Reweighted Factor Selection for Slms-rl1 Algorithm under Gaussian Mixture Noise Environments

The sign least mean square with reweighted L1-norm constraint (SLMS-RL1) algorithm is an attractive sparse channel estimation method among Gaussian mixture model (GMM) based algorithms for use in impulsive noise environments. The channel sparsity can be exploited by SLMS-RL1 algorithm based on appropriate reweighted factor, which is one of key parameters to adjust the sparse constraint for SLMS-RL1 algorithm. However, to the best of the authors' knowledge, a reweighted factor selection scheme has not been developed. This paper proposes a Monte-Carlo (MC) based reweighted factor selection method to further strengthen the performance of SLMS-RL1 algorithm. To validate the performance of SLMS-RL1 using the proposed reweighted factor, simulations results are provided to demonstrate that convergence speed can be reduced by increasing the channel sparsity, while the steady-state MSE performance only slightly changes with different GMM impulsive-noise strengths.


Introduction
Adaptive filtering algorithms have been widely applied for multipath channel estimation, especially in broadband wireless systems [1][2][3][4][5][6][7], where the broadband signals are vulnerable to multipath fading as well as additive noises [8][9][10].Hence, channel state information (CSI) is necessary for coherent demodulation [11].Based on the classical Gaussian noise model, second-order statistics based least mean square (LMS) algorithm has been widely used to estimate channels due to its simplicity and robustness [1,2].However, the performance of LMS is usually degraded by impulsive noise [12], which is common in broadband wireless systems and can be described by the Gaussian mixture noise model (GMM) [13].Thus, it is necessary to develop robust channel estimation algorithms in the presence of GMM impulsive noise.In [1], a standard sign least mean square (SLMS) algorithm was proposed to suppress impulsive noise.In [14], Jiang et al. proposed a sophisticated robust matched filtering algorithm in ℓ -space to realize time delay estimation (TDE) and joint delay-Doppler estimation (JDDE) for target localization.On the other hand, wireless channels can be often modeled as sparse or cluster-sparse and hence many of channel coefficients are zero [15][16][17][18][19].However, standard SLMS algorithm does not exploit sparse channel structure information, while some potential performance gain could be obtained by adopting advanced adaptive channel estimation algorithms.
To exploit channel sparsity as well as to mitigate GMM impulsive noises, some state-of-the-art channel estimation algorithms using linear programming [20,21] and Bayesian learning [22] have been investigated.However, these algorithms often have high computational complexity.It is well known that the fast channel estimation algorithm is one of important factors to design wireless communication systems.Hence, a fast adaptive sparse channel estimation algorithm, i.e., SLMS with reweighted L1-norm constraint (SLMS-RL1) algorithm was proposed in [23].In our previous work, we focused on the convergence analysis without considering the reweighted factor selection, where the empirical parameter is set as 0.005 [6].However, the reweighted factor is one of critical parameters to balance estimation performance and sparsity exploitation.To this end, this paper proposes a Monte-Carlo (MC) based selection method to select suitable reweighted factor for SLMS-RL1 algorithm.Numerical simulations are provided to evaluate the performance of the SLMS-RL1 algorithm using the proposed reweighted factor.
The rest of the paper is organized as follows.In Section 2, we introduce GMM noise model and review of SLMS-RL1 algorithm.By analyzing the convergence performance of LMS-RL1 algorithm, the important problem of the reweighted factor selection is point out.In Section 3, the MC-based selection method is proposed to select an appropriate reweighted factor for SLMS-RL1 algorithm.In Section 4, numerical simulations are provided to demonstrate the effectiveness of SLMS-RL1 with the proposed reweighted factor.Finally, Section 5 concludes this paper.

Review of SLMS-RL1 Algorithm
Consider an additive noise interference channel, which is modeled by the unknown N-length finite impulse response (FIR) vector , , … , at discrete time index .The ideal received signal is expressed as (1) where , 1 , … , 1 is the input signal vector of the most recent input samples; is an -dimensional column vector of the unknown system that we wish to estimate, and is impulsive noise which can be described by Gaussian mixture model (GMM) [13] distribution as where ≫ 1 denotes impulsive-noise strength and 0, denotes the Gaussian distributions with zero mean and variance , and the is the mixture parameter to control the impulsive noise level.According to Equation (2), one can find that stronger impulsive noises can be described by larger noise variance as well as larger mixture parameter .According to Equation (2), variance of GMM is obtained.

