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Iterative Hard Thresholding with Combined Variable Step Size & Momentum-Based Estimator for Wireless Communication Systems with Dynamic Sparse Channels^{ †}

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## Abstract

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## 1. Introduction

- We propose an efficient dynamic channel estimator that is developed by deriving a combined variable step size mechanism and variable momentum and incorporating this into the traditional iterative hard thresholding. The estimator is named Iterative Hard Thresholding with Combined Variable Step Size and Momentum (IHT-wCVSSnM)-based estimator.
- We present a comparative study of the proposed IHT-wCVSSnM-based estimator with some other estimators when employed in a broadband wireless communication system that is operating in a dynamic sparse wireless channel.

## 2. Wireless System Model with Sparse Channel

## 3. The Proposed IHT-wCVSSnM-Based Estimator for Dynamic Sparse Wireless Channel

## 4. Computer Simulation Results and Computational Complexity Costs

Algorithm 1 Proposed IHT-wCVSSnM-based Estimator for Wireless Communication systems with Dynamic Sparse Channels |

Input:$Y=\left[{y}^{\left[1\right]},\text{}{y}^{\left[2\right]},\dots ,\text{}{y}^{\left[T\right]}\right]$; ${\Psi}^{\left[t\right]}=\Psi =\left[{\phi}_{1},\text{}{\phi}_{2},\dots \text{}{\phi}_{N}\text{}\right]$; $\Im $; $P$; $T$;Itr-Max; $\epsilon ={10}^{-5}$: stopping tolerance; $\mu $: step size Output: Reconstructed sparse channel: $\widehat{H}=\left[{\widehat{h}}^{\left[1\right]},\text{}{\widehat{h}}^{\left[2\right]},\text{}\dots ,\text{}{\widehat{h}}^{\left[T\right]}\right]$Stage 1, Initialization of the Proposed IHT-wCVSSnM-based estimator:${\widehat{h}}_{k=0}^{\left[t\right]}=zeros\left(P,\text{}1\right)$; $k=0$: Iteration counter ${r}^{\left[t\right]\left[k-1\right]}={y}^{\left[t\right]}$: residue error; ${\xi}_{k=0}^{\left[t\right]}=\varnothing $: Path delay support set of ${\widehat{h}}^{\left[t\right]}$ ${\left({\xi}_{k}^{\left[t\right]}\right)}^{c}=ones\left(N,\text{}1\right)$; Stage 2, Iteration section of the Proposed IHT-wCVSSnM-based estimator:for $t=1\text{}to\text{}T$ do1. While $(k<Itr\_Max)$ $k=k+1$2. ${\mathcal{S}}_{k}={\xi}_{k}^{\left[t\right]}\cup supp\left({\wp}_{{\mathfrak{M}}_{P}}\left({\nabla}_{{\xi}_{k}^{c\left[t\right]}}f\left({\widehat{h}}_{k}^{\left[t\right]}\right)\right)\right)$ 3. ${\mu}_{k}=argmi{n}_{\mu}{\Vert f\left({\widehat{h}}_{k}^{\left[t\right]}-0.5\mu {\nabla}_{{\mathcal{S}}_{k}}f\left({\widehat{h}}_{k}^{\left[t\right]}\right)\right)\Vert}_{2}^{2}$ $=argmi{n}_{\mu}{\Vert {y}^{\left[t\right]}-{\Psi}^{\left[t\right]}\left({\widehat{h}}_{k}^{\left[t\right]}-0.5\mu {\nabla}_{{\mathcal{S}}_{k}}f\left({\widehat{h}}_{k}^{\left[t\right]}\right)\right)\Vert}_{2}^{2}$ $=\frac{\Vert {\nabla}_{{\mathcal{S}}_{k}}f\left({\widehat{h}}_{k}^{\left[t\right]}\right){\Vert}_{2}^{2}}{\Vert {\Psi}^{\left[t\right]}{\nabla}_{{\mathcal{S}}_{k}}f\left({\widehat{h}}_{k}^{\left[t\right]}\right){\Vert}_{2}^{2}}$ 4. ${\eta}_{k}=argmi{n}_{\eta}\Vert {y}^{\left[t\right]}-{\Psi}^{\left[t\right]}{\widehat{h}}_{k+1}^{\left[t\right]}{\Vert}_{2}^{2}=\frac{\langle {y}^{\left[t\right]}-{\Psi}^{\left[t\right]}{\widehat{h}}_{k}^{\left[t\right]},\text{}{\Psi}^{\left[t\right]}{\widehat{h}}_{k}^{\left[t\right]}-{\Psi}^{\left[t\right]}{\widehat{h}}_{k-1}^{\left[t\right]}\rangle}{\Vert {\Psi}^{\left[t\right]}{\widehat{h}}_{k}^{\left[t\right]}-{\Psi}^{\left[t\right]}{\widehat{h}}_{k-1}^{\left[t\right]}{\Vert}_{2}^{2}}$ 5. ${z}_{k}^{\left[t\right]}={\wp}_{{\mathfrak{M}}_{P}}\left({\widehat{h}}_{k}^{\left[t\right]}-0.5{\mu}_{k}{\nabla}_{{\mathcal{S}}_{k}}f\left({\widehat{h}}_{k}^{\left[t\right]}\right)\right)$ 6. ${\widehat{h}}_{k+1}^{\left[t\right]}={z}_{k}^{\left[t\right]}+{\eta}_{k}\left({z}_{k}^{\left[t\right]}-{z}_{k-1}^{\left[t\right]}\right)$ 7. ${r}^{\left[t\right]\left[k\right]}\leftarrow {y}^{\left[t\right]}-{\Psi}^{\left[t\right]}{\widehat{h}}_{k+1}^{\left[t\right]}$ Stage 3, Stopping Criterion for the Proposed IHT-wCVSSnM-based estimator:8. Until the stopping criterion is met:9. $\Vert {\widehat{h}}_{k}^{\left[t\right]}-{\widehat{h}}_{k+1}^{\left[t\right]}\Vert \le \epsilon \Vert {\widehat{h}}_{k+1}^{\left[t\right]}\Vert $ or Itr-Max 10. end while11. Return ${\widehat{h}}^{\left[t\right]}={\widehat{h}}_{k+1}^{\left[t\right]}$ |

