PAPR降低方法_通信仿真_3G/4G/5G开发

<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/627384f07ec58a2c0130b3e1/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/627384f07ec58a2c0130b3e1/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">arXiv:1511.09013v1 [cs.IT] 29 Nov 2015</div><div class="t m1 x2 h3 y2 ff1 fs1 fc1 sc0 ls0 ws0">1</div><div class="t m1 x3 h4 y3 ff1 fs2 fc1 sc0 ls0 ws0">An Efﬁcient Bayesian PAPR Reduction Method for</div><div class="t m1 x4 h4 y4 ff1 fs2 fc1 sc0 ls0 ws0">OFDM-Based Massive MIMO Systems</div><div class="t m1 x5 h5 y5 ff1 fs3 fc1 sc0 ls0 ws0">Hengyao Bao, Jun Fang, Zhi Chen, Hongbin Li, Senior Member, IEEE, and Shaoqian Li, Fellow, IEEE</div><div class="t m1 x6 h6 y6 ff3 fs4 fc1 sc0 ls0 ws0">Abstract—We consider the problem of peak-to-average power</div><div class="t m1 x7 h6 y7 ff4 fs4 fc1 sc0 ls0 ws0">ratio (PAPR) reduction in orthogonal frequency-division multi-</div><div class="t m1 x7 h6 y8 ff4 fs4 fc1 sc0 ls0 ws0">plexing (OFDM) based massive multiple-input multiple-output</div><div class="t m1 x7 h6 y9 ff4 fs4 fc1 sc0 ls0 ws0">(MIMO) downlink systems. Speciﬁcally, given a set of symbol</div><div class="t m1 x7 h6 ya ff4 fs4 fc1 sc0 ls0 ws0">vectors to be transmitted to K users, the problem is to ﬁnd an</div><div class="t m1 x7 h6 yb ff4 fs4 fc1 sc0 ls0 ws0">OFDM-modulated signal that has a low PAPR and meanwhile en-</div><div class="t m1 x7 h6 yc ff4 fs4 fc1 sc0 ls0 ws0">ables multiuser interference (MUI) cancelation. Unlike previous</div><div class="t m1 x7 h6 yd ff4 fs4 fc1 sc0 ls0 ws0">works that tackled the problem using convex optimization, we</div><div class="t m1 x7 h6 ye ff4 fs4 fc1 sc0 ls0 ws0">take a Bayesian approach and develop an efﬁcient PAPR reduc-</div><div class="t m1 x7 h6 yf ff4 fs4 fc1 sc0 ls0 ws0">tion method by exploiting the redundant degrees-of- freedom of</div><div class="t m1 x7 h6 y10 ff4 fs4 fc1 sc0 ls0 ws0">the transmit array. The sought-after signal is treated as a random</div><div class="t m1 x7 h6 y11 ff4 fs4 fc1 sc0 ls0 ws0">vector with a hierarchical truncated Gaussian mixture prior,</div><div class="t m1 x7 h6 y12 ff4 fs4 fc1 sc0 ls0 ws0">which has the potential to encourage a low PAPR signal with</div><div class="t m1 x7 h6 y13 ff4 fs4 fc1 sc0 ls0 ws0">most of its samples concentrated on the boundaries. A variational</div><div class="t m1 x7 h6 y14 ff4 fs4 fc1 sc0 ls0 ws0">expectation-maximization (EM) strategy is developed to obtain</div><div class="t m1 x7 h6 y15 ff4 fs4 fc1 sc0 ls0 ws0">estimates of the hyperparameters associated with the prior model,</div><div class="t m1 x7 h6 y16 ff4 fs4 fc1 sc0 ls0 ws0">along with the signal. In addition, the generalized approximate</div><div class="t m1 x7 h6 y17 ff4 fs4 fc1 sc0 ls0 ws0">message passing (GAMP) is embedded into the variational EM</div><div class="t m1 x7 h6 y18 ff4 fs4 fc1 sc0 ls0 ws0">framework, which results in a signiﬁcant reduction in computa-</div><div class="t m1 x7 h6 y19 ff4 fs4 fc1 sc0 ls0 ws0">tional complexity of the proposed algorithm. Simulation results</div><div class="t m1 x7 h6 y1a ff4 fs4 fc1 sc0 ls0 ws0">show our proposed algorithm achieves a substantial performance</div><div class="t m1 x7 h6 y1b ff4 fs4 fc1 sc0 ls0 ws0">improvement over existing methods in terms of both the PAPR</div><div class="t m1 x7 h6 y1c ff4 fs4 fc1 sc0 ls0 ws0">reduction and computational complexity.</div><div class="t m1 x6 h6 y1d ff3 fs4 fc1 sc0 ls0 ws0">Index Terms—Massive MIMO-OFDM, PAPR reduction, vari-</div><div class="t m1 x7 h6 y1e ff4 fs4 fc1 sc0 ls0 ws0">ational EM, GAMP.</div><div class="t m1 x8 h7 y1f ff1 fs5 fc1 sc0 ls0 ws0">I. INTRODUCTION</div><div class="t m1 x6 h7 y20 ff1 fs5 fc1 sc0 ls0 ws0">Massive multiple-input multiple-output (MIMO), also</div><div class="t m1 x7 h7 y21 ff1 fs5 fc1 sc0 ls0 ws0">known as large-scale or very-large MIMO, is a promising</div><div class="t m1 x7 h7 y22 ff1 fs5 fc1 sc0 ls0 ws0">technology to meet the ever growing demands for higher</div><div class="t m1 x7 h7 y23 ff1 fs5 fc1 sc0 ls0 ws0">throughput and better quality-of-service of next-generation</div><div class="t m1 x7 h7 y24 ff1 fs5 fc1 sc0 ls0 ws0">wireless communication systems [1]. Massive MIMO sys-</div><div class="t m1 x7 h7 y25 ff1 fs5 fc1 sc0 ls0 ws0">tems are those that are equipped with a large number of</div><div class="t m1 x7 h7 y26 ff1 fs5 fc1 sc0 ls0 ws0">antennas at the base station (BS) simultaneously serving a</div><div class="t m1 x7 h7 y27 ff1 fs5 fc1 sc0 ls0 ws0">much smaller number of single-antenna users sharing the same</div><div class="t m1 x7 h7 y28 ff1 fs5 fc1 sc0 ls0 ws0">time-frequency bandwidth. In addition to higher throughput,</div><div class="t m1 x7 h7 y29 ff1 fs5 fc1 sc0 ls0 ws0">massive MIMO systems also have the potential to improve</div><div class="t m1 x7 h7 y2a ff1 fs5 fc1 sc0 ls0 ws0">the energy efﬁciency and enable the use of inexpensive, low-</div><div class="t m1 x7 h7 y2b ff1 fs5 fc1 sc0 ls0 ws0">power components. Hence, it is expected that massive MIMO</div><div class="t m1 x7 h7 y2c ff1 fs5 fc1 sc0 ls0 ws0">will bring radical changes to future wireless communication</div><div class="t m1 x7 h7 y2d ff1 fs5 fc1 sc0 ls0 ws0">systems.</div><div class="t m1 x6 h7 y2e ff1 fs5 fc1 sc0 ls0 ws0">In practice, broadband wireless communications may suf-</div><div class="t m1 x7 h7 y2f ff1 fs5 fc1 sc0 ls0 ws0">fer from frequency-selective fading. Orthogonal frequency-</div><div class="t m1 x7 h7 y30 ff1 fs5 fc1 sc0 ls0 ws0">division multiplexing (OFDM), a scheme of encoding digital</div><div class="t m1 x9 h8 y31 ff1 fs6 fc1 sc0 ls0 ws0">Hengyao Bao, Jun Fang, Zhi Chen, and Shaoqian Li are with the National</div><div class="t m1 x7 h8 y32 ff1 fs6 fc1 sc0 ls0 ws0">Key Laboratory of Science and Technology on Communications, University</div><div class="t m1 x7 h8 y33 ff1 fs6 fc1 sc0 ls0 ws0">of Electronic Science and Technology of China, Chengdu 611731, China,</div><div class="t m1 x7 h8 y34 ff1 fs6 fc1 sc0 ls0 ws0">Email: JunFang@uestc.edu.cn, chenzhi@uestc.edu.cn, lsq@uestc.edu.cn</div><div class="t m1 x9 h8 y35 ff1 fs6 fc1 sc0 ls0 ws0">Hongbin Li is with the Department of Electrical and Computer Engineering,</div><div class="t m1 x7 h8 y36 ff1 fs6 fc1 sc0 ls0 ws0">Stevens Institute of Technology, Hoboken, NJ 07030, USA, E-mail: Hong-</div><div class="t m1 x7 h8 y37 ff1 fs6 fc1 sc0 ls0 ws0">bin.Li@stevens.edu</div><div class="t m1 x9 h8 y38 ff1 fs6 fc1 sc0 ls0 ws0">This work was supported in part by the National Science Foundation of</div><div class="t m1 x7 h8 y39 ff1 fs6 fc1 sc0 ls0 ws0">China under Grant 61172114.