Block Sparse Bayesian Learning Research Articles

Estimating the direction of arrival of spatially spread sources using block-sparse Bayesian learning with an extended dictionary.

Estimating the direction of arrival (DOA) of spatially spread sources is a significant challenge in array signal processing. This work introduces an effective method within the sparse Bayesian framework to tackle this issue. A spatially spread source is modeled using a multi-dimensional Slepian signal subspace that expands the dictionary and results in a block-sparse structured solution. By taking advantage of block-sparse Bayesian learning, parameter estimation becomes feasible. A complex Gaussian posterior is derived under a multi-snapshot block-sparse framework with a complex Gaussian prior and varying noise conditions. The hyperparameters are estimated using the expectation-maximization algorithm. Through numerical tests and sea test data evaluations, the proposed method shows superior energy focusing for spatially spread signals. Under limited snapshots and challenging signal-to-noise ratios, the current method can still offer precise DOA determination for spatially spread sources.

Sound field reconstruction using sparse Bayesian learning equivalent source method with hyperparametric-coupled prior

Block sparse Bayesian learning (SBL) equivalent source method (ESM) suffers from the block structure mismatch problem that leads to poor sound field reconstruction accuracy for block sparse sources. To address this issue, this paper proposes to employ a hyperparametric-coupled prior in the SBL-ESM. In the method, a hierarchical SBL model is established based on the ESM, and then the coupling constraint is imposed to the hyper-prior of equivalent source strengths by a projection transformation. The proposed method can produce a coupling effect, such that the sparsity of equivalent source strengths is controlled not only by each corresponding hyper-parameter, but also by their immediate neighbors. Therefore, it enables to promote block sparse solutions and reconstruct the sound fields of block sparse sources well without the prior information of block partition. Numerical simulations of linear-block sources are used to verify the effectiveness and superiority of the proposed method, and the effects of sound power ratio, signal-to-noise ratio and reconstruction distance on the reconstructed results are investigated. Subsequently, the performance of the proposed method is further verified through an experiment of a composite loudspeaker and an experiment of the mixed sources consisting of a small loudspeaker and an impacted steel plate.

Fast electrical impedance tomography based on sparse Bayesian learning

Electrical impedance tomography (EIT) is a severely under-determined and ill-posed inverse problem. Therefore, based on the compressed sensing theory and the block sparse Bayesian learning (BSBL) model, a novel accelerated EIT sparse imaging algorithm is presented. The key feature of the proposed algorithm is that the convex–concave procedure (CCP) method is used to optimize the non-convex cost function of the block sparse Bayesian learning model which improves the convergence speed and reduces the time complexity of the algorithm. The proposed method can adaptively explore and utilize the inter-block sparsity and intra-block structural correlation of non-sparse signals without any a priori information, thereby sparse imaging is performed on non-sparse physiological data effectively. The results comparing with other methods show that the proposed algorithm not only reconstruct inclusions with different shapes and conductivity applicably and effectively but also reduce the running time of the algorithm.

Improved block sparse Bayesian learning based DOA estimation for massive MIMO systems

Direction-of-arrival (DOA) estimation is crucial for suppressing inter-cell interference in massive multiple-input multiple-output (MIMO) systems. However, conventional algorithms are unsuitable for low signal-to-noise ratio (SNR) and a small number of snapshots. Consequently, in an attempt to achieve fast reconstruction speed while ensuring high accuracy, this study proposes an improved block sparse Bayesian learning-based DOA estimation algorithm for two-dimension massive MIMO systems. Using the sparse characteristics and inherent structure of the signal, the DOA estimation problem is transformed to a joint task of sparse signal reconstruction and parameter optimization. Subsequently, the estimation accuracy is improved through complete excavation and learning of the signal temporal structure. In addition, to avoid the direct solution of the posterior probability distribution, reduce the number of iterations and increase the reconstruction speed, variational inference is used to approximate the Bayesian model and solve the joint problem through parameter learning. Moreover, the auxiliary angle is introduced to avoid extra angle matching, which further reduces the complexity. Simulations are performed to verify the proposed method and the results prove its superiority when applied to massive MIMO systems compared with some existing DOA estimation algorithms.