## 5. Conclusions and Future Research Directions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- He, X.; Song, R.; Zhu, W.-P. Pilot Allocation for Distributed-Compressed-Sensing-Based Sparse Channel Estimation in MIMO-OFDM Systems. IEEE Trans. Veh. Technol.
**2016**, 65, 2990–3004. [Google Scholar] [CrossRef] - Qi, C.; Yue, G.; Wu, L.; Huang, Y.; Nallanathan, A. Pilot Design Schemes for Sparse Channel Estimation in OFDM Systems. IEEE Trans. Veh. Technol.
**2014**, 64, 1493–1505. [Google Scholar] [CrossRef] - Masood, M.; Afify, L.H.; Al-Naffouri, T.Y. Efficient Coordinated Recovery of Sparse Channels in Massive MIMO. IEEE Trans. Signal Process.
**2014**, 63, 104–118. [Google Scholar] [CrossRef] [Green Version] - Candes, E. Compressive Sampling. Int. Congr. Math.
**2006**, 3, 1433–1452. [Google Scholar] - Candes, E.; Tao, T. The Dantzig Selector: Statistical estimation when p is much larger than n. Ann. Stat.
**2007**, 35, 2313–2351. [Google Scholar] - Li, J.; Ai, B.; He, R.; Yang, M.; Zhong, Z.; Hao, Y.; Shi, G. The 3D Spatial Non-Stationarity and Spherical Wavefront in Massive MIMO Channel Measurement. In Proceedings of the 2018 10th International Conference on Wireless Communications and Signal Processing (WCSP), Hangzhou, China, 18–20 October 2018; pp. 1–6. [Google Scholar] [CrossRef]
- Chen, J.; Yin, X.; Cai, X.; Wang, S. Measurement-Based Massive MIMO Channel Modeling for Outdoor LoS and NLoS Environments. IEEE Access
**2017**, 5, 2126–2140. [Google Scholar] [CrossRef] - Zhang, P.; Chen, J.; Yang, X.; Ma, N.; Zhang, Z. Recent Research on Massive MIMO Propagation Channels: A Survey. IEEE Commun. Mag.
**2018**, 56, 22–29. [Google Scholar] [CrossRef] - Rappaport, T.S.; Heath, R.W., Jr.; Daniels, R.C.; Murdock, J.N. Book Millimeter-Wave Wireless Communications; Pearson Education: London, UK, 2014. [Google Scholar]
- Heath, R.W.; Gonzalez-Prelcic, N.; Rangan, S.; Roh, W.; Sayeed, A.M. An Overview of Signal Processing Techniques for Millimeter Wave MIMO Systems. IEEE J. Sel. Top. Signal Process.
**2016**, 10, 436–453. [Google Scholar] [CrossRef] - Kocic, M.; Brady, D.; Merriam, S. Reduced-complexity RLS estimation for shallow-water channels. In Proceedings of the IEEE Symposium on Autonomous Underwater Vehicle Technology, Cambridge, MA, USA, 5–6 June 1994; pp. 165–170. [Google Scholar]
- Cotter, S.F.; Rao, B. Matching pursuit based decision-feedback equalizers. In Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing (Cat. No.00CH37100), Istanbul, Turkey, 19–24 April 2002; Volume 5, pp. 2713–2716. [Google Scholar] [CrossRef]
- Cotter, S.; Rao, B. Sparse channel estimation via matching pursuit with application to equalization. IEEE Trans. Commun.