</div><div class="t m1 xa h7 y3a ff1 fs5 fc1 sc0 ls0 ws0">data on multiple carrier frequencies, has been widely used</div><div class="t m1 xa h7 y3b ff1 fs5 fc1 sc0 ls0 ws0">to deal with frequency-selective fading. However, a major</div><div class="t m1 xa h7 y3c ff1 fs5 fc1 sc0 ls0 ws0">problem associated with the OFDM is that it is subject to</div><div class="t m1 xa h7 y3d ff1 fs5 fc1 sc0 ls0 ws0">a high peak-to-average power ratio (PAPR) owing to the</div><div class="t m1 xa h7 y3e ff1 fs5 fc1 sc0 ls0 ws0">independent phases of the sub-carriers [2]. To avoid out-</div><div class="t m1 xa h7 y3f ff1 fs5 fc1 sc0 ls0 ws0">of-band radiation and signal distortion, handling this high</div><div class="t m1 xa h7 y40 ff1 fs5 fc1 sc0 ls0 ws0">PAPR requires a high-resolution digital-to-analog converter</div><div class="t m1 xa h7 y41 ff1 fs5 fc1 sc0 ls0 ws0">(DAC) and a linear power ampliﬁer (PA) at the transmitter,</div><div class="t m1 xa h7 y42 ff1 fs5 fc1 sc0 ls0 ws0">which is not only expensive but also power-inefﬁcient [3]. The</div><div class="t m1 xa h7 y43 ff1 fs5 fc1 sc0 ls0 ws0">situation deteriorates when the number of antennas is large,</div><div class="t m1 xa h7 y44 ff1 fs5 fc1 sc0 ls0 ws0">leaving such systems impractical. Therefore, it is of crucial</div><div class="t m1 xa h7 y45 ff1 fs5 fc1 sc0 ls0 ws0">importance to reduce the PAPR of massive MIMO-OFDM</div><div class="t m1 xa h7 y46 ff1 fs5 fc1 sc0 ls0 ws0">systems to facilitate low-cost and power-efﬁcient hardware</div><div class="t m1 xa h7 y47 ff1 fs5 fc1 sc0 ls0 ws0">implementations.</div><div class="t m1 xb h7 y48 ff1 fs5 fc1 sc0 ls0 ws0">Many techniques have been developed for PAPR reduction</div><div class="t m1 xa h7 y49 ff1 fs5 fc1 sc0 ls0 ws0">in single-input single-output (SISO) OFDM wireless systems.</div><div class="t m1 xa h7 y4a ff1 fs5 fc1 sc0 ls0 ws0">The most prominent are clipping [4], tone reservation (TR)</div><div class="t m1 xa h7 y4b ff1 fs5 fc1 sc0 ls0 ws0">[5], active constellation extension (ACE) [6], selected mapping</div><div class="t m1 xa h7 y4c ff1 fs5 fc1 sc0 ls0 ws0">(SLM) [7], partial transmission sequence (PTS) [8] and others.</div><div class="t m1 xa h7 y4d ff1 fs5 fc1 sc0 ls0 ws0">For a detailed overview, we refer readers to [3], [9]. Although</div><div class="t m1 xa h7 y4e ff1 fs5 fc1 sc0 ls0 ws0">these PAPR-reduction schemes can be extended to point-to-</div><div class="t m1 xa h7 y4f ff1 fs5 fc1 sc0 ls0 ws0">point MIMO systems easily [9]–[11], extension to the multi-</div><div class="t m1 xa h7 y50 ff1 fs5 fc1 sc0 ls0 ws0">user (MU) MIMO downlink is not straightforward, mainly be-</div><div class="t m1 xa h7 y51 ff1 fs5 fc1 sc0 ls0 ws0">cause joint receiver-side signal processing is almost impossible</div><div class="t m1 xa h7 y52 ff1 fs5 fc1 sc0 ls0 ws0">in practice as the users are distributed. Recently, a new PAPR</div><div class="t m1 xa h7 y53 ff1 fs5 fc1 sc0 ls0 ws0">reduction method [12] was developed for massive MIMO-</div><div class="t m1 xa h7 y54 ff1 fs5 fc1 sc0 ls0 ws0">OFDM systems. The proposed scheme utilizes the redundant</div><div class="t m1 xa h7 y55 ff1 fs5 fc1 sc0 ls0 ws0">degrees-of-freedom (DoFs) resulting from the large number</div><div class="t m1 xa h7 y56 ff1 fs5 fc1 sc0 ls0 ws0">of antennas at the BS to achieve joint multiuser interference</div><div class="t m1 xa h7 y57 ff1 fs5 fc1 sc0 ls0 ws0">(MUI) cancelation and PAPR reduction. Speciﬁcally, the prob-</div><div class="t m1 xa h7 y58 ff1 fs5 fc1 sc0 ls0 ws0">lem was formulated as a linear constrained ℓ</div><div class="t m1 xc h9 y59 ff7 fs1 fc1 sc0 ls0 ws0">∞</div><div class="t m1 xd h7 y58 ff1 fs5 fc1 sc0 ls0 ws0">optimization</div><div class="t m1 xa h7 y5a ff1 fs5 fc1 sc0 ls0 ws0">problem and a fast iterative truncation algorithm (FITRA)</div><div class="t m1 xa h7 y5b ff1 fs5 fc1 sc0 ls0 ws0">was developed in [12]. However, the FITRA algorithm shows</div><div class="t m1 xa h7 y5c ff1 fs5 fc1 sc0 ls0 ws0">to have a fairly low convergence rate. Also, the algorithm</div><div class="t m1 xa h7 y5d ff1 fs5 fc1 sc0 ls0 ws0">employs a regularization parameter to achieve balance between</div><div class="t m1 xa h7 y5e ff1 fs5 fc1 sc0 ls0 ws0">the PAPR reduction and the MUI cancelation (i.e. data ﬁt-</div><div class="t m1 xa h7 y5f ff1 fs5 fc1 sc0 ls0 ws0">ting error). The choice of the regularization parameter may</div><div class="t m1 xa h7 y60 ff1 fs5 fc1 sc0 ls0 ws0">be tricky in practice. On the other hand, the regularization</div><div class="t m1 xa h7 y61 ff1 fs5 fc1 sc0 ls0 ws0">parameter may be seen instead as an additional degree of</div><div class="t m1 xa h7 y62 ff1 fs5 fc1 sc0 ls0 ws0">freedom that allows to regulate the operation of the algorithm.</div><div class="t m1 xa h7 y63 ff1 fs5 fc1 sc0 ls0 ws0">In [13], a peak signal clipping scheme was employed to reduce</div><div class="t m1 xa h7 y64 ff1 fs5 fc1 sc0 ls0 ws0">the PAPR and some of the antennas at the BS are reserved to</div><div class="t m1 xa h7 y65 ff1 fs5 fc1 sc0 ls0 ws0">compensate for peak-clipping signals. This method has a lower</div><div class="t m1 xa h7 y66 ff1 fs5 fc1 sc0 ls0 ws0">computational complexity. But it achieves only a mild PAPR</div><div class="t m1 xa h7 y67 ff1 fs5 fc1 sc0 ls0 ws0">reduction and those antennas reserved for compensation may</div><div class="t m1 xa h7 y68 ff1 fs5 fc1 sc0 ls0 ws0">incur large PAPRs.