Training-Signal-Design-for-Correlated-MassiveMIMO_Maasive MIMO_The Signal

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/624f870674bc5c0105423c8d/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624f870674bc5c0105423c8d/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 2017 1135</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls2 ws0">Training Signal Design for Correlated Massive</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls3 ws0">MIMO Channel Estimation</div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls4 ws0">Mojtaba Soltanalian, Member, IEEE, Mohammad Mahdi Naghsh, Member, IEEE, Naﬁseh Shariati,</div><div class="t m0 x5 h4 y5 ff1 fs2 fc0 sc0 ls8 ws0">Petre Stoica, Fellow, IEEE, and Babak Hassibi</div><div class="t m0 x6 h5 y6 ff3 fs3 fc0 sc0 lsa ws0">Abstract—In this paper, we propose a new approach to the</div><div class="t m0 x1 h5 y7 ff4 fs3 fc0 sc0 lsc ws0">design of training sequences that can be used for an accurate</div><div class="t m0 x1 h5 y8 ff4 fs3 fc0 sc0 lsd ws0">estimation of multi-input multi-output channels. The proposed</div><div class="t m0 x1 h5 y9 ff4 fs3 fc0 sc0 lse ws0">method is particularly instrumental in training sequence designs</div><div class="t m0 x1 h5 ya ff4 fs3 fc0 sc0 lsf ws0">that deal with three key challenges: 1) arbitrary channel and</div><div class="t m0 x1 h5 yb ff4 fs3 fc0 sc0 ls10 ws0">noise statistics that do not follow speciﬁc models, 2) limitations</div><div class="t m0 x1 h5 yc ff4 fs3 fc0 sc0 ls11 ws0">on the properties of the transmit signals, including total power,</div><div class="t m0 x1 h5 yd ff4 fs3 fc0 sc0 ls12 ws0">per-antenna power, having a constant-modulus, discrete-phase, or</div><div class="t m0 x1 h5 ye ff4 fs3 fc0 sc0 ls13 ws0">low peak-to-average-power ratio, and 3) signal design for large-</div><div class="t m0 x1 h5 yf ff4 fs3 fc0 sc0 ls14 ws0">scale or massive antenna arrays. Several numerical examples are</div><div class="t m0 x1 h5 y10 ff4 fs3 fc0 sc0 lsc ws0">provided to examine the proposed method.</div><div class="t m0 x6 h5 y11 ff3 fs3 fc0 sc0 ls15 ws0">Index Terms—Channel estimation, massive MIMO, peak-to-</div><div class="t m0 x1 h5 y12 ff4 fs3 fc0 sc0 ls17 ws0">average-power ratio (PAR), signal design, training sequence.</div><div class="t m0 x7 h6 y13 ff1 fs4 fc0 sc0 ls18 ws0">I. INTRODUCTION</div><div class="t m0 x1 h7 y14 ff4 fs6 fc0 sc0 ls1a ws0">A</div><div class="t m0 x8 h6 y15 ff1 fs4 fc0 sc0 ls1b ws0">CCURATE knowledge of multi-input multi-output</div><div class="t m0 x8 h6 y16 ff1 fs4 fc0 sc0 ls1c ws0">(MIMO) channel state information (CSI) plays an impor-</div><div class="t m0 x1 h6 y17 ff1 fs4 fc0 sc0 ls1d ws0">tant role in exploiting the full potential of MIMO communica-</div><div class="t m0 x1 h6 y18 ff1 fs4 fc0 sc0 ls1e ws0">tion systems. A practical way to acquire instantaneous CSI of</div><div class="t m0 x1 h6 y19 ff1 fs4 fc0 sc0 ls1e ws0">the MIMO channel is training-based channel estimation which</div><div class="t m0 x1 h6 y1a ff1 fs4 fc0 sc0 ls1f ws0">relies on transmitting an aprioriknown sequence, namely</div><div class="t m0 x1 h6 y1b ff1 fs4 fc0 sc0 ls1a ws0">a training or pilot sequence, to the receiver, and estimating</div><div class="t m0 x1 h6 y1c ff1 fs4 fc0 sc0 ls24 ws0">the instantaneous channel coefﬁcients based on the received</div><div class="t m0 x1 h6 y1d ff1 fs4 fc0 sc0 ls25 ws0">signal [1]–[13]. This channel estimation scheme is illustrated</div><div class="t m0 x1 h6 y1e ff1 fs4 fc0 sc0 ls21 ws0">in Fig. 1.