**2002**, 50, 374–377. [Google Scholar] [CrossRef] [Green Version] - Hu, D.; Wang, X.; He, L. A New Sparse Channel Estimation and Tracking Method for Time-Varying OFDM Systems. IEEE Trans. Veh. Technol.
**2013**, 62, 4648–4653. [Google Scholar] [CrossRef] - Hijazi, H.; Ros, L. Polynomial Estimation of Time-Varying Multipath Gains with Intercarrier Interference Mitigation in OFDM Systems. IEEE Trans. Veh. Technol.
**2009**, 58, 140–151. [Google Scholar] [CrossRef] - Sahoo, S.K.; Makur, A. Signal Recovery from Random Measurements via Extended Orthogonal Matching Pursuit. IEEE Trans. Signal Process.
**2015**, 63, 2572–2581. [Google Scholar] [CrossRef] - Dai, W.; Milenkovic, O. Subspace Pursuit for Compressive Sensing Signal Reconstruction. IEEE Trans. Inf. Theory
**2009**, 55, 2230–2249. [Google Scholar] [CrossRef] [Green Version] - Needell, D.; Tropp, J. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal.
**2009**, 26, 301–321. [Google Scholar] [CrossRef] [Green Version] - Dai, L.; Wang, J.; Wang, Z.; Tsiaflakis, P.; Moonen, M. Spectrum- and Energy-Efficient OFDM Based on Simultaneous Multi-Channel Reconstruction. IEEE Trans. Signal Process.
**2013**, 61, 6047–6059. [Google Scholar] [CrossRef] - Gao, Z.; Dai, L.; Wang, Z. Structured compressive sensing based superimposed pilot design in downlink large-scale MIMO systems. Electron. Lett.
**2014**, 50, 896–898. [Google Scholar] [CrossRef] [Green Version] - Ji, S.; Xue, Y.; Carin, L. Bayesian Compressive Sensing. IEEE Trans. Signal Process.
**2008**, 56, 2346–2356. [Google Scholar] [CrossRef] - Karseras, E.; Leung, K.; Dai, W. Tracking dynamic sparse signals using Hierarchical Bayesian Kalman filters. In Proceedings of the 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, BC, Canada, 26–31 May 2013; pp. 6546–6550. [Google Scholar] [CrossRef] [Green Version]
- Zhu, X.; Dai, L.; Gui, G.; Dai, W.; Wang, Z.; Adachi, F. Structured Matching Pursuit for Reconstruction of Dynamic Sparse Channels. In Proceedings of the 2015 IEEE Global Communications Conference (GLOBECOM), San Diego, CA, USA, 6–10 December 2015; pp. 1–5. [Google Scholar] [CrossRef] [Green Version]
- Zhu, X.; Dai, L.; Dai, W.; Wang, Z.; Moonen, M. Tracking a dynamic sparse channel via differential orthogonal matching pursuit. In Proceedings of the 2015 IEEE Military Communications Conference (MILCOM), Tampa, FL, USA, 26–28 October 2015; pp. 792–797. [Google Scholar] [CrossRef] [Green Version]
- Zhang, X.; Gui, L.; Qin, Q.; Gong, B. Dynamic sparse channel estimation over doubly selective channels: Differential simultaneous orthogonal matching pursuit. In Proceedings of the 2016 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB), Nara, Japan, 1–3 June 2016; pp. 1–6. [Google Scholar] [CrossRef]
- Zhang, X.; Gui, L.; Gong, B.; Xiong, J.; Qin, Q. Dynamic sparse channel estimation over doubly selective channels for large-scale MIMO systems. In Proceedings of the 2017 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB), Cagliari, Italy, 7–9 June 2017; pp. 1–7. [Google Scholar] [CrossRef]
- Lim, S.H.; Choi, J.W.; Shim, B. Greedy Sparse Channel Estimation for Millimeter Wave Communications. In Proceedings of the 2018 IEEE Region 10 Conference (TENCON), Jeju, Korea, 28–31 October 2018; pp. 1628–1632. [Google Scholar] [CrossRef]
- Jiang, W.; Wang, X.; Tong, F. Dynamic compressed sensing estimation of time varying underwater acoustic channel. In Proceedings of the 2017 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), Xiamen, China, 22–25 October 2017; pp. 1–4. [Google Scholar] [CrossRef]