</div><div class="t m1 xb h7 y69 ff1 fs5 fc1 sc0 ls0 ws0">In this paper, we develop a novel Bayesian approach to</div><div class="t m1 xa h7 y6a ff1 fs5 fc1 sc0 ls0 ws0">address the joint PAPR reduction and MUI cancelation prob-</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a></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/627384f07ec58a2c0130b3e1/bg2.jpg"><div class="t m1 x2 h3 y2 ff1 fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x7 h7 y6b ff1 fs5 fc1 sc0 ls0 ws0">lem for downlink multi-user massive MIMO-OFDM systems.</div><div class="t m1 x7 h7 y6c ff1 fs5 fc1 sc0 ls0 ws0">Speciﬁcally, MUI cancelation can be formulated as an un-</div><div class="t m1 x7 h7 y6d ff1 fs5 fc1 sc0 ls0 ws0">derdetermined linear inverse problem which admits numerous</div><div class="t m1 x7 h7 y6e ff1 fs5 fc1 sc0 ls0 ws0">solutions. To search for a low PAPR solution, a hierarchical</div><div class="t m1 x7 h7 y6f ff1 fs5 fc1 sc0 ls0 ws0">truncated Gaussian mixture prior model is proposed and</div><div class="t m1 x7 h7 y70 ff1 fs5 fc1 sc0 ls0 ws0">assigned to the unknown signal (i.e. solution). This hierar-</div><div class="t m1 x7 h7 y71 ff1 fs5 fc1 sc0 ls0 ws0">chical prior has the potential to encourage a quasi-constant</div><div class="t m1 x7 h7 y72 ff1 fs5 fc1 sc0 ls0 ws0">magnitude solution with as many entries as possible lying</div><div class="t m1 x7 h7 y73 ff1 fs5 fc1 sc0 ls0 ws0">on the truncated boundaries, thus resulting in a low PAPR.</div><div class="t m1 x7 h7 y74 ff1 fs5 fc1 sc0 ls0 ws0">A variational expectation-maximization (EM) algorithm is de-</div><div class="t m1 x7 h7 y75 ff1 fs5 fc1 sc0 ls0 ws0">veloped to obtain estimates of the hyperparameters associated</div><div class="t m1 x7 h7 y76 ff1 fs5 fc1 sc0 ls0 ws0">with the prior model, along with the signal. In addition, the</div><div class="t m1 x7 h7 y77 ff1 fs5 fc1 sc0 ls0 ws0">generalized approximate message passing (GAMP) technique</div><div class="t m1 x7 h7 y78 ff1 fs5 fc1 sc0 ls0 ws0">[14] is employed to facilitate the algorithm development</div><div class="t m1 x7 h7 y79 ff1 fs5 fc1 sc0 ls0 ws0">in the expectation step. This GAMP technique also helps</div><div class="t m1 x7 h7 y7a ff1 fs5 fc1 sc0 ls0 ws0">signiﬁcantly reduce the computational complexity of the pro-</div><div class="t m1 x7 h7 y7b ff1 fs5 fc1 sc0 ls0 ws0">posed algorithm. Simulation results show that the proposed</div><div class="t m1 x7 h7 y7c ff1 fs5 fc1 sc0 ls0 ws0">method presents a substantial improvement over the FITRA</div><div class="t m1 x7 h7 y7d ff1 fs5 fc1 sc0 ls0 ws0">algorithm in terms of both PAPR reduction and computational</div><div class="t m1 x7 h7 y7e ff1 fs5 fc1 sc0 ls0 ws0">complexity.</div><div class="t m1 x6 h7 y7f ff1 fs5 fc1 sc0 ls0 ws0">During the review process of the current work, it was</div><div class="t m1 x7 h7 y80 ff1 fs5 fc1 sc0 ls0 ws0">brought to our attention that an efﬁcient approximate message</div><div class="t m1 x7 h7 y81 ff1 fs5 fc1 sc0 ls0 ws0">passing (AMP)-based Bayesian method was recently proposed</div><div class="t m1 x7 h7 y82 ff1 fs5 fc1 sc0 ls0 ws0">[15] for PAPR reduction for massive MIMO systems, which</div><div class="t m1 x7 h7 y83 ff1 fs5 fc1 sc0 ls0 ws0">can be extended to the case with OFDM modulation. The</div><div class="t m1 x7 h7 y84 ff1 fs5 fc1 sc0 ls0 ws0">rationale behind our work and the above work are similar:</div><div class="t m1 x7 h7 y85 ff1 fs5 fc1 sc0 ls0 ws0">both methods cast the PAPR reduction problem as a Bayesian</div><div class="t m1 x7 h7 y86 ff1 fs5 fc1 sc0 ls0 ws0">inference problem and employ priors to promote solutions with</div><div class="t m1 x7 h7 y87 ff1 fs5 fc1 sc0 ls0 ws0">constant envelopes. The prior distributions employed by these</div><div class="t m1 x7 h7 y88 ff1 fs5 fc1 sc0 ls0 ws0">two works, however, are very different. The prior proposed in</div><div class="t m1 x7 h7 y89 ff1 fs5 fc1 sc0 ls0 ws0">[15] assigns each coefﬁcient to a random point on a circle with</div><div class="t m1 x7 h7 y8a ff1 fs5 fc1 sc0 ls0 ws0">a certain radius on the complex plane. Unlike our work, this</div><div class="t m1 x7 h7 y8b ff1 fs5 fc1 sc0 ls0 ws0">prior only encourages entries of the obtained solution to be</div><div class="t m1 x7 h7 y8c ff1 fs5 fc1 sc0 ls0 ws0">close to the boundary but cannot guarantee that they exactly</div><div class="t m1 x7 h7 y8d ff1 fs5 fc1 sc0 ls0 ws0">lie on the boundary points.</div><div class="t m1 x6 h7 y8e ff1 fs5 fc1 sc0 ls0 ws0">The rest of this paper is organized as follows. In Section II,</div><div class="t m1 x7 h7 y8f ff1 fs5 fc1 sc0 ls0 ws0">we introduce the data model, basic assumptions, and the PAPR</div><div class="t m1 x7 h7 y90 ff1 fs5 fc1 sc0 ls0 ws0">reduction problem. A new hierarchical Bayesian prior model</div><div class="t m1 x7 h7 y91 ff1 fs5 fc1 sc0 ls0 ws0">is proposed in Section III, and an efﬁcient Bayesian algorithm</div><div class="t m1 x7 h7 y92 ff1 fs5 fc1 sc0 ls0 ws0">is developed in Section IV. Simulation results are provided in</div><div class="t m1 x7 h7 y93 ff1 fs5 fc1 sc0 ls0 ws0">Section V, followed by concluding remarks in Section VI.