Joint AMP-SBL Algorithms for Device Activity Detection and Channel Estimation in Massive MIMO mMTC Systems

We consider the problem of detecting active devices and estimating their channels in the uplink of a massive machine type communication (mMTC) network. The base station (BS) in this mMTC system is equipped with massive multiple input multiple output (mMIMO) technology, and the channels across its antennas are correlated, an aspect ignored by most of the existing mMTC works. We propose three Bayesian learning algorithms which exploit channel spatial correlation, and comprehensively outperform several existing state-of-the-art algorithms, which do not exploit it. The proposed algorithms perform better in terms of the normalized mean squared error, activity error rate, and spectral efficiency (SE). The first correlated vector approximate message passing (AMP) algorithm has the best performance, but fails when the BS does not have knowledge of channel large scale fading and device activity probability. The second block sparse Bayesian learning (B-SBL) algorithm overcomes these limitations, but has a high complexity. The third combined AMP-BSBL algorithm retains the advantages of B-SBL, but with a much reduced complexity. We show that all three channel estimators can be represented as plug-in linear minimum mean squared estimators. This crucially helps us in deriving a common lower bound on the SE of mMIMO mMTC systems.

Covariance-Free Variational Bayesian Learning for Correlated Block Sparse Signals

We consider the problem of estimating channel in massive machine type communication (mMTC) systems. The sparse device activity in a mMTC system makes the channel block-sparse, with intra-block correlation. Block-sparse Bayesian learning (B-SBL) is a powerful framework for estimating such signals. The existing B-SBL algorithms become computationally expensive for high-dimensional problems, which is common in mMTC systems. This is because of large number of devices in a mMTC system, they invert a large-dimensional matrix to calculate the covariance matrix. To address this problem, we exploit variational Bayesian inference, and design a novel covariance-free variational B-SBL algorithm which inverts multiple small-sized block matrices, instead of inverting a complete big-sized matrix. The complexity is further reduced by avoiding explicit computation of the covariance matrix. The proposed algorithm, instead of performing costly matrix inversions, solves multiple linear systems to calculate an unbiased estimate of the posterior statistics. The proposed algorithm is numerically shown to estimate the mMTC channel with a much lesser complexity, and that too without compromising the reconstruction performance.

Probability-Based Diagnostic Imaging of Fatigue Damage in Carbon Fiber Composites Using Sparse Representation of Lamb Waves

Carbon fiber composites are commonly used in aerospace and other fields due to their excellent properties, and fatigue damage will occur in the process of service. Damage imaging can be performed using damage probability imaging methods to obtain the fatigue damage condition of carbon fiber composites. At present, the damage factor commonly used in the damage probability imaging algorithm has low contrast and poor anti-noise performance, which leads to artifacts in the imaging and misjudgment of the damaged area. Therefore, this paper proposes a fatigue damage probability imaging method for carbon fiber composite materials based on the sparse representation of Lamb wave signals. Based on constructing the Lamb wave dictionary, a fast block sparse Bayesian learning algorithm is used to represent the Lamb wave signals sparsely, and the definition of Lamb wave sparse representing the damage factor calculates the damage probability of the monitoring area and then images the fatigue damage of the carbon fiber composite materials. The imaging research was carried out using the fatigue monitoring experiment data of NASA’s carbon fiber composite materials. The results show that the proposed damage factor can clearly distinguish the damaged area from the undamaged area and has strong noise immunity. Compared with the energy damage factor and the cross-correlation damage factor, the error percentages are reduced by at least 58.63%, 28.11%, and 8.43% for signal-to-noise ratios of 6 dB, 3 dB, and 0.1 dB, respectively, after adding noise to the signal. The results can more accurately reflect the real location and area of fatigue damage in carbon fiber composites.