</div><div class="t m0 x6 h6 y1f ff1 fs4 fc0 sc0 ls26 ws0">Massive MIMO is an emerging technological concept</div><div class="t m0 x1 h6 y20 ff1 fs4 fc0 sc0 ls27 ws0">in communications where a large number of antennas is</div><div class="t m0 x9 h8 y21 ff1 fs5 fc0 sc0 ls28 ws0">Manuscript received October 13, 2015; revised February 27, 2016 and</div><div class="t m0 x1 h8 y22 ff1 fs5 fc0 sc0 ls29 ws0">September 3, 2016; accepted November 28, 2016. Date of publication</div><div class="t m0 x1 h8 y23 ff1 fs5 fc0 sc0 ls28 ws0">December 14, 2016; date of current version February 9, 2017. This work</div><div class="t m0 x1 h8 y24 ff1 fs5 fc0 sc0 ls2a ws0">was supported in part by the European Research Council, in part by the</div><div class="t m0 x1 h8 y25 ff1 fs5 fc0 sc0 ls2b ws0">Swedish Research Council, in part by the U.S. National Science Foundation</div><div class="t m0 x1 h8 y26 ff1 fs5 fc0 sc0 ls2c ws0">under Grant CNS-0932428, Grant CCF-1018927, Grant CCF-1423663, and</div><div class="t m0 x1 h8 y27 ff1 fs5 fc0 sc0 ls2d ws0">Grant CCF-1409204, in part by Qualcomm Inc., in part by the NASA’s Jet</div><div class="t m0 x1 h8 y28 ff1 fs5 fc0 sc0 ls2e ws0">Propulsion Laboratory through the President and Director’s Fund, in part</div><div class="t m0 x1 h8 y29 ff1 fs5 fc0 sc0 ls2f ws0">by King Abdulaziz University, and in part by the King Abdullah University</div><div class="t m0 x1 h8 y2a ff1 fs5 fc0 sc0 ls30 ws0">of Science and Technology. The associate editor coordinating the review of</div><div class="t m0 x1 h8 y2b ff1 fs5 fc0 sc0 ls32 ws0">this paper and approving it for publication was F. Boccardi. (Corresponding</div><div class="t m0 x1 h9 y2c ff2 fs5 fc0 sc0 ls28 ws0">author: Mojtaba Soltanalian.)</div><div class="t m0 x9 h8 y2d ff1 fs5 fc0 sc0 ls33 ws0">M. Soltanalian is with the Department of Electrical and Computer Engi-</div><div class="t m0 x1 h8 y2e ff1 fs5 fc0 sc0 ls2f ws0">neering, University of Illinois at Chicago, Chicago, IL 60607 USA (e-mail:</div><div class="t m0 x1 h8 y2f ff1 fs5 fc0 sc0 ls28 ws0">msol@uic.edu).</div><div class="t m0 x9 h8 y30 ff1 fs5 fc0 sc0 ls34 ws0">M. M. Naghsh is with the Department of Electrical and Computer Engi-</div><div class="t m0 x1 h8 y31 ff1 fs5 fc0 sc0 ls35 ws0">neering, Isfahan University of Technology, Isfahan 84156-83111, Iran.</div><div class="t m0 x9 h8 y32 ff1 fs5 fc0 sc0 ls36 ws0">N. Shariati was with the Department of Signal Processing, KTH Royal</div><div class="t m0 x1 h8 y33 ff1 fs5 fc0 sc0 ls2a ws0">Institute of Technology, 10044 Stockholm, Sweden. She is now with Ericsson</div><div class="t m0 x1 h8 y34 ff1 fs5 fc0 sc0 ls37 ws0">Research, 16483 Stockholm, Sweden.</div><div class="t m0 x9 h8 y35 ff1 fs5 fc0 sc0 ls38 ws0">P. Stoica is with the Department of Information Technology, Uppsala</div><div class="t m0 x1 h8 y36 ff1 fs5 fc0 sc0 ls35 ws0">University, 75237 Uppsala, Sweden.</div><div class="t m0 x9 h8 y37 ff1 fs5 fc0 sc0 ls34 ws0">B. Hassibi is with the Department of Electrical Engineering, California</div><div class="t m0 x1 h8 y38 ff1 fs5 fc0 sc0 ls2a ws0">Institute of Technology, Pasadena, CA 91125 USA.</div><div class="t m0 x9 h8 y39 ff1 fs5 fc0 sc0 ls39 ws0">Color versions of one or more of the ﬁgures in this paper are available</div><div class="t m0 x1 h8 y3a ff1 fs5 fc0 sc0 ls3a ws0">online at http://ieeexplore.ieee.org.</div><div class="t m0 x9 h8 y3b ff1 fs5 fc0 sc0 ls2f ws0">Digital Object Identiﬁer 10.1109/TWC.2016.2639485</div><div class="t m0 xa h8 y3c ff1 fs5 fc0 sc0 ls35 ws0">Fig. 1. MIMO training for channel estimation: The optimized training signals</div><div class="t m0 xa h8 y3d ff1 fs5 fc0 sc0 ls29 ws0">are transmitted during the training phase in order to estimate the instantaneous</div><div class="t m0 xa h8 y3e ff1 fs5 fc0 sc0 ls3c ws0">channel information based on the received signal and the known input.