</div><div class="t m1 x6 h7 y94 ff2 fs5 fc1 sc0 ls0 ws0">Notations: Lowercase boldface is used for column vectors</div><div class="t m1 x7 h7 y95 ff8 fs5 fc1 sc0 ls0 ws0">x, and uppercase for matrices X. The superscripts (·)</div><div class="t m1 xe ha y96 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x7 h7 y97 ff1 fs5 fc1 sc0 ls0 ws0">and (·)</div><div class="t m1 xf ha y98 ffb fs1 fc1 sc0 ls0 ws0">H</div><div class="t m1 x10 h7 y97 ff1 fs5 fc1 sc0 ls0 ws0">represent the transpose and conjugate transpose,</div><div class="t m1 x7 h7 y99 ff1 fs5 fc1 sc0 ls0 ws0">respectively. kxk</div><div class="t m1 x11 h9 y9a ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x12 h7 y99 ff1 fs5 fc1 sc0 ls0 ws0">is used to denote the ℓ</div><div class="t m1 x13 h9 y9a ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x14 h7 y99 ff1 fs5 fc1 sc0 ls0 ws0">norm of vector</div><div class="t m1 x7 h7 y9b ff8 fs5 fc1 sc0 ls0 ws0">x, and kxk</div><div class="t m1 x15 h9 y9c ff7 fs1 fc1 sc0 ls0 ws0">∞</div><div class="t m1 x16 h7 y9b ff1 fs5 fc1 sc0 ls0 ws0">stands for the ℓ</div><div class="t m1 x17 h9 y9c ff7 fs1 fc1 sc0 ls0 ws0">∞</div><div class="t m1 x18 h7 y9b ff1 fs5 fc1 sc0 ls0 ws0">norm, ℓ</div><div class="t m1 x19 h9 y9d ffd fs1 fc1 sc0 ls0 ws0">f∞</div><div class="t m1 x1a h7 y9b ff1 fs5 fc1 sc0 ls0 ws0">norm is deﬁne</div><div class="t m1 x7 h7 y9e ff1 fs5 fc1 sc0 ls0 ws0">as kxk</div><div class="t m1 xf ha y9f ffb fs1 fc1 sc0 ls0 ws0">ℓ</div><div class="t m1 x1b hb ya0 ffd fs7 fc1 sc0 ls0 ws0">f∞</div><div class="t m1 x1c hc ya1 ff9 fs5 fc1 sc0 ls0 ws0">= max{kℜ{x}k</div><div class="t m1 x1d h9 ya2 ff7 fs1 fc1 sc0 ls0 ws0">∞</div><div class="t m1 x1e hc ya1 ff6 fs5 fc1 sc0 ls0 ws0">, kℑ{x}k</div><div class="t m1 x1f h9 ya2 ff7 fs1 fc1 sc0 ls0 ws0">∞</div><div class="t m1 x20 h7 ya1 ffa fs5 fc1 sc0 ls0 ws0">}, with ℜ{x} and</div><div class="t m1 x7 h7 ya3 ffa fs5 fc1 sc0 ls0 ws0">ℑ{x} denoting the real and imaginary part of x, respectively.</div><div class="t m1 x7 hd ya4 ff8 fs5 fc1 sc0 ls0 ws0">F</div><div class="t m1 x9 ha ya5 ffb fs1 fc1 sc0 ls0 ws0">N</div><div class="t m1 x21 h7 ya4 ff1 fs5 fc1 sc0 ls0 ws0">denotes the N × N unitary discrete Fourier transform</div><div class="t m1 x7 h7 ya6 ff1 fs5 fc1 sc0 ls0 ws0">(DFT) matrix. The N ×N identity matrix and the M ×N all-</div><div class="t m1 x7 h7 ya7 ff1 fs5 fc1 sc0 ls0 ws0">zeros matrix are denoted by I</div><div class="t m1 x22 ha ya8 ffb fs1 fc1 sc0 ls0 ws0">N</div><div class="t m1 x23 h7 ya7 ff1 fs5 fc1 sc0 ls0 ws0">and 0</div><div class="t m1 x24 h9 ya8 ffb fs1 fc1 sc0 ls0 ws0">M×N</div><div class="t m1 x25 h7 ya7 ff1 fs5 fc1 sc0 ls0 ws0">, respectively. We</div><div class="t m1 x7 h7 ya9 ff1 fs5 fc1 sc0 ls0 ws0">denote the pdf of Gaussian random variable x with mean µ and</div><div class="t m1 x7 h7 yaa ff1 fs5 fc1 sc0 ls0 ws0">variance σ</div><div class="t m1 x26 h9 yab ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x15 h7 yaa ff1 fs5 fc1 sc0 ls0 ws0">as N(x; µ, σ</div><div class="t m1 x27 h9 yab ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x28 h7 yaa ff9 fs5 fc1 sc0 ls0 ws0">), for the special case of N(x; 0, 1),</div><div class="t m1 x7 h7 yac ff1 fs5 fc1 sc0 ls0 ws0">we write the cdf as Φ(x). The symbol ⊗ denotes the Kronecker</div><div class="t m1 x7 h7 yad ff1 fs5 fc1 sc0 ls0 ws0">product.</div><div class="t m1 x29 h7 yae ff1 fs5 fc1 sc0 ls0 ws0">II. SYSTEM MODEL AND PROBLEM FORMULATION</div><div class="t m1 xb h7 yaf ff1 fs5 fc1 sc0 ls0 ws0">We ﬁrst introduce the system model of OFDM based mas-</div><div class="t m1 xa h7 yb0 ff1 fs5 fc1 sc0 ls0 ws0">sive MIMO systems. Then we discuss some recent research</div><div class="t m1 xa h7 yb1 ff1 fs5 fc1 sc0 ls0 ws0">on PAPR reduction for multi-user massive MIMO-OFDM</div><div class="t m1 xa h7 yb2 ff1 fs5 fc1 sc0 ls0 ws0">systems.</div><div class="t m1 xa he yb3 ff2 fs5 fc1 sc0 ls0 ws0">A. System Model</div><div class="t m1 xb h7 yb4 ff1 fs5 fc1 sc0 ls0 ws0">The system model of the OFDM-based massive MIMO</div><div class="t m1 xa h7 yb5 ff1 fs5 fc1 sc0 ls0 ws0">downlink scenario is depicted in Fig. 1, where the BS is</div><div class="t m1 xa h7 yb6 ff1 fs5 fc1 sc0 ls0 ws0">assumed to have M transmit antennas and serve K indepen-</div><div class="t m1 xa h7 yb7 ff1 fs5 fc1 sc0 ls0 ws0">dent single-antenna users (K ≪ M ), and the total number of</div><div class="t m1 xa h7 yb8 ff1 fs5 fc1 sc0 ls0 ws0">OFDM tones is N . In practice, the set of tones available are</div><div class="t m1 xa h7 yb9 ff1 fs5 fc1 sc0 ls0 ws0">divided into two sets T and T</div><div class="t m1 x2a h9 yba ff7 fs1 fc1 sc0 ls0 ws0">C</div><div class="t m1 x2b h7 yb9 ff1 fs5 fc1 sc0 ls0 ws0">, where the tones in set T are</div><div class="t m1 xa h7 ybb ff1 fs5 fc1 sc0 ls0 ws0">used for data transmission and the tones in its complementary</div><div class="t m1 xa h7 ybc ff1 fs5 fc1 sc0 ls0 ws0">set T</div><div class="t m1 x2c h9 ybd ff7 fs1 fc1 sc0 ls0 ws0">C</div><div class="t m1 x2d h7 ybc ff1 fs5 fc1 sc0 ls0 ws0">are used for guard band (unused tones at both ends of</div><div class="t m1 xa h7 ybe ff1 fs5 fc1 sc0 ls0 ws0">the spectrum). Hence, for each tone n ∈ T , the corresponding</div><div class="t m1 xa h7 ybf ff6 fs5 fc1 sc0 ls0 ws0">K × 1 vector s</div><div class="t m1 x2e ha yc0 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x2f h7 ybf ff1 fs5 fc1 sc0 ls0 ws0">comprises the symbols for K users, which</div><div class="t m1 xa h7 yc1 ff1 fs5 fc1 sc0 ls0 ws0">are usually chosen from a complex-valued signal alphabet B.</div><div class="t m1 xa h7 yc2 ff1 fs5 fc1 sc0 ls0 ws0">We normalize the data vector to satisfy E{ks</div><div class="t m1 x30 ha yc3 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x31 hc yc2 ffa fs5 fc1 sc0 ls0 ws0">k</div><div class="t m1 xd h9 yc4 ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 xd h9 yc5 ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x32 h7 yc2 ffa fs5 fc1 sc0 ls0 ws0">} = 1. For</div><div class="t m1 xa h7 yc6 ff1 fs5 fc1 sc0 ls0 ws0">each tone n ∈ T</div><div class="t m1 x33 h9 yc7 ff7 fs1 fc1 sc0 ls0 ws0">C</div><div class="t m1 x34 h7 yc6 ff1 fs5 fc1 sc0 ls0 ws0">, we set s</div><div class="t m1 x35 ha yc8 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x2a hf yc6 ff9 fs5 fc1 sc0 ls0 ws0">= 0</div><div class="t m1 x36 h9 yc8 ffb fs1 fc1 sc0 ls0 ws0">K×1</div><div class="t m1 x37 h7 yc6 ff1 fs5 fc1 sc0 ls0 ws0">such that no signal is</div><div class="t m1 xa h7 yc9 ff1 fs5 fc1 sc0 ls0 ws0">transmitted in the guard band.