- Carvajal, R.; Godoy, B.I.; Agüero, J.C.; Goodwin, G.C. EM-based sparse channel estimation in OFDM systems. In Proceedings of the 2012 IEEE 13th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Cesme, Turkey, 4–6 November 2015; pp. 530–534. [Google Scholar] [CrossRef]
- Qin, Q.; Gui, L.; Gong, B.; Luo, S. Sparse Channel Estimation for Massive MIMO-OFDM Systems over Time-Varying Channels. IEEE Access
**2018**, 6, 33740–33751. [Google Scholar] [CrossRef] - Buyuksar, A.B.; Senol, H.; Erkucuk, S.; Cirpan, H.A. Data-aided autoregressive sparse channel tracking for OFDM systems. In Proceedings of the 2016 International Symposium on Wireless Communication Systems (ISWCS), Poznan, Poland, 20–23 September 2016; pp. 424–428. [Google Scholar] [CrossRef]
- Beena, A.O.; Pillai, S.S.; Vijayakumar, N. An Improved Adaptive Sparse Channel Estimation Method for Next Generation Wireless Broadband. In Proceedings of the 2018 International Conference on Wireless Communications, Signal Processing and Networking (WiSPNET), Chennai, India, 22–24 March 2018; pp. 1–5. [Google Scholar] [CrossRef]
- Ma, J.; Zhang, S.; Li, H.; Gao, F.; Jin, S. Sparse Bayesian Learning for the Time-Varying Massive MIMO Channels: Acquisition and Tracking. IEEE Trans. Commun.
**2018**, 67, 1925–1938. [Google Scholar] [CrossRef] - Shahmansoori, A. Sparse Bayesian Multi-Task Learning of Time-Varying Massive MIMO Channels with Dynamic Filtering. IEEE Wirel. Commun. Lett.
**2020**, 9, 871–874. [Google Scholar] [CrossRef] [Green Version] - Zhang, M.; Zhou, X.; Wang, C. Time-Varying Sparse Channel Estimation Based on Adaptive Average and MSE Optimal Threshold in STBC MIMO-OFDM Systems. IEEE Access
**2020**, 8, 177874–177895. [Google Scholar] [CrossRef] - Zhang, Y.; Venkatesan, R.; Dobre, O.A.; Li, C. Efficient Estimation and Prediction for Sparse Time-Varying Underwater Acoustic Channels. IEEE J. Ocean. Eng.
**2019**, 45, 1112–1125. [Google Scholar] [CrossRef] - Oyerinde, O.O.; Flizikowski, A.; Marciniak, T. Iterative Hard Thresholding with Memory-based Dynamic Sparse Wireless Channel Estimator. In Proceedings of the 14th International Conference on Signal Processing and Communication Systems (ICSPCS), Adelaide, Australia, 14–16 December 2020; pp. 1–5. [Google Scholar]
- Oyerinde, O.O. An overview of channel estimation schemes based on regularized adaptive algorithms for OFDM-IDMA systems. Digit. Signal Process.
**2018**, 75, 168–183. [Google Scholar] [CrossRef] - Kyrillidis, A.; Cevher, V. Recipes on hard thresholding methods. In Proceedings of the 2011 4th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), San Juan, PR, USA, 13–16 December 2011; pp. 353–356. [Google Scholar] [CrossRef] [Green Version]
- Cevher, V. On accelerated hard thresholding methods for sparse approximation. In Wavelets and Sparsity XIV; International Society for Optics and Photonics: Bellingham, WA, USA, 2011; Volume 8138, pp. 1–6. [Google Scholar]
- Vithanage, C.; Denic, S.; Sandell, M. Robust Linear Channel Estimation Methods for Per-Subcarrier Transmit Antenna Selection. IEEE Trans. Commun.
**2011**, 59, 2018–2028. [Google Scholar] [CrossRef]