</div><div class="t m0 xa h6 y3f ff1 fs4 fc0 sc0 ls3d ws0">employed at the base stations. Massive MIMO systems have</div><div class="t m0 xa h6 y40 ff1 fs4 fc0 sc0 ls3e ws0">exhibited superior performance with respect to different qual-</div><div class="t m0 xa h6 y41 ff1 fs4 fc0 sc0 ls1d ws0">ity metrics, including capacity, transmit energy efﬁciency,</div><div class="t m0 xa h6 y42 ff1 fs4 fc0 sc0 ls3f ws0">latency, and robustness to jamming [14]–[19]. Signal design</div><div class="t m0 xa h6 y43 ff1 fs4 fc0 sc0 ls40 ws0">for massive MIMO deals with many streams of signals or,</div><div class="t m0 xa h6 y44 ff1 fs4 fc0 sc0 ls41 ws0">equivalently, many design variables. Such a large number of</div><div class="t m0 xa h6 y45 ff1 fs4 fc0 sc0 ls42 ws0">degrees of freedom paves the way for a better quality of signal</div><div class="t m0 xa h6 y46 ff1 fs4 fc0 sc0 ls3f ws0">design, but at the same time, makes the design computationally</div><div class="t m0 xa h6 y47 ff1 fs4 fc0 sc0 ls1d ws0">expensive. Hence, it is no surprise that a considerable attention</div><div class="t m0 xa h6 y48 ff1 fs4 fc0 sc0 ls41 ws0">has been paid to efﬁcient methods that can handle the signal</div><div class="t m0 xa h6 y49 ff1 fs4 fc0 sc0 ls1e ws0">design complexity arising from such large-scale arrays of</div><div class="t m0 xa h6 y4a ff1 fs4 fc0 sc0 ls1b ws0">antennas (see, e.g. [11], [20]–[27], and the references therein).</div><div class="t m0 xb h6 y4b ff1 fs4 fc0 sc0 ls1e ws0">This work is concerned with an efﬁcient design of training</div><div class="t m0 xa h6 y4c ff1 fs4 fc0 sc0 ls43 ws0">signals for channel estimation in massive MIMO systems.</div><div class="t m0 xa h6 y4d ff1 fs4 fc0 sc0 ls44 ws0">We discuss the problem formulation and the contributions of</div><div class="t m0 xa h6 y4e ff1 fs4 fc0 sc0 ls1d ws0">the paper in the following. Notations and their deﬁnitions are</div><div class="t m0 xa h6 y4f ff1 fs4 fc0 sc0 ls1d ws0">provided in Table I.</div><div class="t m0 xa ha y50 ff2 fs4 fc0 sc0 ls45 ws0">A. Problem Formulation</div><div class="t m0 xb h6 y51 ff1 fs4 fc0 sc0 ls46 ws0">We consider a narrowband block fading point-to-point</div><div class="t m0 xa h6 y52 ff1 fs4 fc0 sc0 ls45 ws0">MIMO communication link equipped with n</div><div class="t m0 xc hb y53 ff2 fs7 fc0 sc0 ls1a ws0">T</div><div class="t m0 xd h6 y54 ff1 fs4 fc0 sc0 ls47 ws0">transmit and</div><div class="t m0 xa ha y55 ff2 fs4 fc0 sc0 ls1a ws0">n</div><div class="t m0 xe hb y56 ff2 fs7 fc0 sc0 ls1a ws0">R</div><div class="t m0 xf hc y57 ff1 fs4 fc0 sc0 ls48 ws0">receive antennas. Let P ∈ C</div><div class="t m0 x10 hb y58 ff2 fs7 fc0 sc0 ls1a ws0">B×n</div><div class="t m0 x11 hd y59 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x12 h6 y57 ff1 fs4 fc0 sc0 ls27 ws0">be a matrix whose</div><div class="t m0 xa h6 y5a ff1 fs4 fc0 sc0 ls49 ws0">rows comprise the training symbols at each transmitter.</div><div class="t m0 x13 he y5b ff1 fs7 fc0 sc0 ls1a ws0">1</div><div class="t m0 xe hf y5c ff1 fs9 fc0 sc0 ls1a ws0">1</div><div class="t m0 xb h8 y5d ff1 fs5 fc0 sc0 ls4a ws0">Note that B ≥ n</div><div class="t m0 x14 h10 y5e ff2 fs9 fc0 sc0 ls1a ws0">T</div><div class="t m0 x15 h8 y5d ff1 fs5 fc0 sc0 ls4b ws0">is a necessary condition for obtaining meaningful</div><div class="t m0 xa h8 y5f ff1 fs5 fc0 sc0 ls4c ws0">channel estimates when a least-squares estimation of the channel is used [2].</div><div class="t m0 xa h8 y60 ff1 fs5 fc0 sc0 ls39 ws0">Although a larger B would be beneﬁcial for obtaining a more accurate estimate</div><div class="t m0 xa h8 y61 ff1 fs5 fc0 sc0 ls2b ws0">of the channel, it was shown in [29] that using a smaller B leads to a larger</div><div class="t m0 xa h8 y62 ff1 fs5 fc0 sc0 ls3c ws0">capacity of the MIMO channel by leaving more time for data transmission.</div><div class="t m0 xa h8 y63 ff1 fs5 fc0 sc0 ls3c ws0">In particular, when an MMSE estimate of the channel is used (which is the</div><div class="t m0 xa h8 y64 ff1 fs5 fc0 sc0 ls4c ws0">case in this paper) the error of channel estimation can be reduced even if B</div><div class="t m0 xa h8 y65 ff1 fs5 fc0 sc0 ls29 ws0">is smaller than the number of transmit antennas.