# A Robust Semi-Blind Receiver for Joint Symbol and Channel Parameter Estimation in Multiple-Antenna Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Tucker Model

## 3. System Model

#### 3.1. Constructed Tucker-2 Model

#### 3.2. Uniqueness Issue

#### 3.3. Identifiability Conditions

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**1.**

## 4. Semi-Blind Receiver

Algorithm 1 The optimized LM algorithm |

First stage: • Compute the LS estimate of $\mathbf{Z}$: $\mathbf{Z}={\mathbf{Y}}_{3}\phantom{\rule{0.277778em}{0ex}}{\left({\mathbf{F}}_{3}\right)}^{\u2020}$; • Rearrange $\mathbf{Z}$ to a rank-one matrix $\Xi $; • Apply the SVD on $\Xi $: $SVD\left(\Xi \right)=\mathbf{U}\Sigma {\mathbf{V}}^{H}$; • Calculate initialization matrices ${\mathbf{S}}^{\left(0\right)}$ and ${\mathbf{H}}^{\left(0\right)}$: ${\mathbf{S}}^{\left(0\right)}=unvec\left({{\mathbf{V}}_{\xb71}^{*}/\phantom{{\mathbf{V}}_{\xb71}^{*}v}\phantom{\rule{0.0pt}{0ex}}v}_{1,1}^{*}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\mathbf{H}}^{\left(0\right)}=unvec\left({\sigma}_{1}{\mathbf{U}}_{\xb71}{v}_{1,1}^{*}\right)$. Second stage: Initialization: Initialize ${\mathbf{u}}^{\left(0\right)}={\left[{\mathbf{u}}_{{\widehat{\mathbf{S}}}^{\left(0\right)}}^{T},{\mathbf{u}}_{{\widehat{\mathbf{H}}}^{\left(0\right)}}^{T}\right]}^{T}$, ${\lambda}^{\left(0\right)}=\mathrm{max}\left(\mathrm{diag}\left({\mathbf{J}}^{\left(0\right)H}{\mathbf{J}}^{\left(0\right)}\right)\right)$ and $\tau =2$; set $\epsilon \phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}{10}^{-5}$ and $i=1$; while$\left|\varphi \left({\mathbf{u}}^{\left(i\right)}\right)-\phantom{\rule{0.166667em}{0ex}}\varphi \left({\mathbf{u}}^{\left(i-1\right)}\right)\right|/\phantom{\left|\varphi \left({\mathbf{u}}^{\left(i\right)}\right)\phantom{\rule{4.pt}{0ex}}-\phantom{\rule{4.pt}{0ex}}\varphi \left({\mathbf{u}}^{\left(i-1\right)}\right)\right|\left|\varphi \left({\mathbf{u}}^{\left(i\right)}\right)\right|}\phantom{\rule{0.0pt}{0ex}}\left|\varphi \left({\mathbf{u}}^{\left(i\right)}\right)\right|\u2a7e\epsilon $doStep 1. Compute ${{\mathbf{J}}^{\left(i\right)}}^{H}{\mathbf{J}}^{\left(i\right)}$ and ${\mathbf{g}}^{\left(i\right)}$ respectively;Step 2. Compute $\Delta {\mathbf{u}}^{\left(i\right)}$: $\Delta {\mathbf{u}}^{\left(i\right)}=-{\left({{\mathbf{J}}^{\left(i\right)}}^{H}{\mathbf{J}}^{\left(i\right)}+{\lambda}^{\left(i\right)}{\mathbf{I}}_{Q}\right)}^{-1}{\mathbf{g}}^{\left(i\right)}$; Step 3. Update ${\mathbf{u}}^{\left(i+1\right)}$: ${\mathbf{u}}^{\left(i+1\right)}={\mathbf{u}}^{\left(i\right)}+\Delta {\mathbf{u}}^{\left(i\right)}$;Step 4. Calculate the gain rate $\alpha $: $\alpha =\frac{\varphi \left({\mathbf{u}}^{\left(i+1\right)}\right)\phantom{\rule{4.pt}{0ex}}-\phantom{\rule{4.pt}{0ex}}\varphi \left({\mathbf{u}}^{\left(i\right)}\right)}{{\delta}^{\left(i\right)}}$, where ${\delta}^{\left(i\right)}\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}{\left({\mathbf{J}}^{\left(i\right)}\Delta {\mathbf{u}}^{\left(i\right)}\right)}^{H}\mathbf{z}\left({\mathbf{u}}^{\left(i\right)}\right)\phantom{\rule{4.pt}{0ex}}+\phantom{\rule{4.pt}{0ex}}$$\frac{1}{2{\u2225{\mathbf{J}}^{\left(i\right)}\Delta {\mathbf{u}}^{\left(i\right)}\u2225}_{F}^{2}}$;Step 5. Update $\lambda $: If $\alpha \u2a7e0$, ${\mathbf{u}}^{\left(i+1\right)}$ is ture, and set ${\lambda}^{\left(i+1\right)}={\lambda}^{\left(i\right)}max\left(1-{\left(2\alpha -1\right)}^{3},\phantom{\rule{0.277778em}{0ex}}1/3\right)$ and $\tau =2$. Otherwise, ${\mathbf{u}}^{\left(i+1\right)}$ is invalid, and set ${\lambda}^{\left(i+1\right)}=\tau {\lambda}^{\left(i\right)}$ and $\tau \leftarrow 2\tau $;Step 6.$i\leftarrow i+1$;endAcquire ${\mathbf{S}}^{\left(\infty \right)}$ and ${\mathbf{H}}^{\left(\infty \right)}$: ${\mathbf{S}}^{\left(\infty \right)}={\left(unvec\left({\mathbf{u}}_{\mathbf{S}}^{\left(\infty \right)}\right)\right)}^{T}$, ${\mathbf{H}}^{\left(\infty \right)}={\left(unvec\left({\mathbf{u}}_{\mathbf{H}}^{\left(\infty \right)}\right)\right)}^{T}$. Compute ${\mathbf{H}}_{new}^{\left(\infty \right)}$: If $L<min\left({M}_{S},{M}_{D}\right)$, ${\mathbf{H}}_{new}^{\left(\infty \right)}=SVP\left({\mathbf{H}}^{\left(\infty \right)}\right)$. Otherwise, ${\mathbf{H}}_{new}^{\left(\infty \right)}={\mathbf{H}}^{\left(\infty \right)}$. Remove the scaling ambiguity: ${\widehat{\mathbf{S}}}^{\left(final\right)}=\frac{{\widehat{\mathbf{S}}}^{\left(\infty \right)}}{{\widehat{s}}_{1,1}^{\left(\infty \right)}}$, ${\widehat{\mathbf{H}}}^{\left(final\right)}={\widehat{s}}_{1,1}^{\left(\infty \right)}{\mathbf{H}}_{new}^{\left(\infty \right)}$. |