**Figure 1.**Typical Sample Spaced Sparse Multipath Channel. Reprinted with permission from [38]. Copyright 2018 Elsevier.

**Figure 2.**Achievable comparative MSE versus signal to noise ratio (SNR) performance of the proposed IHT-wCVSSnM-based estimator with other channel estimators when the degree of temporal correlation is set as $L=8$.

**Figure 3.**Achievable comparative MSE versus temporal correlation degree of dynamic sparse channel performance of the proposed IHT-wCVSSnM-based estimator with other channel estimators (

**a**) SNR = 5 dB, (

**b**) SNR = 15 dB.

**Figure 4.**CPU running time versus temporal correlation degree of dynamic sparse channel comparison for all the channel estimators (

**a**) SNR = 5 dB, (

**b**) SNR = 15 dB.

Reference | Contributions |
---|---|

[11] | The paper focused on estimation technique for underwater acoustic (UWA) channels where the channels impulses’ responses are said to possess large delays and Doppler spreads with few significant echoes. The proposed method (a flexible complexity-based recursive least-squares-based estimator) employed single-carrier waveforms for the estimation of doubly-selective single antenna channels, while neglecting the weakest taps. |

[12] | The problem of equalization of sparse channels with large delay was addressed in this paper. By taking advantage of the prior knowledge of the sparsity of the channel the authors proposed the use of a Matching Pursuit (MP) algorithm to obtain the nonzero weights in the channel response of the system. |

[13] | The papers considered communication problems that involve the estimation and equalization of channels with a large delay spread but with small nonzero support. By exploiting the sparse nature of the channel through the use of a matching pursuit (MP) algorithm, the authors developed a technique of obtaining near to accurate channel estimates for the system. |