</div><div class="t m1 xb h7 yca ff1 fs5 fc1 sc0 ls0 ws0">Since cooperative detection among users is often impossi-</div><div class="t m1 xa h7 ycb ff1 fs5 fc1 sc0 ls0 ws0">ble, precoding must be performed at the BS to remove multi-</div><div class="t m1 xa h7 ycc ff1 fs5 fc1 sc0 ls0 ws0">user interference (MUI). Usually, the signal vector on the nth</div><div class="t m1 xa h7 ycd ff1 fs5 fc1 sc0 ls0 ws0">tone is linearly precoded as</div><div class="t m1 x38 hd yce ff8 fs5 fc1 sc0 ls0 ws0">w</div><div class="t m1 x39 ha ycf ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x3a hf yce ff9 fs5 fc1 sc0 ls0 ws0">= P</div><div class="t m1 x3b ha ycf ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x3c hd yce ff8 fs5 fc1 sc0 ls0 ws0">s</div><div class="t m1 x3d ha ycf ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x3e h7 yce ff6 fs5 fc1 sc0 ls0 ws0">, (1)</div><div class="t m1 xa h7 yd0 ff1 fs5 fc1 sc0 ls0 ws0">where w</div><div class="t m1 x3f ha yd1 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x40 hc yd0 ffa fs5 fc1 sc0 ls0 ws0">∈ C</div><div class="t m1 x41 h9 yd2 ffb fs1 fc1 sc0 ls0 ws0">M×1</div><div class="t m1 x42 h7 yd0 ff1 fs5 fc1 sc0 ls0 ws0">is the precoded vector that contains</div><div class="t m1 xa h7 yd3 ff1 fs5 fc1 sc0 ls0 ws0">symbols to be transmitted on the nth sub-carrier through</div><div class="t m1 xa h7 yd4 ff1 fs5 fc1 sc0 ls0 ws0">the M antennas respectively, and P</div><div class="t m1 x43 ha yd5 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x44 hc yd4 ffa fs5 fc1 sc0 ls0 ws0">∈ C</div><div class="t m1 x45 h9 yd6 ffb fs1 fc1 sc0 ls0 ws0">M×K</div><div class="t m1 x46 h7 yd4 ff1 fs5 fc1 sc0 ls0 ws0">represents</div><div class="t m1 xa h7 yd7 ff1 fs5 fc1 sc0 ls0 ws0">the precoding matrix for the nth OFDM tone. Zero-forcing</div><div class="t m1 xa h7 yd8 ff1 fs5 fc1 sc0 ls0 ws0">(ZF) precoding and minimum-mean square-error (MMSE)</div><div class="t m1 xa h7 yd9 ff1 fs5 fc1 sc0 ls0 ws0">precoding are two classical precoding schemes. The former</div><div class="t m1 xa h7 yda ff1 fs5 fc1 sc0 ls0 ws0">aims at removing MUI completely, while the latter tries to</div><div class="t m1 xa h7 ydb ff1 fs5 fc1 sc0 ls0 ws0">achieve balance between the MUI cancellation and the noise</div><div class="t m1 xa h7 ydc ff1 fs5 fc1 sc0 ls0 ws0">enhancement. In this paper, we consider the ZF procoding</div><div class="t m1 xa h7 ydd ff1 fs5 fc1 sc0 ls0 ws0">scheme. Note that since K ≪ M, the ZF precoding matrix</div><div class="t m1 xa h7 yde ff1 fs5 fc1 sc0 ls0 ws0">has an inﬁnite number of forms, among which the most widely</div><div class="t m1 xa h7 ydf ff1 fs5 fc1 sc0 ls0 ws0">used is</div><div class="t m1 x2f hd ye0 ff8 fs5 fc1 sc0 ls0 ws0">P</div><div class="t m1 x47 h9 ye1 ffc fs1 fc1 sc0 ls0 ws0">zf</div><div class="t m1 x47 ha ye2 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x48 hf ye0 ff9 fs5 fc1 sc0 ls0 ws0">= H</div><div class="t m1 x49 ha ye1 ffb fs1 fc1 sc0 ls0 ws0">H</div><div class="t m1 x49 ha ye2 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x4a hf ye0 ff9 fs5 fc1 sc0 ls0 ws0">(H</div><div class="t m1 x4b ha ye3 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x36 hd ye0 ff8 fs5 fc1 sc0 ls0 ws0">H</div><div class="t m1 x4c ha ye1 ffb fs1 fc1 sc0 ls0 ws0">H</div><div class="t m1 x4c ha ye2 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x4d hf ye0 ff9 fs5 fc1 sc0 ls0 ws0">)</div><div class="t m1 x37 h9 ye4 ff7 fs1 fc1 sc0 ls0 ws0">−1</div><div class="t m1 x4e h7 ye0 ff6 fs5 fc1 sc0 ls0 ws0">, (2)</div><div class="t m1 xa h7 ye5 ff1 fs5 fc1 sc0 ls0 ws0">where H</div><div class="t m1 x4f ha ye6 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x50 hc ye5 ffa fs5 fc1 sc0 ls0 ws0">∈ C</div><div class="t m1 x2f h9 ye7 ffb fs1 fc1 sc0 ls0 ws0">K×M</div><div class="t m1 x51 h7 ye5 ff1 fs5 fc1 sc0 ls0 ws0">denotes the MIMO channel matrix</div><div class="t m1 xa h7 ye8 ff1 fs5 fc1 sc0 ls0 ws0">associated with the nth tone. Here we assume the channel</div><div class="t m1 xa h7 ye9 ff1 fs5 fc1 sc0 ls0 ws0">matrix H</div><div class="t m1 x4f ha yea ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x52 h7 ye9 ff1 fs5 fc1 sc0 ls0 ws0">, ∀n to be known at the transmitter, which can be</div><div class="t m1 xa h7 yeb ff1 fs5 fc1 sc0 ls0 ws0">acquired by exploiting the channel reciprocity of time division</div><div class="t m1 xa h7 yec ff1 fs5 fc1 sc0 ls0 ws0">duplexing (TDD) systems (i.e., the downlink channel is the</div><div class="t m1 xa h7 yed ff1 fs5 fc1 sc0 ls0 ws0">transpose of the uplink channel).</div><div class="t m1 xb h7 yee ff1 fs5 fc1 sc0 ls0 ws0">After precoding, all precoded vectors w</div><div class="t m1 x53 ha yef ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x54 h7 yee ff1 fs5 fc1 sc0 ls0 ws0">are reordered to</div><div class="t m1 xa h7 yf0 ff6 fs5 fc1 sc0 ls0 ws0">M antennas for OFDM modulation,</div><div class="t m1 x2e hf yf1 ff9 fs5 fc1 sc0 ls0 ws0">[a</div><div class="t m1 x2f h9 yf2 ffc fs1 fc1 sc0 ls0 ws0">1</div><div class="t m1 x55 hc yf1 ffa fs5 fc1 sc0 ls0 ws0">·· · a</div><div class="t m1 x51 ha yf2 ffb fs1 fc1 sc0 ls0 ws0">M</div><div class="t m1 x56 hf yf1 ff9 fs5 fc1 sc0 ls0 ws0">] = [w</div><div class="t m1 x57 h9 yf2 ffc fs1 fc1 sc0 ls0 ws0">1</div><div class="t m1 x58 hc yf1 ffa fs5 fc1 sc0 ls0 ws0">· · · w</div><div class="t m1 x59 ha yf2 ffb fs1 fc1 sc0 ls0 ws0">N</div><div class="t m1 x5a hf yf1 ff9 fs5 fc1 sc0 ls0 ws0">]</div><div class="t m1 x5b ha yf3 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x45 h7 yf1 ff6 fs5 fc1 sc0 ls0 ws0">, (3)</div><div class="t m1 xa h7 yf4 ff1 fs5 fc1 sc0 ls0 ws0">where a</div><div class="t m1 x5c ha yf5 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x5d hc yf4 ffa fs5 fc1 sc0 ls0 ws0">∈ C</div><div class="t m1 x5e h9 yf6 ffb fs1 fc1 sc0 ls0 ws0">N×1</div><div class="t m1 x5f h7 yf4 ff1 fs5 fc1 sc0 ls0 ws0">represents the frequency-domain signal to</div><div class="t m1 xa h7 yf7 ff1 fs5 fc1 sc0 ls0 ws0">be transmitted from the mth antenna. The time-domain signals</div><div class="t m1 xa h7 yf8 ff1 fs5 fc1 sc0 ls0 ws0">are obtained through the inverse discrete Fourier transform</div><div class="t m1 xa h7 yf9 ff1 fs5 