## 5. Extension to Multi-User Massive Mimo Systems

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Algorithm 1**takes advantage of this low-rank constraint $rank\left({\mathbf{H}}^{\left(m\right)}\right)\u2a7d{L}_{m}$ to further improve the estimation accuracy of the channel.

## 6. Simulation Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Jin, S.; McKay, M.R.; Wong, K.K.; Li, X. Low-SNR capacity of multiple-antenna systems with statistical channel-state information. IEEE Trans. Veh. Technol.
**2010**, 59, 2874–2884. [Google Scholar] [CrossRef] - Shafi, M.; Molisch, A.F.; Smith, P.J.; Haustein, T.; Zhu, P.; De Silva, P.; Tufvesson, F.; Benjebbour, A.; Wunder, G. 5G: A tutorial overview of standards, trials, challenges, deployment, and practice. IEEE J. Sel. Areas Commun.
**2017**, 35, 1201–1221. [Google Scholar] [CrossRef] - Collins, A.; Polyanskiy, Y. Coherent multiple-antenna block-fading channels at finite blocklength. IEEE Trans. Inf. Theory
**2019**, 65, 380–405. [Google Scholar] [CrossRef] - Lahat, D.; Adali, T.; Jutten, C. Multimodal data fusion: An overview of methods, challenges, and prospects. Proc. IEEE
**2015**, 103, 1449–1477. [Google Scholar] [CrossRef] - Sidiropoulos, N.D.; De Lathauwer, L.; Fu, X.; Huang, K.; Papalexakis, E.E.; Faloutsos, C. Tensor decomposition for signal processing and machine learning. IEEE Trans. Signal Process.
**2017**, 65, 3551–3582. [Google Scholar] [CrossRef] - Freitas, W.d.C.; Favier, G.; de Almeida, A.L. Tensor-Based Joint Channel and Symbol Estimation for Two-Way MIMO Relaying Systems. IEEE Signal Process. Lett.
**2019**, 26, 227–231. [Google Scholar] [CrossRef] - Harshman, R.A. Foundations of the PARAFAC Procedure: Models and Conditions for an “Explanatory” Multimodal Factor Analysis. UCLA Working Pap. Phon.
**1970**, 16, 1–84. [Google Scholar] - Sidiropoulos, N.D.; Budampati, R.S. Khatri–Rao space-time codes. IEEE Trans. Signal Process.
**2002**, 50, 2396–2407. [Google Scholar] [CrossRef] - Liu, K.; Da Costa, J.P.C.; So, H.C.; De Almeida, A.L. Semi-blind receivers for joint symbol and channel estimation in space-time-frequency MIMO-OFDM systems. IEEE Trans. Signal Process.
**2013**, 61, 5444–5457. [Google Scholar] [CrossRef] - Rong, Y.; Khandaker, M.R.; Xiang, Y. Channel estimation of dual-hop MIMO relay system via parallel factor analysis. IEEE Trans. Wirel. Commun.
**2012**, 11, 2224–2233. [Google Scholar] [CrossRef] - Du, J.; Yuan, C.; Zhang, J. Low complexity PARAFAC-based channel estimation for non-regenerative MIMO relay systems. IET Commun.
**2014**, 8, 2193–2199. [Google Scholar] [CrossRef] - Du, J.; Yuan, C.; Zhang, J. Semi-blind parallel factor based receiver for joint symbol and channel estimation in amplify-and-forward multiple-input multiple-output relay systems. IET Commun.
**2015**, 9, 737–744. [Google Scholar] [CrossRef] - Ximenes, L.R.; Favier, G.; de Almeida, A.L. Semi-blind receivers for non-regenerative cooperative MIMO communications based on nested PARAFAC modeling. IEEE Trans. Signal Process.
**2015**, 63, 4985–4998. [Google Scholar] [CrossRef] - Zhou, Z.; Fang, J.; Yang, L.; Li, H.; Chen, Z.; Li, S. Channel estimation for millimeter-wave multiuser MIMO systems via PARAFAC decomposition. IEEE Trans. Wirel. Commun.
**2016**, 15, 7501–7516. [Google Scholar] [CrossRef] - Zhou, Z.; Fang, J.; Yang, L.; Li, H.; Chen, Z.; Blum, R.S. Low-rank tensor decomposition-aided channel estimation for millimeter wave MIMO-OFDM systems. IEEE J. Sel. Areas Commun.
**2017**, 35, 1524–1538. [Google Scholar] [CrossRef] - Wei, X.; Peng, W.; Chen, D.; Ng, D.W.K.; Jiang, T. Joint Channel Parameter Estimation in Multi-cell Massive MIMO System. IEEE Trans. Commun.
**2019**. [Google Scholar] [CrossRef] - Comon, P.; Luciani, X.; De Almeida, A.L. Tensor decompositions, alternating least squares and other tales. J. Chemom.
**2009**, 23, 393–405. [Google Scholar] [CrossRef] [Green Version] - Tomasi, G.; Bro, R. A comparison of algorithms for fitting the PARAFAC model. Comput. Stat. Data Anal.
**2006**, 50, 1700–1734. [Google Scholar] [CrossRef] - Nion, D.; De Lathauwer, L. A block component model-based blind DS-CDMA receiver. IEEE Trans. Signal Process.
**2008**, 56, 5567–5579. [Google Scholar] [CrossRef] - De Almeida, A.L.; Favier, G.; Ximenes, L.R. Space-time-frequency (STF) MIMO communication systems with blind receiver based on a generalized PARATUCK2 model. IEEE Trans. Signal Process.