[14] | The authors considered orthogonal frequency-division multiplexing (OFDM) system that is based on a dynamic parametric channel model. The channel model is assumed to be parameterized by a small number of distinct paths that are characterized by time-varying path delay and path gain. For this system, the authors proposed a sparse channel estimation and tracking method using the polynomial basis expansion model of [15]. |

[16] | To recover a d-dimensional m-sparse signal with high probability, the author proposed an extended Orthogonal Matching Pursuit (OMP)-based channel estimationa scheme that brings the required number of measurements for OMP closer to Basic Pursuit (BP)-based estimator. |

[17] | The reconstruction of sparse signals with and without noisy perturbations was considered in this paper. To achieve this, the authors developed a recovery technique for sparse signals sampled by employing the subspace pursuit (SP) algorithm. |

[18] | The authors proposed a signal reconstruction algorithm that is based on the OMP algorithm. The new algorithm is called CoSaMP and it incorporates several other ideas from some other previous works. The new ideas were incorporated to both accelerate the algorithm and provide strong guarantees that OMP failed to provide. |

[19] | The authors proposed a structured compressive sensing (SCS)-aided time-domain synchronous-OFDM scheme using wireless channel properties. These properties include the channel sparsity and the slow time-varying path delays which are usually not considered in conventional OFDM schemes. |

[20] | In this paper, the author proposed structured subspace pursuit (SSP) algorithm to simultaneously recover multiple channels with low pilot overhead, in comparison with other methods, in downlink large-scale MIMO systems. |

[21] | Bayesian compressive sensing technique was proposed for the reconstruction of compressible signals on some linear basis. From this, the reconstruction of the signal can be executed accurately using only a small number of basis-function coefficients associated with the linear basis. |

[22] | The reconstruction of time-varying signals for which the support is assumed to be sparse is considered in this paper. A Hierarchical Bayesian Kalman (HBK) filter-based estimator is used by the authors for the reconstruction of a time-varying sparse channel. |

[23] | The authors proposed a structured compressive sensing algorithm named structured matching pursuit (SMP) for the reconstruction of dynamic sparse channels in broadband wireless communication systems. This is achieved by using temporal correlations associated with time-varying sparse channels for the reconstruction. |

[24] | The tracking of a dynamic sparse channel in a broadband wireless communication system was considered in this paper. The authors proposed a dynamic CS algorithm named differential orthogonal matching pursuit (D-OMP) based on the standard OMP algorithm to track a dynamic sparse channel. |

[25] | The authors proposed a differential simultaneous orthogonal matching pursuit (DSOMP) algorithm-based joint multi-symbol channel estimation to estimate dynamic channel parameters. The authors took advantage of the complex exponential basis expansion model (CE-BEM) in the time domain and exploiting the channel sparsity in the delay domain. |

[26] | Differential block simultaneous orthogonal matching pursuit (DBSOMP) algorithm based on jointly sparsity in different complex basis expansion mode (CE-BEM) order was proposed to estimate a dynamic sparse channel in the massive MIMO systems with better recovery performance and lower computational complexity. |

[27] | A new greedy channel estimation technique that is capable of tracking dynamic sparse signals is proposed for millimeter-wave (mmWave) communication systems. Some of the signals tracked by estimation techniques include time-varying angle of departure (AoD), angle of arrival (AoA), and channel gain amplitudes of mmWave channel. |

[28] | Kalman Filtered Compressed Sensing (KF-CS) estimation of the time-varying underwater acoustic channel is studied in this paper. The authors modeled the time-varying underwater acoustic (UWA) channels as sparse. This consists of both constant and time-varying supports. The KF-CS-based estimator is then employed to enhance the underwater acoustic communication systems’ performance. |

[29] | The problem of estimating sparse communication channels in the OFDM system is considered in this paper. The modified likelihood function’s maximization for the system is performed with the aid of the Expectation–Maximization (EM) algorithm. |