</div><div class="t m0 x16 h8 y66 ff1 fs5 fc0 sc0 ls4e ws0">1536-1276 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.</div><div class="t m0 x17 h8 y67 ff1 fs5 fc0 sc0 ls31 ws0">See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624f870674bc5c0105423c8d/bg2.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls51 ws0">1136 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 2017</div><div class="t m0 x18 h8 y68 ff1 fs5 fc0 sc0 ls53 ws0">TABLE I</div><div class="t m0 x19 h8 y69 ff1 fs5 fc0 sc0 ls1a ws0">N</div><div class="t m0 x1a h11 y6a ff1 fsa fc0 sc0 ls54 ws0">OTATIONS</div><div class="t m0 x1 h6 y6b ff1 fs4 fc0 sc0 ls3e ws0">The training phase can be described by the equation,</div><div class="t m0 x1b h12 y6c ff4 fs4 fc0 sc0 ls1a ws0">Y = HP</div><div class="t m0 x1c hb y6d ff2 fs7 fc0 sc0 ls1a ws0">T</div><div class="t m0 x1d h12 y6e ff5 fs4 fc0 sc0 ls1a ws0">+ N, (1)</div><div class="t m0 x1 hc y6f ff1 fs4 fc0 sc0 ls1a ws0">in which H ∈ C</div><div class="t m0 x1e hb y70 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x1f hd y71 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x20 hb y70 ff5 fs7 fc0 sc0 ls1a ws0">×n</div><div class="t m0 x1a hd y71 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x1c h6 y72 ff1 fs4 fc0 sc0 ls1c ws0">denotes the MIMO channel</div><div class="t m0 x1 hc y73 ff1 fs4 fc0 sc0 ls25 ws0">with H(k, l) being the gain of the MIMO path between</div><div class="t m0 x1 h12 y74 ff1 fs4 fc0 sc0 ls58 ws0">receiver k and transmitter l. We assume vec(H) ∼ CN (0, R)</div><div class="t m0 x1 hc y75 ff1 fs4 fc0 sc0 ls5c ws0">where R ∈ C</div><div class="t m0 x21 hb y76 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x22 hd y77 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x23 hb y76 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x16 hd y77 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x24 hb y76 ff5 fs7 fc0 sc0 ls1a ws0">×n</div><div class="t m0 x1b hd y77 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x17 hb y76 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x25 hd y77 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x18 h6 y78 ff1 fs4 fc0 sc0 ls41 ws0">is the channel covariance matrix.</div><div class="t m0 x1 hc y79 ff1 fs4 fc0 sc0 ls5d ws0">Moreover, N ∈ C</div><div class="t m0 x24 hb y7a ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x26 hd y7b ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x27 hb y7a ff5 fs7 fc0 sc0 ls1a ws0">×B</div><div class="t m0 x19 h6 y7c ff1 fs4 fc0 sc0 ls5e ws0">represents the noise matrix whose</div><div class="t m0 x1 h6 y7d ff1 fs4 fc0 sc0 ls3f ws0">columns consist of (possibly colored) Gaussian noise, i.e.,</div><div class="t m0 x1 h12 y7e ff7 fs4 fc0 sc0 ls1a ws0">vec(N) ∼ CN (0, M) with M ∈ C</div><div class="t m0 x28 hb y7f ff2 fs7 fc0 sc0 ls5f ws0">Bn</div><div class="t m0 x29 hd y80 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x2a hb y7f ff5 fs7 fc0 sc0 ls1a ws0">×Bn</div><div class="t m0 x2b hd y80 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x2c h6 y81 ff1 fs4 fc0 sc0 ls55 ws0">denoting</div><div class="t m0 x1 h6 y82 ff1 fs4 fc0 sc0 ls60 ws0">the noise covariance matrix. The received data is denoted</div><div class="t m0 x1 hc y83 ff1 fs4 fc0 sc0 ls21 ws0">by Y ∈ C</div><div class="t m0 x2d hb y84 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x2e hd y85 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x2f hb y84 ff5 fs7 fc0 sc0 ls1a ws0">×B</div><div class="t m0 x30 h13 y86 ff7 fs4 fc0 sc0 ls1a ws0">.