多种MIMO检测算法,附有简单的注释。

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多种MIMO检测算法,包含ZF/MMSE/OSIC/SQRD/SD/SDR半正定/LR格基规约辅助检测算法,参考Erik G. Larsson的论文及多篇文献复现的代码,附有简单的注释。采用BPSK调制,可仿真2x2、4x4的MIMO系统。运行平台为Matlab R2021a或以前的版本等,其中SDR算法还可以使用cvx工具箱,由于需要额外安装所以注释了,有需要的可以反注释
MIMO.rar
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内容介绍
<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://csdnimg.cn/release/download_crawler_static/css/base.min.css"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/css/fancy.min.css"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/19105117/raw.css"><script src="https://csdnimg.cn/release/download_crawler_static/js/compatibility.min.js"></script><script src="https://csdnimg.cn/release/download_crawler_static/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://csdnimg.cn/release/download_crawler_static/19105117/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">IEEE SIGNAL PROCESSING MAGAZINE <span class="ff2 fc1">[<span class="fc0">91</span>]</span> MA<span class="_ _0"></span>Y 2009</div><div class="t m1 x2 h3 y2 ff3 fs1 fc2 sc0 ls0 ws0">[</div><div class="t m2 x3 h4 y3 ff4 fs2 fc0 sc0 ls1 ws0">lecture </div><div class="t m3 x4 h5 y4 ff2 fs2 fc3 sc0 ls2 ws0">NOTES</div><div class="t m1 x5 h3 y2 ff3 fs1 fc2 sc0 ls0 ws0">]</div><div class="t m3 x6 h6 y5 ff5 fs3 fc0 sc0 ls3 ws1"> <span class="_ _1"></span>Dig<span class="_ _2"></span>ital O<span class="_ _2"></span>bject Id<span class="_ _2"></span>e<span class="_ _2"></span>ntif<span class="_ _2"></span>i<span class="_ _2"></span>er 10.1109<span class="_ _2"></span>/<span class="_ _2"></span>M<span class="_ _2"></span>SP.<span class="_ _2"></span>20<span class="_ _2"></span>0<span class="_ _2"></span>9.<span class="_ _2"></span>93<span class="_ _2"></span>2126</div><div class="t m3 x7 h7 y6 ff3 fs4 fc3 sc0 ls0 ws0">I</div><div class="t m3 x8 h8 y7 ff6 fs5 fc0 sc0 ls4 ws2">n communications, the receiver<span class="_ _0"></span> </div><div class="t m3 x8 h8 y8 ff6 fs5 fc0 sc0 ls5 ws3">often observes a linear superposition </div><div class="t m3 x8 h8 y9 ff6 fs5 fc0 sc0 ls6 ws4">of separately transmitted informa-</div><div class="t m3 x8 h8 ya ff6 fs5 fc0 sc0 ls7 ws5">tion symbols. From the receiver&#8217;<span class="_ _0"></span>s </div><div class="t m3 x8 h8 yb ff6 fs5 fc0 sc0 ls8 ws5">perspective, the problem is then to </div><div class="t m3 x6 h8 yc ff6 fs5 fc0 sc0 ls5 ws6">separate the transmitted symbols. This is </div><div class="t m3 x6 h8 yd ff6 fs5 fc0 sc0 ls9 ws7">basically an inverse problem with a<span class="_ _0"></span> </div><div class="t m3 x6 h8 ye ff6 fs5 fc0 sc0 lsa ws5">finite-alphabet constraint. This lecture </div><div class="t m3 x6 h8 yf ff6 fs5 fc0 sc0 lsb ws4">will present an accessible overview of </div><div class="t m3 x6 h8 y10 ff6 fs5 fc0 sc0 lsc ws0">state-of-the-art solutions to this problem. </div><div class="t m3 x6 h9 y11 ff3 fs5 fc0 sc0 lsd ws0">RELEV<span class="_ _0"></span>ANCE</div><div