[30] | By exploiting the sparsity in the delay domain, and high correlation in the spatial domain massive MIMO-OFDM systems, the authors proposed a sparse channel estimation scheme for massive MIMO-OFDM downlink transmission over time-varying channels. Specifically, the authors employed a quasi-block simultaneous orthogonal matching pursuit (QBSO) algorithm for the proposed sparse channel estimation scheme. |

[31] | Space Alternated Generalized Expectation Maximization Maximum a Posteriori(SAGE-MAP)-based channel estimator is proposed for tracking sparse channels of Orthogonal Frequency Division Multiplexing (OFDM) systems. The channel model considered is the Autoregressive (AR) modeled time-varying sparse channels. The technique employed is based on amended mean square error (MSE) optimal threshold (IMOT) and adaptive multi-frame averaging (AMA) schemes |

[32] | An Adaptive Channel Estimation (ACE) technique that exploited the sparsity in time-varying broadband wireless channels is proposed. The estimator is named Variable Step Size Sign Data Sign Error NLMS (VSS-SDSENLMS)-based estimator and it is used to track sparse channels in the considered system. |

[33] | Both downlink (DL) and uplink (UL) channel estimation for the time-varying massive MIMO networks is studied in this paper. An expectation maximization-based sparse Bayesian learning framework is developed to learn the model parameters of the sparse virtual channel. |

[34] | The authors investigated the estimation of the sparse multi-user massive MIMO channels via multi-task (MT)-sparse Bayesian learning (SBL) that is employed in learning dynamic sparse channels in the uplink paths of multi-user massive MIMO-OFDM systems. Specifically, the dynamic information of the sparse channel is used to initialize the hyper-parameters in the multi-task (MT)-sparse Bayesian learning (SBL) MT-SBL procedure for the next time step. |

[35] | A new method for channel estimation in space-time block coding (STBC) multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. This is achieved by using the sparsity and the inherent temporal correlation of the time-varying wireless channel. Specifically, an adaptive multi-frame averaging (AMA) and improved mean square error (MSE) optimal threshold (IMOT)-based channel estimation method is proposed by the authors. |

[36] | The authors investigated the estimation and prediction of the sparse time-varying channel in underwater acoustic (UWA) communication systems, in which they proposed a decision-directed-based sparse adaptive predictor that works in the delay-Doppler domain for dynamic UWA channels. The proposed technique extrapolates the channel knowledge estimated from a block of training symbols, and the predicted channel is used to decode consecutive data blocks. |

[37] | The authors proposed a channel estimator that is based on iterative hard thresholding (IHT) algorithm. This is achieved by using the temporal correlation that is associated with the dynamic sparse wireless channel. The proposed estimator is named the iterative hard thresholding with memory (IHT-wM)-based estimator. The proposed estimator also employs its memory term to enhance channel estimation procedure in the system. |

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**MDPI and ACS Style**

Oyerinde, O.O.; Flizikowski, A.; Marciniak, T.
Iterative Hard Thresholding with Combined Variable Step Size & Momentum-Based Estimator for Wireless Communication Systems with Dynamic Sparse Channels. *Electronics* **2021**, *10*, 842.
https://doi.org/10.3390/electronics10070842

**AMA Style**

Oyerinde OO, Flizikowski A, Marciniak T.
Iterative Hard Thresholding with Combined Variable Step Size & Momentum-Based Estimator for Wireless Communication Systems with Dynamic Sparse Channels. *Electronics*. 2021; 10(7):842.
https://doi.org/10.3390/electronics10070842

**Chicago/Turabian Style**

Oyerinde, Olutayo Oyeyemi, Adam Flizikowski, and Tomasz Marciniak.
2021. "Iterative Hard Thresholding with Combined Variable Step Size & Momentum-Based Estimator for Wireless Communication Systems with Dynamic Sparse Channels" *Electronics* 10, no. 7: 842.
https://doi.org/10.3390/electronics10070842