fc1 sc0 ls0 ws0">(IDFT), i.e., ˆa</div><div class="t m1 x60 ha yfa ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x33 hf yf9 ff9 fs5 fc1 sc0 ls0 ws0">= F</div><div class="t m1 x61 ha yfb ffb fs1 fc1 sc0 ls0 ws0">H</div><div class="t m1 x61 ha yfc ffb fs1 fc1 sc0 ls0 ws0">N</div><div class="t m1 x42 hd yf9 ff8 fs5 fc1 sc0 ls0 ws0">a</div><div class="t m1 x62 ha yfa ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x56 h7 yf9 ff1 fs5 fc1 sc0 ls0 ws0">, ∀m. Then, a cyclic preﬁx (CP) is</div><div class="t m1 xa h7 yfd ff1 fs5 fc1 sc0 ls0 ws0">added to the time-domain samples of each antenna to elimi-</div><div class="t m1 xa h7 yfe ff1 fs5 fc1 sc0 ls0 ws0">nate intersymbol interference (ISI). Finally, these samples are</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/627384f07ec58a2c0130b3e1/bg3.jpg"><div class="t m1 x2 h3 y2 ff1 fs1 fc1 sc0 ls0 ws0">3</div><div class="t m1 x7 h8 yff ff1 fs6 fc1 sc0 ls0 ws0">Fig. 1. System model for the downlink of OFDM based massive MIMO, with N OFDM tones,M transmit antennas and K independent single-antenna users.</div><div class="t m1 x7 h7 y100 ff1 fs5 fc1 sc0 ls0 ws0">converted to analog signals and transmitted via the frequency-</div><div class="t m1 x7 h7 y101 ff1 fs5 fc1 sc0 ls0 ws0">selective channel.</div><div class="t m1 x6 h7 y102 ff1 fs5 fc1 sc0 ls0 ws0">At the receivers, after removing the CPs of the received</div><div class="t m1 x7 h7 y103 ff1 fs5 fc1 sc0 ls0 ws0">signals, the DFT is performed to obtain the frequency-domain</div><div class="t m1 x7 h7 y104 ff1 fs5 fc1 sc0 ls0 ws0">signals. The receive vector consisting of K users’ signals can</div><div class="t m1 x7 h7 y105 ff1 fs5 fc1 sc0 ls0 ws0">be described as</div><div class="t m1 x63 hd y106 ff8 fs5 fc1 sc0 ls0 ws0">r</div><div class="t m1 x12 ha y107 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x64 hf y106 ff9 fs5 fc1 sc0 ls0 ws0">= H</div><div class="t m1 x65 ha y107 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x66 hd y106 ff8 fs5 fc1 sc0 ls0 ws0">w</div><div class="t m1 x22 ha y107 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x17 hf y106 ff9 fs5 fc1 sc0 ls0 ws0">+ e</div><div class="t m1 x67 ha y107 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x68 h7 y106 ff6 fs5 fc1 sc0 ls0 ws0">, ∀n (4)</div><div class="t m1 x7 h7 y108 ff1 fs5 fc1 sc0 ls0 ws0">where r</div><div class="t m1 x1b ha y109 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x69 hc y108 ffa fs5 fc1 sc0 ls0 ws0">∈ C</div><div class="t m1 x6a h9 y10a ffb fs1 fc1 sc0 ls0 ws0">K×1</div><div class="t m1 x12 h7 y108 ff1 fs5 fc1 sc0 ls0 ws0">denotes the receive vector associated with</div><div class="t m1 x7 h7 y10b ff1 fs5 fc1 sc0 ls0 ws0">the nth tone, and e</div><div class="t m1 x6b ha y10c ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x6c hc y10b ffa fs5 fc1 sc0 ls0 ws0">∈ C</div><div class="t m1 x65 h9 y10d ffb fs1 fc1 sc0 ls0 ws0">K×1</div><div class="t m1 x6d h7 y10b ff1 fs5 fc1 sc0 ls0 ws0">is the receiver noise and has</div><div class="t m1 x7 h7 y10e ff1 fs5 fc1 sc0 ls0 ws0">i.i.d. circularly symmetric complex Gaussian entries with zero-</div><div class="t m1 x7 h7 y10f ff1 fs5 fc1 sc0 ls0 ws0">mean and variance N</div><div class="t m1 x6e ha y110 ffb fs1 fc1 sc0 ls0 ws0">o</div><div class="t m1 x6f h7 y10f ff1 fs5 fc1 sc0 ls0 ws0">. If the ZF precoding scheme is used, by</div><div class="t m1 x7 h7 y111 ff1 fs5 fc1 sc0 ls0 ws0">combining (1), (2) and (4), the received signal vector equals</div><div class="t m1 x7 h7 y112 ff1 fs5 fc1 sc0 ls0 ws0">to r</div><div class="t m1 x70 ha y113 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x71 hf y112 ff9 fs5 fc1 sc0 ls0 ws0">= s</div><div class="t m1 x1c ha y113 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x72 hf y112 ff9 fs5 fc1 sc0 ls0 ws0">+ e</div><div class="t m1 x73 ha y113 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x74 h7 y112 ff1 fs5 fc1 sc0 ls0 ws0">, ∀n, which means the MUI is perfectly</div><div class="t m1 x7 h7 y114 ff1 fs5 fc1 sc0 ls0 ws0">removed.</div><div class="t m1 x7 he y115 ff2 fs5 fc1 sc0 ls0 ws0">B. Peak-to-Average Power Ratio (PAPR) Reduction</div><div class="t m1 x6 h7 y116 ff1 fs5 fc1 sc0 ls0 ws0">OFDM modulation typically exhibits a large dynamic range</div><div class="t m1 x7 h7 y117 ff1 fs5 fc1 sc0 ls0 ws0">because the phases of the sub-carriers are independent of each</div><div class="t m1 x7 h7 y118 ff1 fs5 fc1 sc0 ls0 ws0">other, which may combine in a constructive or destructive</div><div class="t m1 x7 h7 y119 ff1 fs5 fc1 sc0 ls0 ws0">manner. To avoid out-of-band radiation and signal distortion,</div><div class="t m1 x7 h7 y11a ff1 fs5 fc1 sc0 ls0 ws0">high-solution DACs and linear power ampliﬁers are required</div><div class="t m1 x7 h7 y11b ff1 fs5 fc1 sc0 ls0 ws0">at the transmitter to accommodate the large peaks of OFDM</div><div class="t m1 x7 h7 y11c ff1 fs5 fc1 sc0 ls0 ws0">signals, which leads to expensive and power-inefﬁcient RF</div><div class="t m1 x7 h7 y11d ff1 fs5 fc1 sc0 ls0 ws0">chains.</div><div class="t m1 x6 h7 y11e ff1 fs5 fc1 sc0 ls0 ws0">PAPR is deﬁned as the ratio of the peak power of the signal</div><div class="t m1 x7 h7 y11f ff1 fs5 fc1 sc0 ls0 ws0">to its average power. Speciﬁcally, the PAPR at the mth transmit</div><div class="t m1 x7 h7 y120 ff1 fs5 fc1 sc0 ls0 ws0">antenna is deﬁned as</div><div class="t m1 x74 h7 y121 ff1 fs5 fc1 sc0 ls0 ws0">PAPR</div><div class="t m1 x27 ha y122 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x75 h10 y121 ff12 fs5 fc1 sc0 ls0 ws0">,</div><div class="t m1 x22 hc y123 ff9 fs5 fc1 sc0 ls0 ws0">2Nkˆa</div><div class="t m1 x76 ha y124 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x77 hc y123 ffa fs5 fc1 sc0 ls0 ws0">k</div><div class="t m1 x78 h9 y125 ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x78 h9 y126 ffd fs1 fc1 sc0 ls0 ws0">f∞</div><div class="t m1 x79 hc y127 ffa fs5 fc1 sc0 ls0 ws0">kˆa</div><div class="t m1 x18 ha y128 