class="t m3 x6 h8 y12 ff6 fs5 fc0 sc0 ls5 ws8">The most important motivating applica-</div><div class="t m3 x6 h8 y13 ff6 fs5 fc0 sc0 lse ws5">tion for the discussion here is receivers </div><div class="t m3 x6 h8 y14 ff6 fs5 fc0 sc0 lsf ws5">for multiple-antenna systems such as </div><div class="t m3 x6 h8 y15 ff6 fs5 fc0 sc0 ls5 ws9">multiple-input, multiple-output (MIMO), </div><div class="t m3 x6 h8 y16 ff6 fs5 fc0 sc0 ls10 ws5">where several transmit antennas simul-</div><div class="t m3 x6 h8 y17 ff6 fs5 fc0 sc0 ls11 ws5">taneously send different data streams. </div><div class="t m3 x6 h8 y18 ff6 fs5 fc0 sc0 ls12 ws4">However<span class="_ _0"></span>, essentially the same problem </div><div class="t m3 x6 h8 y19 ff6 fs5 fc0 sc0 ls13 ws5">occurs in systems where the channel </div><div class="t m3 x6 h8 y1a ff6 fs5 fc0 sc0 ls14 ws5">itself introduces time- or frequency-dis-</div><div class="t m3 x6 h8 y1b ff6 fs5 fc0 sc0 ls15 ws5">persion, in multiuser detection, and in </div><div class="t m3 x6 h8 y1c ff6 fs5 fc0 sc0 ls5 ws0">cancellation of crosstalk. </div><div class="t m3 x6 h9 y1d ff3 fs5 fc0 sc0 lsd ws0">PREREQUISITES</div><div class="t m3 x6 h8 y1e ff6 fs5 fc0 sc0 ls16 wsa">General mathematical maturity is<span class="_ _0"></span> </div><div class="t m3 x6 h8 y1f ff6 fs5 fc0 sc0 ls17 wsb">required along with knowledge of basic </div><div class="t m3 x6 h8 y20 ff6 fs5 fc0 sc0 ls5 ws0">linear algebra and probability<span class="_ _0"></span>. </div><div class="t m3 x6 h9 y21 ff3 fs5 fc0 sc0 lsd ws0">PROBLEM ST<span class="_ _0"></span>A<span class="_ _3"></span>TEMENT</div><div class="t m3 x6 h8 y22 ff6 fs5 fc0 sc0 ls0 wsc">Concisely<span class="_ _0"></span>, the problem is to recover the </div><div class="t m3 x6 h8 y23 ff6 fs5 fc0 sc0 ls18 ws0">vector </div><div class="c x9 y24 w2 ha"><div class="t m3 x0 hb y25 ff7 fs5 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 xa hc y26 ff8 fs5 fc0 sc0 ls0 ws0">[<span class="_ _4"> </span><span class="ff9 fs6">R</span></div><div class="c x9 y24 w2 ha"><div class="t m3 xb hd y27 ff5 fs7 fc0 sc0 ls0 ws0">n</div></div><div class="t m3 xc h8 y26 ff6 fs5 fc0 sc0 ls18 ws5"> from an observation of </div><div class="t m3 x6 h8 y28 ff6 fs5 fc0 sc0 ls0 ws0">the form </div><div class="t m3 x6 h8 y29 ff6 fs5 fc0 sc0 ls0 ws0"> </div><div class="c xa y2a w3 he"><div class="t m3 x0 hb y2b ff7 fs5 fc0 sc0 ls0 ws0">y</div></div><div class="t m3 xd hb y29 ffa fs5 fc0 sc0 ls0 ws0">5<span class="_ _5"> </span><span class="ff7 ls19">Hs<span class="_ _5"> </span></span>1<span class="_ _5"> </span><span class="ff7">e<span class="ff6">,</span></span></div><div class="t m4 xe hf y29 ffb fs5 fc0 sc0 ls1a ws0"> </div><div class="c xa y2a w3 he"><div class="t m3 xf hb y2b ff7 fs5 fc0 sc0 ls0 ws0">y</div></div><div class="t m3 x10 hc y29 ff8 fs5 fc0 sc0 ls0 ws0">[<span class="_ _4"> </span><span class="ff9 fs6">R</span></div><div class="c xa y2a w3 he"><div class="t m3 x11 hd y2c