Correlated Bayesian Co-Training for Virtual Metrology

A rising challenge in manufacturing data analysis is training robust regression models using limited labeled data. In this work, we investigate a semi-supervised regression scenario, where a manufacturing process operates on multiple mutually correlated states. We exploit this inter-state correlation to improve regression accuracy by developing a novel co-training method, namely Correlated Bayesian Co-training (CBCT). CBCT adopts a block Sparse Bayesian Learning framework to enhance multiple individual regression models which share the same support. Additionally, CBCT casts a unified prior distribution on both the coefficient magnitude and the inter-state correlation. The model parameters are estimated using maximum-a-posteriori estimation (MAP), while hyper-parameters are estimated using the expectation-maximization (EM) algorithm. Experimental results from two industrial examples shows that CBCT successfully leverages inter-state correlation to reduce the modeling error by up to 79.40%, compared to other conventional approaches. This suggests that CBCT is of great value to multi-state manufacturing applications.

BSBL-Based Auxiliary Vehicle Position Analysis in Smart City Using Distributed MEC and UAV-Deployed IoT

Smart city enters the new 3.0 era, and Internet of Things (IoT) perform as the urban neural network in smart city. In the industrial areas of smart city, the IoT focuses on industrial applications, such as logistics and environmental monitoring. In this work, an auxiliary position analysis framework composed of IoT and distributed mobile-edge computing (MEC) is proposed to analyze the position of vehicles in the industrial areas of smart city. In the proposed framework, IoT are deployed by multiple unmanned aerial vehicles (UAVs) equipped with uniform linear array (ULA), which receives the signal emitted by vehicles for obtaining the direction of arrival (DOA), and the distributed MEC provides computing and synchronization services to auxiliary positioning. The DOA estimation is a key issue for auxiliary positioning in the proposed framework. To realize DOA estimation with unknown mutual coupling (MC) existing in IoT nodes, a novel block sparse Bayesian learning (SBL) algorithm is developed. In the developed algorithm, the unknown MC existing between sensors in each IoT nodes is first fused with the signal by parameterizing steering vector. Then, a block SBL (BSBL) procedure is presented to perform DOA estimation by using the inherent block sparse structure in the equivalent signal obtained after fusion. Benefiting from the fusion of MC and signal, the developed DOA estimation algorithm does not require a separate estimation of the unknown MC and also does not cause the loss of the array aperture. Based on the DOA estimation information, the position of vehicles in the industrial environment is effectively analyzed through weighted multiple cross-locations. Synthetic data set simulation is carried out to verify that the vehicle positions in smart city can be efficiently analyzed and estimated based on the presented framework and algorithm.

Bayesian Image Reconstruction Using Weighted Laplace Prior for Lung Respiratory Monitoring With Electrical Impedance Tomography

The poor spatial resolution of electrical impedance tomography (EIT) limits its use in medical devices because of its highly nonlinear and ill-posed nature. A novel hierarchical block sparse Bayesian learning (BSBL) method is designed for lung respiratory monitoring with EIT. It is its excellent modeling capability and noise robustness that allows BSBL to adaptively explore and exploit the internal conductivity distribution, e.g., block sparsity and intra-block correlation. First, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -nearest neighbor (KNN) strategy is employed to automatically group each block based on the clustering property and incorporated to constrain the intra-block correlation matrix based on the spatial distance. Second, a three-layer hierarchical BSBL model using weighted Laplace (WL) prior is considered to enhance the recovery performance. Finally, an efficient bound optimization (BO) method is performed for Bayesian inference, avoiding the tedious parameter adjustment. This leads to improvements in recovery performance, algorithm robustness, and computational efficiency. Moreover, root mean square error (RMSE), image correlation coefficient (ICC), and relative size coverage ratio (RCR) are applied for quantitative comparisons of image quality. Numerical simulations and in vivo lung respiratory experiments showed that the proposed KNN-BSBL-WL method outperforms existing referenced methods in terms of reconstruction accuracy and computational time. On average, KNN-BSBL-WL obtains the RMSE <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$= 0.07\,\,\pm \,\,0.02$ </tex-math></inline-formula> , ICC <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$= 0.96\,\,\pm \,\,0.01$ </tex-math></inline-formula> , and RCR <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$= 0.98\,\,\pm \,\,0.05$ </tex-math></inline-formula> , which are closest to the true values of the recorded snapshots obtained in the experiments. Meanwhile, the average time of KNN-BSBL-WL is 0.16 s, achieving an acceleration ratio close to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 3$ </tex-math></inline-formula> compared with the referenced KNN-BSBL method. Therefore, the proposed method is an efficient and robust means of imaging reconstruction, which further improves the feasibility of EIT in respiratory clinical applications.