</div><div class="t m0 x6 hc y87 ff1 fs4 fc0 sc0 ls1d ws0">The key goal is to design the matrix P in order to produce</div><div class="t m0 x1 hc y88 ff1 fs4 fc0 sc0 ls3d ws0">an accurate estimate of the channel H. To accomplish this</div><div class="t m0 x1 h6 y89 ff1 fs4 fc0 sc0 ls1a ws0">goal, we consider the minimization of the mean-squared</div><div class="t m0 x1 h6 y8a ff1 fs4 fc0 sc0 ls45 ws0">error (MSE) of the channel estimate when the minimum</div><div class="t m0 x1 h6 y8b ff1 fs4 fc0 sc0 ls60 ws0">mean-square error (MMSE) estimator is used to estimate the</div><div class="t m0 x1 h6 y8c ff1 fs4 fc0 sc0 ls1c ws0">instantaneous channel coefﬁcients. The MSE of the channel</div><div class="t m0 x1 h6 y8d ff1 fs4 fc0 sc0 ls61 ws0">estimate can be formulated as [6], [8], [30],</div><div class="t m0 x31 h6 y8e ff1 fs4 fc0 sc0 ls62 ws0">MSE = tr</div><div class="t m0 x32 h14 y8f ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x33 h14 y90 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x34 h12 y8e ff4 fs4 fc0 sc0 ls1a ws0">R</div><div class="t m0 x16 he y91 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x27 h12 y92 ff5 fs4 fc0 sc0 ls1a ws0">+ (P ⊗ I</div><div class="t m0 x1c hb y93 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x35 hd y94 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x36 h13 y92 ff7 fs4 fc0 sc0 ls1a ws0">)</div><div class="t m0 x37 hb y91 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x38 hc y92 ff3 fs4 fc0 sc0 ls1a ws0">M</div><div class="t m0 x39 he y91 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x3a h12 y92 ff7 fs4 fc0 sc0 ls1a ws0">(P ⊗ I</div><div class="t m0 x3b hb y93 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x3c hd y94 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x3d h13 y92 ff7 fs4 fc0 sc0 ls1a ws0">)</div><div class="t m0 x2b h14 y95 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x3e he y96 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x3f h14 y97 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x40 h6 y98 ff7 fs4 fc0 sc0 ls1a ws0">. (2)</div><div class="t m0 x1 h6 y99 ff1 fs4 fc0 sc0 ls1e ws0">Several analytical and computational approaches to acquiring</div><div class="t m0 x1 hc y9a ff1 fs4 fc0 sc0 ls63 ws0">channel and noise covariance matrices (R and M)aredis-</div><div class="t m0 x1 h6 y9b ff1 fs4 fc0 sc0 ls56 ws0">cussed in [1], [4], [31], and [32].</div><div class="t m0 x1 ha y9c ff2 fs4 fc0 sc0 ls66 ws0">B. Contributions of This Work</div><div class="t m0 x6 h6 y9d ff1 fs4 fc0 sc0 ls25 ws0">In the literature, the optimized training sequence is usually</div><div class="t m0 x1 h6 y9e ff1 fs4 fc0 sc0 ls55 ws0">found by minimizing the MSE criterion under a total power</div><div class="t m0 x41 he y9f ff1 fs7 fc0 sc0 ls1a ws0">2</div><div class="t m0 x1 h6 ya0 ff1 fs4 fc0 sc0 ls1c ws0">constraint. Moreover, in order to make the problem tractable</div><div class="t m0 x1 h6 ya1 ff1 fs4 fc0 sc0 ls45 ws0">(or to obtain a closed-form solution), the covariance matrices</div><div class="t m0 x1 h6 ya2 ff1 fs4 fc0 sc0 ls23 ws0">associated with the channel and noise are usually assumed</div><div class="t m0 x1 h6 ya3 ff1 fs4 fc0 sc0 ls67 ws0">to possess special structures, e.g. to satisfy the Kronecker</div><div class="t m0 x1 h6 ya4 ff1 fs4 fc0 sc0 ls68 ws0">model or a diagonal matrix structure [1], [3], [5], [6], [8].</div><div class="t m0 x1 h6 ya5 ff1 fs4 fc0 sc0 ls1e ws0">The contributions of this paper can be summarized as follows:</div><div class="t m0 x9 hf ya6 ff1 fs9 fc0 sc0 ls1a ws0">2</div><div class="t m0 x6 h8 ya7 ff1 fs5 fc0 sc0 ls69 ws0">Please see (27) and the related discussions for details.</div><div class="t m0 xb h6 ya8 ff5 fs7 fc0 sc0 ls1a ws0">• The problem formulation is cast in a rather general form;</div><div class="t m0 x42 h6 ya9 ff1 fs4 fc0 sc0 ls6a ws0">i.e. no special structure is assumed for the covariance</div><div class="t m0 x42 h6 yaa ff1 fs4 fc0 sc0 ls68 ws0">matrices of the channel and noise.