ff5 fs7 fc0 sc0 ls0 ws0">m</div></div><div class="t m3 x12 h8 y29 ff6 fs5 fc0 sc0 ls5 wsd">, (1) </div><div class="t m3 x6 hb y2d ff6 fs5 fc0 sc0 ls0 ws0">where <span class="_ _5"> </span><span class="ff7">H<span class="_ _4"> </span><span class="ff8">[</span></span></div><div class="t m3 x13 h10 y2e ff9 fs6 fc0 sc0 ls0 ws0">R</div><div class="t m3 x14 h11 y2f ff5 fs7 fc0 sc0 ls0 ws0">m<span class="_ _2"></span><span class="ffa">3<span class="_ _6"> </span></span>n</div><div class="t m3 x15 h8 y2e ff6 fs5 fc0 sc0 ls0 wse"> is a known (typically<span class="_ _0"></span>, </div><div class="t m3 x6 h8 y30 ff6 fs5 fc0 sc0 ls1b ws5">estimated beforehand) channel matrix </div><div class="t m3 x6 hb y31 ff6 fs5 fc0 sc0 ls1c ws0">and <span class="_ _4"> </span><span class="ff7 ls0">e<span class="_ _4"> </span><span class="ff8">[</span></span></div><div class="t m3 x16 h10 y32 ff9 fs6 fc0 sc0 ls0 ws0">R</div><div class="c x17 y33 w4 ha"><div class="t m3 xb hd y34 ff5 fs7 fc0 sc0 ls0 ws0">m</div></div><div class="t m3 x18 h8 y32 ff6 fs5 fc0 sc0 ls1c wsf"> represents noise. We<span class="_ _3"></span> </div><div class="t m3 x6 hb y35 ff6 fs5 fc0 sc0 ls1d ws10">assume that <span class="ff7 ls0 ws0">e<span class="_ _4"> </span><span class="ffc">,<span class="_ _4"> </span><span class="ff5">N</span></span></span></div><div class="t m3 x19 h12 y36 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x1a hb y35 ffe fs5 fc0 sc0 ls0 ws0">0<span class="ff6 ls8">, </span><span class="ffa">s<span class="ff7">I</span></span></div><div class="t m3 x1b h12 y36 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x1c h8 y35 ff6 fs5 fc0 sc0 ls1d ws10">. The ele-</div><div class="t m3 x6 h8 y37 ff6 fs5 fc0 sc0 ls1e ws5">ments of </div><div class="c x1d y38 w5 h13"><div class="t m3 x0 hb y25 ff7 fs5 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x1e h8 y39 ff6 fs5 fc0 sc0 ls1e ws5">, say <span class="_ _2"></span><span class="ff5 ls0 ws0">s</span></div><div class="c x15 y3a w6 h14"><div class="t m3 x1f hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 x20 h8 y39 ff6 fs5 fc0 sc0 ls1e ws5">, belong to a finite<span class="_ _7"></span> </div><div class="t m3 x6 h8 y3c ff6 fs5 fc0 sc0 ls1f ws0">alphabet </div><div class="c xa y3d w7 h15"><div class="t m3 x0 h16 y3e fff fs5 fc0 sc0 ls0 ws0">S</div></div><div class="t m3 xd h8 y3d ff6 fs5 fc0 sc0 ls1f ws5"> of size </div><div class="c x20 y3f w8 h17"><div class="t m3 x0 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 x21 h16 y3d fff fs5 fc0 sc0 ls0 ws0">S</div><div class="c x20 y3f w8 h17"><div class="t m3 x22 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 x23 h8 y3d ff6 fs5 fc0 sc0 ls15 ws4">. Hence there are </div><div class="c x6 y41 w9 h17"><div class="t m3 x0 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 x24 h16 y42 fff fs5 fc0 sc0 ls0 ws0">S</div><div class="c x6 y41 w9 h17"><div class="t m3 x22 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 xf hd y43 