Block Sparse Bayesian Learning Based Joint User Activity Detection and Channel Estimation in Grant-Free MIMO-NOMA

In the massive machine type of communication (mMTC), grant-free non-orthogonal multiple access (NOMA) is receiving more and more attention because it can skip the complex grant process to allocate non-orthogonal resources to serve more users. To address the limited wireless resources and substantial connection challenges, combining grant-free NOMA and multiple-input multiple-output (MIMO) is crucial to further improve the system’s capacity. In the grant-free MIMO-NOMA system, the base station should obtain the relevant information of the user before data detection. Thus, user activity detection (UAD) and channel estimation (CE) are two problems that should be solved urgently. In this paper, we fully consider the sparse characteristics of signals and the spatial correlation between multiple antennas in the grant-free MIMO-NOMA system. Then, we propose a spatial correlation block sparse Bayesian learning (SC-BSBL) algorithm to address the joint UAD and CE problems. First, by fully mining the block sparsity of signals in the grant-free MIMO-NOMA system, we model the joint UAD and CE problem as a three-dimensional block sparse signal recovery problem. Second, we derive the cost function based on the hierarchical Bayesian theory and spatial correlation. Finally, to estimate the channel and the set of active users, we optimize the cost function with fast marginal likelihood maximization. The simulation results indicate that, compared with the existing algorithms, SC-BSBL can always fully use the signal sparsity and spatial correlation to accurately complete UAD and CE under various user activation probabilities, SNRs, and the number of antennas. The normalized mean square error of CE can be reduced to 0.01, and the UAD error rate can be less than 10−5.

OTFS Transceiver Design and Sparse Doubly-Selective CSI Estimation in Analog and Hybrid Beamforming Aided mmWave MIMO Systems

Orthogonal time frequency space (OTFS) waveform based millimeter wave (mmWave) MIMO systems are capable of achieving high data rates in high-mobility scenarios. Hence, transceivers are designed for both analog beamforming (AB) and hybrid beamforming (HB), where we commence by deriving the delay-Doppler (DD)-domain input-output relationship considering a delay-Doppler-angular domain channel model. Subsequently, a novel two-stage procedure is developed for transmit beamformer (TBF)/ precoder (TPC) and receiver combiner (RC) design, and for estimating the DD-domain’s equivalent channel state information (CSI). The key feature of the proposed framework is that the RF TBF/ TPC and RC design maximizes the directional beamforming gains. It is also demonstrated that the low-dimensional baseband CSI of the DD-domain becomes sparse for mmWave-AB MIMO OTFS systems, and block-sparse for mmWave-HB MIMO OTFS systems. Subsequently, Bayesian learning (BL) and block-sparse BL (BS-BL) solutions are developed for improved CSI estimation. We also derive the Bayesian Cramer-Rao lower bounds (BCRLB) for benchmarking the mean-squared-error (MSE) of the CSI estimates. Finally, our simulation results demonstrate the improved efficacy of the proposed transceiver designs and confirm the enhanced CSI estimation performance of the BL-based schemes over other competing sparse signal recovery schemes.

DOA Estimation for Massive MIMO Systems with Unknown Mutual Coupling Based on Block Sparse Bayesian Learning.

Obtaining accurate angle parameters using direction-of-arrival (DOA) estimation algorithms is crucial for acquiring channel state information (CSI) in massive multiple-input multiple-output (MIMO) systems. However, the performance of the existing algorithms deteriorates severely due to mutual coupling between antenna elements in practical engineering. Therefore, for solving the array mutual coupling, the array output signal vector is modeled by mutual coupling coefficients and the DOA estimation problem is transformed into block sparse signal reconstruction and parameter optimization in this paper. Then, a novel sparse Bayesian learning (SBL)-based algorithm is proposed, in which the expectation-maximum (EM) algorithm is used to estimate the unknown parameters iteratively, and the convergence speed of the algorithm is enhanced by utilizing the approximate approximation. Moreover, considering the off-grid error caused by discretization processes, the grid refinement is carried out using the polynomial roots to realize the dynamic update of the grid points, so as to improve the DOA estimation accuracy. Simulation results show that compared with the existing algorithms, the proposed algorithm is more robust to mutual coupling and off-grid error and can obtain better estimation performance.