**2013**, 61, 1895–1909. [Google Scholar] [CrossRef] - De Almeida, A.L.; Favier, G.; Mota, J.C. Space–time spreading–multiplexing for MIMO wireless communication systems using the PARATUCK-2 tensor model. Signal Process.
**2009**, 89, 2103–2116. [Google Scholar] [CrossRef] - Favier, G.; Da Costa, M.N.; De Almeida, A.L.; Romano, J.M.T. Tensor space–time (TST) coding for MIMO wireless communication systems. Signal Process.
**2012**, 92, 1079–1092. [Google Scholar] [CrossRef] - Favier, G.; de Almeida, A.L. Tensor space-time-frequency coding with semi-blind receivers for MIMO wireless communication systems. IEEE Trans. Signal Process.
**2014**, 62, 5987–6002. [Google Scholar] [CrossRef] - Da Costa, M.N.; Favier, G.; Romano, J.M.T. Tensor modelling of MIMO communication systems with performance analysis and Kronecker receivers. Signal Process.
**2018**, 145, 304–316. [Google Scholar] [CrossRef] - Tucker, L.R. Some mathematical notes on three-mode factor analysis. Psychometrika
**1966**, 31, 279–311. [Google Scholar] [CrossRef] - Favier, G.; de Almeida, A.L. Overview of constrained PARAFAC models. EURASIP J. Adv. Signal Process.
**2014**, 2014, 142. [Google Scholar] [CrossRef] [Green Version] - Du, J.; Yuan, C.; Hu, Z.; Lin, H. A novel tensor-based receiver for joint symbol and channel estimation in two-hop cooperative MIMO relay systems. IEEE Commun. Lett.
**2015**, 19, 1961–1964. [Google Scholar] [CrossRef] - Chen, Y.; Han, D.; Qi, L. New ALS methods with extrapolating search directions and optimal step size for complex-valued tensor decompositions. IEEE Trans. Signal Process.
**2011**, 59, 5888–5898. [Google Scholar] [CrossRef] - Van Loan, C.F.; Pitsianis, N. Approximation with Kronecker products. In Linear Algebra for Large Scale and Real-Time Applications; Springer: Dordrecht, The Netherlands, 1993; pp. 293–314. [Google Scholar]
- Du, J.; Tian, P.; Lin, H. Tucker-2 Model Based Scheme for Joint Signal Detection and Channel Estimation in MIMO Systems. J. Beijing Univ. Posts Telecommun.
**2016**, 39, 6–10. [Google Scholar] - Jain, P.; Meka, R.; Dhillon, I.S. Guaranteed rank minimization via singular value projection. In Proceedings of the Advances in Neural Information Processing Systems, Vancouver, BC, Canada, 6–9 December 2010; pp. 937–945. [Google Scholar]
- Shen, W.; Dai, L.; Shim, B.; Mumtaz, S.; Wang, Z. Joint CSIT acquisition based on low-rank matrix completion for FDD massive MIMO systems. IEEE Commun. Lett.
**2015**, 19, 2178–2181. [Google Scholar] [CrossRef] - Alkhateeb, A.; El Ayach, O.; Leus, G.; Heath, R.W. Hybrid precoding for millimeter wave cellular systems with partial channel knowledge. In Proceedings of the 2013 Information Theory and Applications Workshop (ITA), San Diego, CA, USA, 10–15 February 2013; pp. 1–5. [Google Scholar]
- Dai, L.; Gao, X.; Quan, J.; Han, S.; Chih-Lin, I. Near-optimal hybrid analog and digital precoding for downlink mmWave massive MIMO systems. In Proceedings of the 2015 IEEE International Conference on Communications (ICC), London, UK, 8–12 June 2015; pp. 1334–1339. [Google Scholar]
- Shiu, D.S.; Foschini, G.J.; Gans, M.J.; Kahn, J.M. Fading correlation and its effect on the capacity of multielement antenna systems. IEEE Trans. Commun.
**2000**, 48, 502–513. [Google Scholar] [CrossRef] - Nion, D.; De Lathauwer, L. An enhanced line search scheme for complex-valued tensor decompositions. Application in DS-CDMA. Signal Process.
**2008**, 88, 749–755. [Google Scholar] [CrossRef] - Alkhateeb, A.; El Ayach, O.; Leus, G.; Heath, R.W. Channel estimation and hybrid precoding for millimeter wave cellular systems. IEEE J. Sel. Top. Signal Process.
**2014**, 8, 831–846. [Google Scholar] [CrossRef] - Hu, C.; Dai, L.; Mir, T.; Gao, Z.; Fang, J. Super-resolution channel estimation for mmWave massive MIMO with hybrid precoding. IEEE Trans. Veh. Technol.
**2018**, 67, 8954–8958. [Google Scholar] [CrossRef] - Srivastava, S.; Mishra, A.; Rajoriya, A.; Jagannatham, A.K.; Ascheid, G. Quasi-Static and Time-Selective Channel Estimation for Block-Sparse Millimeter Wave Hybrid MIMO Systems: Sparse Bayesian Learning (SBL) Based Approaches. IEEE Trans. Signal Process.
**2019**, 67, 1251–1266. [Google Scholar] [CrossRef]