</div><div class="t m0 xb h6 yab ff5 fs7 fc0 sc0 ls1a ws0">• The proposed optimization approach can handle not only</div><div class="t m0 x42 h6 yac ff1 fs4 fc0 sc0 ls46 ws0">the total training power but also more complicated signal</div><div class="t m0 x42 h6 yad ff1 fs4 fc0 sc0 ls47 ws0">constraints that are typically of interest at the transmit-</div><div class="t m0 x42 h6 yae ff1 fs4 fc0 sc0 ls56 ws0">side of the communication system [17], [33]–[35]. Such</div><div class="t m0 x42 h6 yaf ff1 fs4 fc0 sc0 ls3f ws0">constraints include constant-modulus, desired peak-to-</div><div class="t m0 x42 h6 yb0 ff1 fs4 fc0 sc0 ls63 ws0">average-power ratio (PAR), and quantized-phase alpha-</div><div class="t m0 x42 h6 yb1 ff1 fs4 fc0 sc0 ls46 ws0">bet restrictions. To the best of our knowledge, such a</div><div class="t m0 x42 h6 yb2 ff1 fs4 fc0 sc0 ls26 ws0">constrained training sequence design was not addressed</div><div class="t m0 x42 h6 yb3 ff1 fs4 fc0 sc0 ls6b ws0">in the literature prior to this work.</div><div class="t m0 xb h6 yb4 ff5 fs7 fc0 sc0 ls1a ws0">• We devise an efﬁcient cyclic method to tackle the</div><div class="t m0 x42 h6 yb5 ff1 fs4 fc0 sc0 ls41 ws0">resulting non-convex design problem. The low compu-</div><div class="t m0 x42 h6 yb6 ff1 fs4 fc0 sc0 ls6c ws0">tational complexity of the devised optimization approach</div><div class="t m0 x42 h6 yb7 ff1 fs4 fc0 sc0 ls45 ws0">makes it a good candidate for usage in massive MIMO</div><div class="t m0 x42 h6 yb8 ff1 fs4 fc0 sc0 ls3f ws0">scenarios [14]–[16], [36].</div><div class="t m0 xb h6 yb9 ff1 fs4 fc0 sc0 ls6d ws0">According to the goals set above, we consider the following</div><div class="t m0 xa h6 yba ff1 fs4 fc0 sc0 ls27 ws0">general form of the design problem</div><div class="t m0 x43 h6 ybb ff1 fs4 fc0 sc0 ls55 ws0">min</div><div class="t m0 x44 h15 ybc ff4 fs7 fc0 sc0 ls1a ws0">P</div><div class="t m0 x45 h6 ybd ff1 fs4 fc0 sc0 ls22 ws0">tr</div><div class="t m0 x46 h14 ybe ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x47 h14 ybf ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x48 h12 ybd ff4 fs4 fc0 sc0 ls1a ws0">R</div><div class="t m0 x49 he yc0 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x4a h12 ybd ff5 fs4 fc0 sc0 ls1a ws0">+ (P ⊗ I</div><div class="t m0 x4b hb yc1 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x4c hd yc2 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x4d h13 ybd ff7 fs4 fc0 sc0 ls1a ws0">)</div><div class="t m0 x4e hb yc0 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x4f hc ybd ff3 fs4 fc0 sc0 ls1a ws0">M</div><div class="t m0 x50 he yc0 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x51 h12 ybd ff7 fs4 fc0 sc0 ls1a ws0">(P ⊗I</div><div class="t m0 x52 hb yc1 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x53 hd yc2 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x54 h13 ybd ff7 fs4 fc0 sc0 ls1a ws0">)</div><div class="t m0 x55 h14 ybf ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x56 he yc3 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x57 h14 yc4 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x43 hc yc5 ff1 fs4 fc0 sc0 ls1a ws0">s.t. P ∈  (3)</div><div class="t m0 xa h6 yc6 ff1 fs4 fc0 sc0 ls5c ws0">where  denotes the constraints imposed on the training</div><div class="t m0 xa h6 yc7 ff1 fs4 fc0 sc0 ls6e ws0">sequences.</div><div class="t m0 xb h6 yc8 ff1 fs4 fc0 sc0 ls1e ws0">The rest of this work is organized as follows. The proposed</div><div class="t m0 xa h6 yc9 ff1 fs4 fc0 sc0 ls23 ws0">approach is presented in Section II. Several numerical exam-</div><div class="t m0 xa h6 yca ff1 fs4 fc0 sc0 ls41 ws0">ples are provided in Section III. Finally, Section IV concludes</div><div class="t m0 xa h6 ycb ff1 fs4 fc0 sc0 ls48 ws0">the paper.