Application of Intelligent Signal Reflection in the Communication Model of an Electronic Controller

This paper analyses the automatic control effect of an electronic controller, combined with artificial intelligence technology in order to improve the automatic control of an electronic controller. In order to improve the automatic control effect of the electronic controller, a novel centralized intelligent reflective surface-assisted millimeter-wave computational imaging scheme based on pixel block division and block sparse signal recovery is proposed in this paper. Meanwhile, this paper proposes a fast block-sparse Bayesian learning algorithm. It combines the GAMP algorithm with machine learning, so it can achieve similar performance with much lower computational complexity than the traditional block sparse Bayesian learning algorithm. The simulation clustering study shows that the electronic controller automation system based on artificial intelligence technology proposed in this paper can effectively improve the control effect of the electronic controller.

Normal mode extraction using sparse Bayesian learning in shallow water

The low-frequency signals propagating in the shallow-water waveguides are dispersive. They are composed of several normal modes according to the normal mode theory. For the vertical array data, the horizontal wavenumbers and the associated multi-frequency modal depth functions were estimated using block sparse Bayesian learning (Niu et al., JASA 2020), while a priori knowledge of sea bottom, moving source, and source locations is not needed. For the impulsive or known-form signals received by one hydrophone, the sparse Bayesian learning (SBL) approach can be also used to extract the modes (Niu et al., JASA 2021). It uses the approximate modal dispersion relation, connecting the horizontal wavenumbers (phase velocities) for multiple frequencies, to build the dictionary matrix for SBL. Different from warping transforms based on the group slowness (or group speed) dispersion curves, mode separation using SBL is performed in frequency domain based on the phase speed (or equivalently horizontal wavenumbers) dispersion relation. The simulation results demonstrate that the proposed approach is adapted to the environment where both the reflected and refracted modes coexist, whereas the performance of the time warping transformation degrades significantly in this scenario.

Broadband Off-Grid DOA Estimation Using Block Sparse Bayesian Learning for Nonuniform Noise Variance

In this article, an extended block sparse Bayesian learning (SBL) is applied to a linear system for estimating the direction of arrival (DOA). DOA estimation using conventional SBL deteriorates when inconsistent noise occurs (e.g., some sensors in the array are malfunctioning). The SBL is extended to accommodate the nonuniform noise variance and shows superior robustness in DOA estimation owing to its selective usage of sensors under normal conditions. Furthermore, a block SBL framework is adopted to exploit the commonality of DOAs over broadband signal frequency components and their correlation at the array. The transformation matrix in the linear system is composed of replicas for preset arrival angles as its basis. Practically, the DOAs deviate from the on-grid angles in the matrix, which results in degradation by basis mismatch. An off-grid SBL incorporating an inconsistent noise variance and block sparsity is formulated to estimate the DOAs reliably. The extended SBL is applied to simulated and measured data (SWellEx-96), and its results are compared with those from conventional approaches. It exhibits better performance in terms of resolution and denoising. In particular, when a part of the array is malfunctioning, the extended SBL can still estimate the DOAs clearly, whereas other approaches fail.

A Regional Smoothing Block Sparse Bayesian Learning Method With Temporal Correlation for Channel Selection in P300 Speller.

The P300-based brain–computer interfaces (BCIs) enable participants to communicate by decoding the electroencephalography (EEG) signal. Different regions of the brain correspond to various mental activities. Therefore, removing weak task-relevant and noisy channels through channel selection is necessary when decoding a specific type of activity from EEG. It can improve the recognition accuracy and reduce the training time of the subsequent models. This study proposes a novel block sparse Bayesian-based channel selection method for the P300 speller. In this method, we introduce block sparse Bayesian learning (BSBL) into the channel selection of P300 BCI for the first time and propose a regional smoothing BSBL (RSBSBL) by combining the spatial distribution properties of EEG. The RSBSBL can determine the number of channels adaptively. To ensure practicality, we design an automatic selection iteration strategy model to reduce the time cost caused by the inverse operation of the large-size matrix. We verified the proposed method on two public P300 datasets and on our collected datasets. The experimental results show that the proposed method can remove the inferior channels and work with the classifier to obtain high-classification accuracy. Hence, RSBSBL has tremendous potential for channel selection in P300 tasks.

