver: (a) three-dimension domain; (b) the code phase; (c) the Doppler frequency.
(a) (b) (c)


## 4.4. Detection Probability for Acquisition Performance

In this subsection, the acquisition performance with CSMC beamformer is assessed in terms of the detection probability. To check the robustness validity of our approach, we consider three sets of parameters withα=1.05, α=1.45, and α=1.85 representing the alpha noises of different impulsive level. The acquisition test statistic and detection metric used in this paper is based on the test detector in [28]. Figure 6 shows the detection probability versus SNR in the presence of impulsive noises. It can be observed from the results shown in Figure 6(a) that the proposed beamforming algorithm enjoys a higher detection probability than most of the conventional beamformers for the value of α=1.85 which indicates lowly impulsive noises. As the value of α decreases in Figures 6(b) and 6(c), which means more highly impulsive noises exist, the proposed algorithms outperform the comparison methods significantly and achieve the highest rate of acquisition for satellite signal, whereas the detection probability of traditional algorithms is deteriorating and fluctuating in these two cases, which demonstrates that these methods are more sensitive to impulsive noises and fail to acquire GNSS signal in severely impulsive noises scenarios.Figure 6
Acquisition performance in terms of detection probability in presence of the alpha noises of different impulsive level: (a)α=1.05; (b) α=1.45; (c) α=1.85.
(a) (b) (c)


## 4.5. The Sparsity of the ULA with CMSC in GNSS Receiver

This section presents three sets of simulations carried out with a 30-element ULA (simulated large number of sensor array utilized in satellite communication system) to explore the sparsity of the proposed beamforming method for GNSS receiver. All the simulations were carried out using 3 distinct levels of norm constraint, according to Table3. The elements represented by white circles are those turned off by the algorithm. The results depicted in Figure 7 indicate that the acquisition of satellite signal with ULA favors the sparse solution produced by CSMC algorithm, the GNSS beamformer synthesized by the CSMC with 25%, 45%, and 65% norm constraints keeps only 16.67%, 26.67%, and 40% of the total number of elements online, and the increasing sparsity was achieved decreasing the norm constraint value. From the results of simulation, we can conclude that the efficient ability of the proposed algorithm to produce sparse solutions and shut down a number of sensors in adaptive antenna arrays is suitable for GNSS receiver system.Table 3
Parameter values for sparsity simulations.
The Number of Elements 30 L1 Norm Constraintt t=wopt1  Norm Constraint Ratio 25%, 45%, 65%Figure 7
The sparsity of GNSS array obtained from employing the CSMC beamformer with different norm constraints: (a)t=25%×wopt1; (b) t=45%×wopt1; (c) t=65%×wopt1.
(a) (b) (c)


## 4.6. The Influence of the Kernel Size and Step Parameter on CSMC Algorithm Performance

To evaluate the effects of all the used parameters on the behavior of the proposed CSMC algorithm and fully justify these parameters and their influence on the results closely examined, we conduct a series of simulations in the context of adaptive beamforming, including simulations on kernel size and step parameter. The mean square error (MSE) measure between the local satellite signal replica and beamformer output is employed to quantify the performance in the literature. A total of10000 iterations were performed for the verification. The simulation experiment environment of this example is the same as Section 4.1. In the first case, for investigating the effect of the kernel size on MSE with different level, we consider five sets of parameters to justify this parameter on our approach. Figure 8(a) illustrates the resultant MSE performance versus the number of iterations for different kernel sizes. It can be observed from the results shown in the figure that choosing a proper value for the kernel size σ is a crucial issue in CSMC algorithm since it directly affects the adaptive learning speed and accuracy. σ=0.5 will lead to a very poor performance so that the kernel size cannot be chosen arbitrarily small. However, the large values of the kernel size σ will lead to a low speed performance; thus, in this paper, we fix σ=3 in the simulation which benefits from both the high speed and the high accuracy, simultaneously. In the next case, we investigate the effects for different values of the step parameter on the CSMC performance. The MSE curve of simulation results for different μ is shown in Figure 8(b). As can be seen, the large step size leads to fast convergence rate, but the steady state MSE of the proposed algorithms increases as the step sizes increase, achieving higher steady state error, while small step size has small misadjustment for adaptive beamforming, but suffers from slow convergence speed. From the above, we can conclude that, to obtain performance improvement and stability with balanced convergence rate, we select μ = 0.01 to guarantee the fast convergence and satisfied stable steady state MSE for the proposed CSMC algorithm.Figure 8
MSE learning curves of the proposed CSMC with different used parameters: (a) MSE for different kernel size; (b) MSE for different step parameter.
(a) (b)


## 5. Conclusion

A novel adaptive beamforming approach is proposed aiming to improve the performance of GNSS signal acquisition in this paper. We incorporate the sparse norm regularization and constraint maximum correntropy criterion into the optimization constraint equation of beamforming to obtain a new adaptive beamformer which is able to mitigate the adverse effect of impulsive noises while allowing flexibility in the energy consumption at GNSS receiver. A detailed mean square analysis of the proposed methods was presented and the stability condition was derived. The numerical simulation results described in the paper indicate that the proposed CSMC beamformer provides better impulsive noises suppression for improved performance of satellite signal acquisition. Furthermore, with the sparsity produced by CSMC, a thinned 30-element ULA is achieved to save energy to GNSS receiver. The proposed algorithms can also be extended to other array geometries and applications in GNSS signal processing. We plan to explore those applications in future work.



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*Source: 6193049-2019-06-18.xml*