</div><div class="t m0 x58 h6 ycc ff1 fs4 fc0 sc0 ls18 ws0">II. T</div><div class="t m0 x59 h6 ycd ff1 fs5 fc0 sc0 ls6f ws0">RAINING SEQUENCE OPTIMIZATION</div><div class="t m0 xb h6 yce ff1 fs4 fc0 sc0 ls45 ws0">In what follows, we propose an iterative cyclic minimization</div><div class="t m0 xa h6 ycf ff1 fs4 fc0 sc0 ls40 ws0">approach to tackle (3). The central idea of cyclic minimization</div><div class="t m0 xa h6 yd0 ff1 fs4 fc0 sc0 ls72 ws0">is to partition the set of optimization variables into two subsets.</div><div class="t m0 xa h6 yd1 ff1 fs4 fc0 sc0 ls3f ws0">The minimization then is done by cyclically ﬁxing one subset</div><div class="t m0 xa h6 yd2 ff1 fs4 fc0 sc0 ls73 ws0">of variables and performing the optimization with respect to</div><div class="t m0 xa h6 yd3 ff1 fs4 fc0 sc0 ls60 ws0">the other, and vice versa. We refer the interested reader to the</div><div class="t m0 xa h6 yd4 ff1 fs4 fc0 sc0 ls68 ws0">brief survey on cyclic minimization presented in [37].</div><div class="t m0 xb h6 yd5 ff1 fs4 fc0 sc0 ls40 ws0">Let</div><div class="t m0 x43 h14 yd6 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x43 h12 yd5 ff3 fs4 fc0 sc0 ls1a ws0">P  P ⊗ I</div><div class="t m0 x49 hb yd7 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x15 hd yd8 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x5a h16 yd9 ff5 fs4 fc0 sc0 ls1a ws0">∈ C</div><div class="t m0 x5b hb yda ff2 fs7 fc0 sc0 ls5f ws0">Bn</div><div class="t m0 x5c hd ydb ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x5d hb yda ff5 fs7 fc0 sc0 ls1a ws0">×n</div><div class="t m0 x5e hd ydb ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x5f hb yda ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x60 hd ydb ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x4f h6 yd9 ff1 fs4 fc0 sc0 ls66 ws0">, and note that using the</div><div class="t m0 xa h6 ydc ff1 fs4 fc0 sc0 ls47 ws0">matrix inversion lemma we have</div><div class="t m0 x61 h17 ydd ffb fs4 fc0 sc0 ls1a ws0">θ </div><div class="t m0 x62 h14 yde ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x14 hc ydd ff3 fs4 fc0 sc0 ls1a ws0">R</div><div class="t m0 x63 he ydf ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x64 h16 ye0 ff5 fs4 fc0 sc0 ls1a ws0">+</div><div class="t m0 x65 h14 ye1 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x66 hc ye0 ff3 fs4 fc0 sc0 ls1a ws0">P</div><div class="t m0 x67 hb ye2 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x68 hc ye0 ff3 fs4 fc0 sc0 ls1a ws0">M</div><div class="t m0 x69 he ydf ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x5f h14 ye3 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x6a hc ye4 ff3 fs4 fc0 sc0 ls1a ws0">P</div><div class="t m0 x10 h14 ye5 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x6b he ye6 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x6c h6 ye0 ff1 fs4 fc0 sc0 ls56 ws0">(4)</div><div class="t m0 x6d hc ye7 ff5 fs4 fc0 sc0 ls1a ws0">= R − R</div><div class="t m0 x6e h14 ye8 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x6f hc ye7 ff3 fs4 fc0 sc0 ls1a ws0">P</div><div class="t m0 x65 hb ye9 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x70 h14 yea ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x71 hc yeb ff3 fs4 fc0 sc0 ls1a ws0">M +</div><div class="t m0 x6a h14 yec ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x4e hc yeb ff3 fs4 fc0 sc0 ls74 ws0">PR</div><div class="t m0 x72 h14 yec ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x50 hc yeb ff3 fs4 fc0 sc0 ls1a ws0">P</div><div class="t m0 x73 hb ye9 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x12 h14 yea ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x74 he yed ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x75 h14 yee ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 xc hc yef ff3 fs4 fc0 sc0 ls75 ws0">PR. (5)</div><div class="t m0 xa h6 yf0 ff1 fs4 fc0 sc0 ls47 ws0">Deﬁne</div><div class="t m0 x42 hc yf1 ff3 fs4 fc0 sc0 ls1a ws0">X </div><div class="t m0 x76 h14 yf2 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x77 