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x68 hc y127 ffa fs5 fc1 sc0 ls0 ws0">k</div><div class="t m1 x77 h9 y129 ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x77 h9 y12a ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x7a h7 y12b ff6 fs5 fc1 sc0 ls0 ws0">, (5)</div><div class="t m1 x7 h7 y12c ff1 fs5 fc1 sc0 ls0 ws0">where the operator k · k</div><div class="t m1 x7b h9 y12d ffc fs1 fc1 sc0 ls0 ws0">2</div><div class="t m1 x7b h9 y12e ffd fs1 fc1 sc0 ls0 ws0">f∞</div><div class="t m1 x7c h7 y12c ff1 fs5 fc1 sc0 ls0 ws0">is used because RF-chains often</div><div class="t m1 x7 h7 y12f ff1 fs5 fc1 sc0 ls0 ws0">process and modulate the real and imaginary part of time-</div><div class="t m1 x7 h7 y130 ff1 fs5 fc1 sc0 ls0 ws0">domain samples independently. It should also be noted that,</div><div class="t m1 x7 h7 y131 ff1 fs5 fc1 sc0 ls0 ws0">we only consider the PAPR of discrete-time OFDM signals</div><div class="t m1 x7 h7 y132 ff1 fs5 fc1 sc0 ls0 ws0">in this paper, one can obtain its continuous-time counterpart</div><div class="t m1 x7 h7 y133 ff1 fs5 fc1 sc0 ls0 ws0">precisely by implementing an L-times oversampling in OFDM</div><div class="t m1 x7 h7 y134 ff1 fs5 fc1 sc0 ls0 ws0">modulation [16].</div><div class="t m1 x7d h3 y135 ff1 fs1 fc1 sc0 ls0 ws0">1</div><div class="t m1 x12 h7 y134 ff1 fs5 fc1 sc0 ls0 ws0">Since many conventional MIMO-OFDM</div><div class="t m1 x7 h7 y136 ff1 fs5 fc1 sc0 ls0 ws0">systems, such as 3GPP LTE [17] and IEEE 802.11.n [18],</div><div class="t m1 x7 h7 y137 ff1 fs5 fc1 sc0 ls0 ws0">disallow such an oversampling operation, here we ignore the</div><div class="t m1 x7 h7 y138 ff1 fs5 fc1 sc0 ls0 ws0">difference as in [12] (i.e., L = 1).</div><div class="t m1 x6 h7 y139 ff1 fs5 fc1 sc0 ls0 ws0">When the number of transmit antennas is larger than the</div><div class="t m1 x7 h7 y13a ff1 fs5 fc1 sc0 ls0 ws0">number of users, numerous ZF precoding matrices are avail-</div><div class="t m1 x7 h7 y13b ff1 fs5 fc1 sc0 ls0 ws0">able. In other words, for a set of s</div><div class="t m1 x7e ha y13c ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x7f h7 y13b ff1 fs5 fc1 sc0 ls0 ws0">, n = 1, ..., N , we have</div><div class="t m1 x9 h11 y13d ff1 fs8 fc1 sc0 ls0 ws0">1</div><div class="t m1 x80 h8 y13e ff1 fs6 fc1 sc0 ls0 ws0">Instead of N -point IDFT, L-times oversampling can be implemented by</div><div class="t m1 x7 h8 y13f ff11 fs6 fc1 sc0 ls0 ws0">LN-point IDFT of the frequency-domain signals with (L−1)N zero-padding.</div><div class="t m1 xa h7 y140 ff1 fs5 fc1 sc0 ls0 ws0">an inﬁnite number of precoded signals w , [w</div><div class="t m1 x81 ha y141 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x81 h9 y142 ffc fs1 fc1 sc0 ls0 ws0">1</div><div class="t m1 x82 hf y140 ff6 fs5 fc1 sc0 ls0 ws0">, ..., w</div><div class="t m1 x83 ha y141 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x83 ha y143 ffb fs1 fc1 sc0 ls0 ws0">N</div><div class="t m1 x84 hf y140 ff9 fs5 fc1 sc0 ls0 ws0">]</div><div class="t m1 x85 ha y141 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 xa h7 y144 ff1 fs5 fc1 sc0 ls0 ws0">that achieve perfect MUI cancelation. Thus there may exist a</div><div class="t m1 xa h7 y145 ff1 fs5 fc1 sc0 ls0 ws0">candidate w whose associated time-domain signals {ˆa</div><div class="t m1 x86 ha y146 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x87 h7 y145 ffa fs5 fc1 sc0 ls0 ws0">} have</div><div class="t m1 xa h7 y147 ff1 fs5 fc1 sc0 ls0 ws0">low PAPRs. In this paper, instead of designing the procoding</div><div class="t m1 xa h7 y148 ff1 fs5 fc1 sc0 ls0 ws0">matrix, we directly search for the signal w to achieve a joint</div><div class="t m1 xa h7 y149 ff1 fs5 fc1 sc0 ls0 ws0">PAPR reduction and MUI cancelation. Speciﬁcally, in order</div><div class="t m1 xa h7 y14a ff1 fs5 fc1 sc0 ls0 ws0">to remove the MUI, the precoded vectors w</div><div class="t m1 x88 ha y14b ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x89 h7 y14a ff1 fs5 fc1 sc0 ls0 ws0">need to satisfy:</div><div class="t m1 x8a hd y14c ff8 fs5 fc1 sc0 ls0 ws0">s</div><div class="t m1 x47 ha y14d ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x8b hf y14c ff9 fs5 fc1 sc0 ls0 ws0">= H</div><div class="t m1 x8c ha y14d ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x8d hd y14c ff8 fs5 fc1 sc0 ls0 ws0">w</div><div class="t m1 x8e ha y14d ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x8f h7 y14c ff6 fs5 fc1 sc0 ls0 ws0">, n ∈ T , (6a)</div><div class="t m1 x90 h12 y14e fff fs5 fc1 sc0 ls0 ws0">0</div><div class="t m1 x91 h9 y14f ffb fs1 fc1 sc0 ls0 ws0">M×1</div><div class="t m1 x39 hf y14e ff9 fs5 fc1 sc0 ls0 ws0">= w</div><div class="t m1 x8e ha y14f ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x8f hc y14e ff6 fs5 fc1 sc0 ls0 ws0">, n ∈ T</div><div class="t m1 x59 ha y150 ffb fs1 fc1 sc0 ls0 ws0">c</div><div class="t m1 x92 h7 y14e ff6 fs5 fc1 sc0 ls0 ws0">. (6b)</div><div class="t m1 xa h7 y151 ff1 fs5 fc1 sc0 ls0 ws0">The whole linear constraints of (6) can be further written as</div><div class="t m1 x39 h7 y152 ff8 fs5 fc1 sc0 ls0 ws0">s = Hw (7)</div><div class="t m1 xa h7 y153 ff1 fs5 fc1 sc0 ls0 ws0">where s ∈ C</div><div class="t m1 x93 h9 y154 ffb fs1 fc1 sc0 ls0 ws0">NK×1</div><div class="t m1 x90 h7 y153 ff1 fs5 fc1 sc0 ls0 ws0">denotes the concatenation of all vectors on</div><div class="t m1 xa h7 y155 ff1 fs5 fc1 sc0 ls0 ws0">the left-hand side of (6), H is a block diagonal matrix with its</div><div class="t m1 xa h7 y156 ff1 fs5 fc1 sc0 ls0 ws0">diagonal blocks equal to H</div><div class="t m1 x49 ha y157 ffb fs1 fc1 sc0 ls0 ws0">n</div><div class="t m1 x94 h7 y156 ff1 fs5 fc1 sc0 ls0 ws0">for n ∈ T and I</div><div class="t m1 x30 ha y157 ffb fs1 fc1 sc0 ls0 ws0">M</div><div class="t m1 x95 h7 y156 ff1 fs5 fc1 sc0 ls0 ws0">for n ∈ T</div><div class="t m1 x85 ha y158 ffb fs1 fc1 sc0 ls0 ws0">c</div><div class="t m1 x96 h7 y156 ff1 fs5 fc1 sc0 ls0 ws0">.