1
(3) Note that will reduce to Gaussian noise model if 0. The objective of the adaptive channel estimation is to perform adaptive estimate of with limited complexity and memory given sequential observation , in the presence of additive GMM noise .According to Equation ( 1), instantaneous estimation error can be written as where is the estimator of at iteration .To obtain the optimal channel estimation, one can construct the ℓ -norm minimization problem as where ‖ ‖ denotes ℓ -norm operator which is defined as ‖ ‖ # | 0 .That is to say, the main function of ‖ ‖ is to find the total number of nonzero coefficients.However, solving the ℓ -norm minimization is a Non-Polynomial (NP) hard problem.Hence, it is necessary to introduce an approximate ℓ -norm minimization function so that Equation ( 5) is solvable.On the adaptive sparse channel estimation, reweighted ℓ -norm (RL1) minimization has a better performance than ℓ -minimization that is usually employed in compressive sensing [24].It is due to the fact that a properly RL1 can approximate the ℓ -norm more accurate than ℓ -norm.Hence, one approach to enforce the sparsity of the solution for the sparse SLMS algorithm is to introduce the RL1 penalty term in thee cost function as RL1-LAE which considers a penalty term proportional to the RL1 of the coefficient vector.Hence, the cost function Equation ( 5) can be revised as where is the weight associated with the penalty term and elements of the diagonal reweighted matrix are devised as where being some positive number and hence 0 for 0, 1, … , 1 .The update equation can be derived by differentiating (6) with respect to the FIR channel vector .Then, the resulting update equation is where .Notice that in Equation (8), since sgn , one can get sgn sgn sgn sgn , where sgn ⋅ denotes sign function, i.e., sgn a/|a| for 0, sgn 0 for 0.

Problem Formulation
Define the misalignment channel vector as ≜ and ≜ as the second moment matrix of , Equation ( 4) can be rewritten as .To verify the performance, the convergence analysis of SLMS-RL1 algorithm is derived via mean convergence and excess MSE.Based on independent assumptions, in [23], the authors derive that SLMS-RL1 is stable if where denotes the maximal eigenvalue of .Then the mean estimation error 1 is derived as where 1 1 and ∞ lim → .Similarly, excess mean square error (MSE) is approximated as where and are defined as and respectively.Here, ‖⋅‖ denotes ℓ -norm constraint.Both mean estimation error and excess MSE imply that the reweighted factor adjusts performance of SLMS-RL1 algorithm.Hence, it is necessary to develop effective method to choose agreeable reweighted factor for further reinforce the proposed SLMS-RL1 algorithm.

Reweighted Factor Selection for SLMS-RL1 Algorithm
MC-based reweighted factor selection method is developed for SLMS-RL1 algorithm in different SNR regimes, impulsive-noise strength 400, mixture parameters 0.1 as well as channel sparsity 4. For achieving average performance, 1000 independent Monte-Carlo runs are adopted.
The simulation setup is configured according to the typical broadband wireless communication system [10].The signal bandwidth is 50 MHz located at the central radio frequency of 2.1 GHz.The maximum delay spread of 0.8 .Hence, the maximum length of channel vector is 80.In addition, each dominant channel tap follows random Gaussian distribution as 0, which is subject to ‖ ‖ 1 and their positions are randomly decided within .To evaluate SLMS-RL1 algorithm using different factors, we adopt the average mean square error (MSE) metric which is defined as where and n are the actual signal vector and reconstruction vector, respectively; ⋅ denotes mathematical expectation operator.The received SNR is defined as / , where is the received power of the pseudo-random (PN) binary sequence for training signal.Detailed parameters for computer simulation are listed in Table 1.First of all, MC based reweighted factor selection method is performed in Figures 1 and 2. Average MSE curves of the SLMS-RL1 algorithm are depicted under two SNR regimes, i.e., ∈ 5 dB, 10 dB .To confirm the effectiveness of the proposed method, standard SLMS [1] is considered as a performance benchmark.As we discussed in Section 2, one can see that the MSE performance of SLMS-RL1 algorithms depends highly on the reweighted factor .In these two figures, the lowest MSE performance of SLMS-RL1 is achieved when reweighted factor is set as 0.005 in two SNR regimes.On the one hand, too big reweighted factor may suppress noise excessively and hence it result in lossy exploitation of channel sparsity.On the other hand, a too small reweighted factor may mitigate noise insufficiently and it causes inefficient exploitation of channel sparsity.Therefore, suitable reweighted factor could balance the noise suppression and channel sparsity exploitation.