hc yf3 ff3 fs4 fc0 sc0 ls76 ws0">RR</div><div class="t m0 x78 h14 yf4 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x6e hc yf3 ff3 fs4 fc0 sc0 ls1a ws0">P</div><div class="t m0 x65 hb yf5 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x79 h14 yf6 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x7a hc yf7 ff3 fs4 fc0 sc0 ls74 ws0">PR M+</div><div class="t m0 x78 h14 yf6 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x78 hc yf7 ff3 fs4 fc0 sc0 ls74 ws0">PR</div><div class="t m0 x5b h14 yf6 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x7b hc yf7 ff3 fs4 fc0 sc0 ls1a ws0">P</div><div class="t m0 x71 hb yf8 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x4b h14 yf9 ff9 fs4 fc0 sc0 ls1a ws0"></div><div class="t m0 x4d h16 yfa ff5 fs4 fc0 sc0 ls1a ws0">∈ C</div><div class="t m0 x6b hb yfb ff7 fs7 fc0 sc0 ls1a ws0">(B+n</div><div class="t m0 x7c hd yfc ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x7d hb yfb ff7 fs7 fc0 sc0 ls1a ws0">)n</div><div class="t m0 x7e hd yfc ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x7f hb yfb ff5 fs7 fc0 sc0 ls1a ws0">×(B+n</div><div class="t m0 x56 hd yfc ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x80 hb yfb ff7 fs7 fc0 sc0 ls1a ws0">)n</div><div class="t m0 x81 hd yfc ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x82 h6 yfd ff7 fs4 fc0 sc0 ls1a ws0">, (6)</div><div class="t m0 x83 hc yfe ff3 fs4 fc0 sc0 ls1a ws0">U  (I</div><div class="t m0 x79 hb yff ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x77 hd y100 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x84 hb yff ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x47 hd y100 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x85 h12 y101 ff4 fs4 fc0 sc0 ls1a ws0">0</div><div class="t m0 x86 hb yff ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x87 hd y100 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x4a hb yff ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x78 hd y100 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x88 hb yff ff5 fs7 fc0 sc0 ls1a ws0">×Bn</div><div class="t m0 x89 hd y100 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x8a h13 y101 ff7 fs4 fc0 sc0 ls1a ws0">)</div><div class="t m0 x5d hb y102 ff2 fs7 fc0 sc0 ls1a ws0">T</div><div class="t m0 x8b h16 y101 ff5 fs4 fc0 sc0 ls1a ws0">∈ C</div><div class="t m0 x8c hb y102 ff7 fs7 fc0 sc0 ls1a ws0">(B+n</div><div class="t m0 x51 hd y103 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x12 hb y102 ff7 fs7 fc0 sc0 ls1a ws0">)n</div><div class="t m0 x8d hd y103 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x8e hb y102 ff5 fs7 fc0 sc0 ls1a ws0">×n</div><div class="t m0 x52 hd y103 ff2 fs8 fc0 sc0 ls1a ws0">T</div><div class="t m0 x8f hb y102 ff2 fs7 fc0 sc0 ls1a ws0">n</div><div class="t m0 x54 hd y103 ff2 fs8 fc0 sc0 ls1a ws0">R</div><div class="t m0 x90 h6 y101 ff7 fs4 fc0 sc0 ls1a ws0">, (7)</div><div class="t m0 xa h6 y104 ff1 fs4 fc0 sc0 ls1c ws0">and observe that [38],</div><div class="t m0 x4a hc y105 ff3 fs4 fc0 sc0 ls1a ws0">U</div><div class="t m0 x88 hb y106 ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x91 hc y107 ff3 fs4 fc0 sc0 ls1a ws0">X</div><div class="t m0 x92 he y106 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x5d hc y107 ff3 fs4 fc0 sc0 ls1a ws0">U = θ</div><div class="t m0 x93 he y106 ff5 fs7 fc0 sc0 ls1a ws0">−1</div><div class="t m0 x11 h6 y107 ff7 fs4 fc0 sc0 ls1a ws0">. (8)</div><div class="t m0 xa h6 y108 ff1 fs4 fc0 sc0 ls59 ws0">The goal is to minimize tr{θ } with respect to the train-</div><div class="t m0 xa hc y109 ff1 fs4 fc0 sc0 ls73 ws0">ing matrix P. To this end, let f (V, P)  tr{V</div><div class="t m0 x94 hb y10a ff2 fs7 fc0 sc0 ls1a ws0">H</div><div class="t m0 x95 hc ya7 ff3 fs4 fc0 sc0 ls77 ws0">XV}</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>