</div><div class="t m1 xa h7 y159 ff1 fs5 fc1 sc0 ls0 ws0">According to (3), the reordering operation can be equivalently</div><div class="t m1 xa h7 y15a ff1 fs5 fc1 sc0 ls0 ws0">written as a linear transformation, i.e.,</div><div class="t m1 x39 h7 y15b ff8 fs5 fc1 sc0 ls0 ws0">a = T w, (8)</div><div class="t m1 xa h7 y15c ff1 fs5 fc1 sc0 ls0 ws0">where a = [a</div><div class="t m1 x97 ha y15d ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x97 h9 y15e ffc fs1 fc1 sc0 ls0 ws0">1</div><div class="t m1 x33 hf y15c ff6 fs5 fc1 sc0 ls0 ws0">, ..., a</div><div class="t m1 x48 ha y15d ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x48 ha y15f ffb fs1 fc1 sc0 ls0 ws0">M</div><div class="t m1 x62 hf y15c ff9 fs5 fc1 sc0 ls0 ws0">]</div><div class="t m1 x98 ha y15d ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x99 h7 y15c ff1 fs5 fc1 sc0 ls0 ws0">, T is a permutation matrix that</div><div class="t m1 xa h7 y160 ff1 fs5 fc1 sc0 ls0 ws0">assigns the M entries of each precoded vector to the M</div><div class="t m1 xa h7 y161 ff1 fs5 fc1 sc0 ls0 ws0">antennas respectively. Recalling ˆa</div><div class="t m1 x36 ha y162 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x9a hf y161 ff9 fs5 fc1 sc0 ls0 ws0">= F</div><div class="t m1 x4e ha y163 ffb fs1 fc1 sc0 ls0 ws0">H</div><div class="t m1 x4e ha y164 ffb fs1 fc1 sc0 ls0 ws0">N</div><div class="t m1 x9b hd y161 ff8 fs5 fc1 sc0 ls0 ws0">a</div><div class="t m1 xc ha y162 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 x31 h7 y161 ff6 fs5 fc1 sc0 ls0 ws0">, ∀m, (7) and</div><div class="t m1 xa h7 y165 ff1 fs5 fc1 sc0 ls0 ws0">(8), we have</div><div class="t m1 x42 hf y166 ff8 fs5 fc1 sc0 ls0 ws0">s = HT</div><div class="t m1 x9c ha y167 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x9d hd y166 ff8 fs5 fc1 sc0 ls0 ws0">F</div><div class="t m1 x9e h13 y168 ff15 fs5 fc1 sc0 ls0 ws0">b</div><div class="t m1 x9e h7 y166 ff8 fs5 fc1 sc0 ls0 ws0">a, (9)</div><div class="t m1 xa h7 y169 ff1 fs5 fc1 sc0 ls0 ws0">where</div><div class="t m1 x9f h10 y16a ff8 fs5 fc1 sc0 ls0 ws0">F , I</div><div class="t m1 xa0 ha y16b ffb fs1 fc1 sc0 ls0 ws0">M</div><div class="t m1 x34 hc y16a ffa fs5 fc1 sc0 ls0 ws0">⊗ F</div><div class="t m1 xa1 ha y16b ffb fs1 fc1 sc0 ls0 ws0">N</div><div class="t m1 x62 h7 y16a ff1 fs5 fc1 sc0 ls0 ws0">, and</div><div class="t m1 xa2 h13 y16c ff15 fs5 fc1 sc0 ls0 ws0">b</div><div class="t m1 xa2 h10 y16a ff8 fs5 fc1 sc0 ls0 ws0">a , [ˆa</div><div class="t m1 x37 ha y16d ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x37 h9 y16e ffc fs1 fc1 sc0 ls0 ws0">1</div><div class="t m1 xa3 hf y16f ff6 fs5 fc1 sc0 ls0 ws0">, ..., ˆa</div><div class="t m1 xa4 ha y170 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 xa4 ha y171 ffb fs1 fc1 sc0 ls0 ws0">M</div><div class="t m1 xd hf y172 ff9 fs5 fc1 sc0 ls0 ws0">]</div><div class="t m1 xa5 ha y173 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 x46 h7 y172 ff1 fs5 fc1 sc0 ls0 ws0">. Given a</div><div class="t m1 xa h7 y174 ff1 fs5 fc1 sc0 ls0 ws0">symbol vector</div><div class="t m1 xa6 h7 y175 ff8 fs5 fc1 sc0 ls0 ws0">s, our goal is to search for a signal</div><div class="t m1 x95 h13 y176 ff15 fs5 fc1 sc0 ls0 ws0">b</div><div class="t m1 x95 h7 y175 ff8 fs5 fc1 sc0 ls0 ws0">a satisfying</div><div class="t m1 xa h7 y177 ff1 fs5 fc1 sc0 ls0 ws0">the above equation (9), and meanwhile its sub-vector ˆa</div><div class="t m1 x87 ha y178 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 xa7 h7 y177 ff1 fs5 fc1 sc0 ls0 ws0">, i.e.</div><div class="t m1 xa h7 y179 ff1 fs5 fc1 sc0 ls0 ws0">the signal to be transmitted at each antenna, having a low</div><div class="t m1 xa h7 y17a ff1 fs5 fc1 sc0 ls0 ws0">PAPR. This problem can be formulated as a minimax problem</div><div class="t m1 xa h7 y17b ff1 fs5 fc1 sc0 ls0 ws0">which minimizes the maximum PAPR among all antennas</div><div class="t m1 xa h7 y17c ff1 fs5 fc1 sc0 ls0 ws0">subject to the linear constraint deﬁned in (9). Nevertheless,</div><div class="t m1 xa h7 y17d ff1 fs5 fc1 sc0 ls0 ws0">this problem, as indicated in [12], is complex to solve. To</div><div class="t m1 xa h7 y17e ff1 fs5 fc1 sc0 ls0 ws0">circumvent the difﬁculty, the minimax problem is replaced by</div><div class="t m1 xa h7 y17f ff1 fs5 fc1 sc0 ls0 ws0">a constrained optimization which minimizes the ℓ</div><div class="t m1 x32 h9 y180 ffd fs1 fc1 sc0 ls0 ws0">f∞</div><div class="t m1 x86 h7 y17f ff1 fs5 fc1 sc0 ls0 ws0">norm of</div><div class="t m1 xa h13 y181 ff15 fs5 fc1 sc0 ls0 ws0">b</div><div class="t m1 xa h7 y182 ff8 fs5 fc1 sc0 ls0 ws0">a, a vector formed by aggregating all time-domain vectors</div><div class="t m1 xa hc y183 ffa fs5 fc1 sc0 ls0 ws0">{ˆa</div><div class="t m1 xa8 ha y184 ffb fs1 fc1 sc0 ls0 ws0">m</div><div class="t m1 xa9 h7 y183 ffa fs5 fc1 sc0 ls0 ws0">} [12]</div><div class="t m1 xaa hc y185 ff9 fs5 fc1 sc0 ls0 ws0">min k</div><div class="t m1 x41 h13 y186 ff15 fs5 fc1 sc0 ls0 ws0">b</div><div class="t m1 x41 hc y185 ff8 fs5 fc1 sc0 ls0 ws0">ak</div><div class="t m1 x5f h9 y187 ffd fs1 fc1 sc0 ls0 ws0">f∞</div><div class="t m1 xab h7 y185 ff1 fs5 fc1 sc0 ls0 ws0">subject to</div><div class="t m1 xac hf y188 ff8 fs5 fc1 sc0 ls0 ws0">s = HT</div><div class="t m1 x88 ha y189 ffb fs1 fc1 sc0 ls0 ws0">T</div><div class="t m1 xad hd y188 ff8 fs5 fc1 sc0 ls0 ws0">F</div><div class="t m1 xae h13 y18a ff15 fs5 fc1 sc0 ls0 ws0">b</div><div class="t m1 xae h7 y188 ff8 fs5 fc1 sc0 ls0 ws0">a. (10)</div><div class="t m1 xa h7 y18b ff1 fs5 fc1 sc0 ls0 ws0">This problem can be further converted into a real-valued form</div><div class="t m1 xa h7 y18c ff1 fs5 fc1 sc0 ls0 ws0">as follows [12]</div><div class="t m1 x60 hc y18d ff9 fs5 fc1 sc0 ls0 ws0">min kxk</div><div class="t m1 xa1 h9 y18e ff7 fs1 fc1 sc0 ls0 ws0">∞</div><div class="t m1 x8c h7 y18d ff1 fs5 fc1 sc0 ls0 ws0">subject to y = Ax, (11)</div><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m2"></div></a></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>