**Figure 3.**Bit error rate (BER) performance of different receivers versus signal-to-noise ratio (SNR).

**Figure 5.**BER and NMSE performance of traditional alternating least squares (T-ALS) and O-LM algorithms for different L and $\rho $.

**Figure 10.**BER performance of the proposed receiver for the multi-user massive multiple-input multiple-output (MIMO) system.

$\mathit{SNR}$ (dB) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$TALS$, $L=8$, $\rho =0$ | 0.3917 | 0.2265 | 0.1348 | 0.0822 | 0.0514 | 0.0318 | 0.0199 | 0.0124 | 0.0077 | 0.0049 | 0.0030 | 0.0019 | 0.0012 |

$O-LM$, $L=8$, $\rho =0$ | 0.3900 | 0.2253 | 0.1355 | 0.0814 | 0.0503 | 0.0317 | 0.0198 | 0.0124 | 0.0077 | 0.0049 | 0.0030 | 0.0019 | 0.0012 |

$TALS$, $L=8$, $\rho =0.8$ | 0.5920 | 0.3113 | 0.1803 | 0.1079 | 0.0666 | 0.0407 | 0.0251 | 0.0156 | 0.0100 | 0.0061 | 0.0039 | 0.0024 | 0.0015 |

$O-LM$, $L=8$, $\rho =0.8$ | 0.5777 | 0.3090 | 0.1814 | 0.1078 | 0.0664 | 0.0404 | 0.0251 | 0.0161 | 0.0098 | 0.0062 | 0.0039 | 0.0024 | 0.0015 |

$TALS$, $L=7$, $\rho =0.8$ | 1.5305 | 0.6519 | 0.3055 | 0.1623 | 0.1004 | 0.0570 | 0.0360 | 0.0221 | 0.0139 | 0.0091 | 0.0051 | 0.0034 | 0.0021 |

$O-LM$, $L=7$, $\rho =0.8$ | 1.4047 | 0.6228 | 0.2849 | 0.1770 | 0.0952 | 0.0580 | 0.0354 | 0.0221 | 0.0136 | 0.0087 | 0.0055 | 0.0034 | 0.0021 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Du, J.; Han, M.; Hua, Y.; Chen, Y.; Lin, H.
A Robust Semi-Blind Receiver for Joint Symbol and Channel Parameter Estimation in Multiple-Antenna Systems. *Electronics* **2019**, *8*, 550.
https://doi.org/10.3390/electronics8050550

**AMA Style**

Du J, Han M, Hua Y, Chen Y, Lin H.
A Robust Semi-Blind Receiver for Joint Symbol and Channel Parameter Estimation in Multiple-Antenna Systems. *Electronics*. 2019; 8(5):550.
https://doi.org/10.3390/electronics8050550

**Chicago/Turabian Style**

Du, Jianhe, Meng Han, Yan Hua, Yuanzhi Chen, and Heyun Lin.
2019. "A Robust Semi-Blind Receiver for Joint Symbol and Channel Parameter Estimation in Multiple-Antenna Systems" *Electronics* 8, no. 5: 550.
https://doi.org/10.3390/electronics8050550