Nonnegative block-sparse Bayesian learning algorithm for EEG brain source localization

Localizing electrical sources on the cortex surface from scalp recorded electroencephalogram (EEG) is challenging, due to ill-posed problem, noises/artifacts contamination, etc. Applying sparse Bayesian learning (SBL) in this field can automatically give sparse solution to ill-posed problem, while most of the SBL-based algorithms require precise or estimated version of noise statistics information. As EEG signals are more likely to stem from locally synchronized neural masses, modeling source block-sparsity on the cortex surface would bring benefits. In this paper, we develop an EEG brain source localization algorithm in SBL framework, with innovative modeling at sensor level as well as source level. For sensor-level modeling, the distribution of sample covariance matrix of multi-electrode measurements is considered, to circumvent the requirement of noise covariance matrix information. The innovation of source-level modeling is that, with block-sparsity prior used, the block-sparse signal reconstruction problem is transformed to an atom-sparse one, in which variance parameters of brain regions are to be estimated. As these parameters are nonnegative, their priors are modeled by nonnegative Gaussian, which was neglected by previous studies, and ultimately a nonnegative block-SBL (NNBSBL) algorithm is proposed in expectation–maximization (EM) approach. Simulations demonstrate that the proposed NNBSBL algorithm has excellent performance in variations of source number, source locations, homoscedastic/heteroscedastic noises, signal-to-noise ratio (SNR), and number of samples, compared to benchmark and state-of-the-art algorithms. The performance of the proposed algorithm is also evaluated through real P300 EEG data, which is proved consistent with the conclusions of P300 source locations in literature.

Compression Reconstruction and Fault Diagnosis of Diesel Engine Vibration Signal Based on Optimizing Block Sparse Bayesian Learning.

It is critical to deploy wireless data transmission technologies remotely, in real-time, to monitor the health state of diesel engines dynamically. The usual approach to data compression is to collect data first, then compress it; however, we cannot ensure the correctness and efficiency of the data. Based on sparse Bayesian optimization block learning, this research provides a method for compression reconstruction and fault diagnostics of diesel engine vibration data. This method’s essential contribution is combining compressive sensing technology with fault diagnosis. To achieve a better diagnosis effect, we can effectively improve the wireless transmission efficiency of the vibration signal. First, the dictionary is dynamically updated by learning the dictionary using singular value decomposition to produce the ideal sparse form. Second, a block sparse Bayesian learning boundary optimization approach is utilized to recover structured non-sparse signals rapidly. A detailed assessment index of the data compression effect is created. Finally, the experimental findings reveal that the approach provided in this study outperforms standard compression methods in terms of compression efficiency and accuracy and its ability to produce the desired fault diagnostic effect, proving the usefulness of the proposed method.

Sound field reconstruction using block sparse Bayesian learning equivalent source method.

Nearfield acoustic holography based on the compressed sensing theory can realize the accurate reconstruction of sound fields with fewer measurement points on the premise that an appropriate sparse basis is obtained. However, for different types of sound sources, the appropriate sparse bases are diverse and should be constructed elaborately. In this paper, a block sparse Bayesian learning (SBL) equivalent source method is proposed for realizing the reconstruction of the sound fields radiated by different types of sources, including the spatially sparse sources, the spatially extended sources, and the mixed ones of the above two, without the elaborate construction of the sparse basis. The proposed method constructs a block sparse equivalent source model and promotes a block sparse solution by imposing a structured prior on the equivalent source model and estimating the posterior of the model by using the SBL, which can achieve the accurate reconstruction of the radiated sound fields of different types of sources simply by adjusting the block size. Numerical simulation and experimental results demonstrate the validity and superiority of the proposed method, and the effects of two key parameters, the block size, and sparsity pruning threshold value are investigated through simulations.