Numerical Simulations
In this section, three examples are given to verify the performance of SLMS-RL1 algorithm by using proposed reweighted factor ε 0.005 in the scenarios of = 10 dB, impulsive-noise strength ∈ 200,400,600,800 , mixture parameters 0.1 as well as channel sparsity ∈ 2,4,8,12,16 .For achieving average performance, 1000 independent Monte-Carlo runs are adopted as well.Detailed parameters for computer simulation are listed in Table 2.In the first example, average MSE curves of different algorithms are depicted in Figure 3.Under the certain circumstance, channel sparsity 4, 10dB, GMM noise with impulsive-noise parameter 400 as well as mixture parameter 0.1, one can find that proposed SLMS-RL1 algorithm can achieve at least 5 dB and 10 dB performance gain in contrast to SLMS algorithm and LMS-type algorithms, respectively.Because SLMS algorithm does not exploit the channel sparsity while LMS-type algorithms do not stable under GMM noise environments.Hence, the proposed SLMS-RL1 can exploit channel sparsity and can keep stability in the presence of GMM noises.
In the second example, average MSE curves of SLMS-RL1 algorithm with respect to channel sparsity are depicted in Figure 4.Under the certain circumstance, e.g., 10dB, GMM noise with impulsive-noise parameter 400 as well as mixture parameter 0.1, one can find that that convergence speed of SLMS-RL1 depends on channel sparsity ( ) while steady-state MSE curves of corresponding algorithms are very close.For different channel sparsity, in other words, the adaptive sparse algorithms may differ from conventional compressive sensing based sparse channel estimation algorithms [15], [16], [25], [26] which depend highly on channel sparsity.Hence, SLMS-RL1 using MC-based reweighted factor is expected to deal with different sparse channels stably even non-sparse cases.In the third example, average MSE curves of SLMS-RL1 algorithm using reweighted selected factor ε 0.005 with respect to impulsive-noise strength are depicted in Figure 5.In addition, average MSE curves of the algorithm with respect to mixture parameter are depicted in Figure 6.In the two figures, one can see that SLMS-RL1 algorithm using 0.005 is stable for different GMM noises with impulsive-noise strength parameters ∈ 200,400,600,800 as well as mixture parameters ∈ 0.05,0.1,0.2,0.4,0.6,0.8,1.0 .The main reason of SLMS-RL1 algorithm is that sign function is utilized to mitigate the GMM impulsive noise.It is worth noting that SLMS-RL1 algorithm may be deteriorated by the enlarging the mixture parameter of impulsive noise.In practical application scenarios, the mixture parameter is very small (less than 0.1).Hence, the proposed reweighted factor for SLMS-RL1 is stable for GMM impulsive-noise.

Conclusions
In this paper, we propose a Monte-Carlo based reweighted factor selection method so that the SLMS-RL1 algorithm can exploit channel sparsity efficiently.Simulation results are provided to illustrate our findings.First of all, SLMS-RL1 can achieve the lowest MSE performance by selecting the reweighted factor as 0.005 in different SNR regimes.Secondly, the convergence speed of SLMS-RL1 can be reduced by increasing the channel sparsity .At last, the steady-state MSE performance of SLMS-RL1 does not change considerably under different GMM impulsive-noise strength .In other words, SLMS-RL1 algorithm using the reweighted factor 0.005 is stable under different GMM impulsive noises.