ff5 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m3 x25 h8 y42 ff6 fs5 fc0 sc0 ls0 ws11"> possible vectors </div><div class="c xe y44 w5 h13"><div class="t m3 x0 hb y25 ff7 fs5 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x23 h8 y42 ff6 fs5 fc0 sc0 ls0 ws11">. For simplicity of </div><div class="t m3 x26 h8 y7 ff6 fs5 fc0 sc0 ls20 ws12">our discussion, we assume that all<span class="_ _0"></span> </div><div class="t m3 x26 h8 y8 ff6 fs5 fc0 sc0 ls0 ws13">quantities are real-valued. This is most-</div><div class="t m3 x26 h8 y9 ff6 fs5 fc0 sc0 ls0 ws14">ly a matter of notation, since </div><div class="c x27 y45 wa h18"><div class="t m3 x0 h10 y3e ff9 fs6 fc0 sc0 ls0 ws0">C</div><div class="t m3 x28 hd y46 ff5 fs7 fc0 sc0 ls0 ws0">n</div></div><div class="t m3 x29 h8 y45 ff6 fs5 fc0 sc0 ls0 ws14"> is iso-</div><div class="t m3 x26 h8 y47 ff6 fs5 fc0 sc0 ls21 ws5">morphic to </div><div class="t m3 x2a h10 y48 ff9 fs6 fc0 sc0 ls0 ws0">R</div><div class="c x2a y48 wb h19"><div class="t m3 x28 hd y46 ff6 fs7 fc0 sc0 ls0 ws0">2</div></div><div class="t m3 x2b hd y49 ff5 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m3 x2c h8 y48 ff6 fs5 fc0 sc0 ls21 ws5">. We also assume that<span class="_ _0"></span> </div><div class="t m3 x26 h8 y4a ff5 fs5 fc0 sc0 ls0 ws0">m</div><div class="c x26 y4b wc h1a"><div class="t m3 x2d h1b y25 ffa fs5 fc0 sc0 ls0 ws0">$</div></div><div class="t m3 x2e h8 y6 ff5 fs5 fc0 sc0 ls0 ws0">n<span class="ff6 ls22 ws5">, that is, (1) is not underdeter<span class="_ _0"></span>-</span></div><div class="t m3 x26 hb y4c ff6 fs5 fc0 sc0 ls23 ws15">mined, and that <span class="_ _6"></span><span class="ff7 ls0 ws0">H<span class="_ _6"></span></span> has full column<span class="_ _0"></span> </div><div class="t m3 x26 h8 y4d ff6 fs5 fc0 sc0 ls0 ws16">rank. This is so with probability one in </div><div class="t m3 x26 h8 y4e ff6 fs5 fc0 sc0 ls0 ws17">most applications. We also assume that </div><div class="t m3 x26 hb y4f ff7 fs5 fc0 sc0 ls0 ws0">H<span class="_ _6"></span><span class="ff6 ls24 ws5"> has no specific structure. If <span class="_ _2"></span></span>H<span class="_ _6"></span><span class="ff6 ls24 ws5"> has </span></div><div class="t m3 x26 h8 y50 ff6 fs5 fc0 sc0 ls0 ws18">structure, for example, if it is a T<span class="_ _3"></span>oeplitz </div><div class="t m3 x26 h8 y51 ff6 fs5 fc0 sc0 ls0 ws19">matrix, then one should use algorithms </div><div class="t m3 x26 h8 y52 ff6 fs5 fc0 sc0 ls0 ws0">that can exploit this structure. </div><div class="t m3 x2f h8 y53 ff6 fs5 fc0 sc0 lsd ws1a">We want to detect </div><div class="c x30 y54 w5 h13"><div class="t m3 x0 hb y25 ff7 fs5 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x31 h8 y55 ff6 fs5 fc0 sc0 lsd ws1a"> in the maxi <span class="_ _8"></span>mum-</div><div class="t m3 x26 h8 y56 ff6 fs5 fc0 sc0 lsd ws1b">likelihood (ML) sense. This is equivalent to </div><div class="t m3 x26 h8 y57 ff6 fs5 fc0 sc0 ls25 ws1c"> The <span class="_ _9"></span>problem:</div><div class="t m4 x32 hf y58 ffb fs5 fc0 sc0 ls0 ws0"> </div><div class="t m3 x33 h8 y58 ff6 fs5 fc0 sc0 ls26 ws0">min</div><div class="c x34 y59 wd h1c"><div class="t m3 x35 h1d y3b ff7 fs7 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x30 h1e y5a ff8 fs7 fc0 sc0 ls0 ws0">[<span class="_ _2"> </span><span class="fff">S<span class="_"> </span><span class="ff6 fs8"> </span></span></div><div class="t m3 x36 h1f y5b ff5 fs9 fc0 sc0 ls0 ws0">n</div><div class="t m3 x37 h12 y58 ffd fs5 fc0 sc0 ls0 ws0">7<span class="_ _6"></span><span class="ff7">y<span class="_ _5"> </span><span class="ffa">2<span class="_ _5"> </span></span><span class="ls19">Hs<span class="_ _a"></span></span></span>7<span class="_ _2"></span><span class="ff6 ls5 ws1d">. (2) </span></div><div class="t m3 x26 h8 y5c ff6 fs5 fc0 sc0 ls27 ws1e">Problem (2) is a finite-alphabet-con-</div><div class="t m3 x26 h8 y5d ff6 fs5 fc0 sc0 ls28 ws1f">strained least-squares (LS) problem,<span class="_ _0"></span> </div><div class="t m3 x26 h8 y5e ff6 fs5 fc0 sc0 ls29 ws20">which is known to be nondeterministic </div><div class="t m3 x26 h8 y5f ff6 fs5 fc0 sc0 ls5 ws21">polynomial-time (NP)-hard. The compli-</div><div class="t m3 x26 h8 y60 ff6 fs5 fc0 sc0 ls2a ws5">cating factor is of course the constraint </div><div class="c x26 y61 we ha"><div class="t m3 x0 hb y25 ff7 fs5 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x38 hc y62 ff8 fs5 fc0 sc0 ls0 ws0">[</div><div class="c x26 y61 we ha"><div class="t m3 x39 h16 y25 fff fs5 fc0 sc0 ls0 ws0">S</div></div><div class="t m3 x3a h20 y62 ff6 fsa fc0 sc0 ls0 ws0"> </div><div class="c x26 y61 we ha"><div class="t m3 x3b hd y27 ff5 fs7 fc0 sc0 ls0 ws0">n</div></div><div class="t m3 x3c h8 y62 ff6 fs5 fc0 sc0 ls5 ws22">, otherwise (2) would be just clas-</div><div class="t m3 x26 h8 y63 ff6 fs5 fc0 sc0 ls5 ws0">sical LS regression. </div><div class="t m3 x26 h9 y64 ff3 fs5 fc0 sc0 lsd ws0">SOLUTIONS</div><div class="t m3 x26 h8 y65 ff6 fs5 fc0 sc0 ls2b ws23">As a preparation, we introduce the<span class="_ _0"></span> </div><div class="t m3 x26 hb y66 ff6 fs5 fc0 sc0 ls5 ws24">QL-decomposition of <span class="_ _6"></span><span class="ff7 ls0 ws0">H</span> : <span class="_ _6"></span><span class="ff7 ls0 ws0">H<span class="_ _5"> </span><span class="ffa">5<span class="_ _5"> </span></span><span class="ls2c">QL</span></span>, where </div><div class="t m3 x26 hb y67 ff7 fs5 fc0 sc0 ls0 ws0">Q<span class="_ _4"> </span><span class="ff8">[</span></div><div class="t m3 x3d h10 y68 ff9 fs6 fc0 sc0 ls0 ws0">R</div><div class="t m3 x3e h11 y69 ff5 fs7 fc0 sc0 ls0 ws0">m<span class="_ _2"></span><span class="ffa">3<span class="_ _6"> </span></span>n</div><div class="t m3 x3f hb y68 ff6 fs5 fc0 sc0 ls23 ws25"> is orthonormal (<span class="ff7 ls0 ws0">Q</span></div><div class="c x40 y6a wf h21"><div class="t m3 x41 hd y6b ff5 fs7 fc0 sc0 ls0 ws0">T</div></div><div class="t m3 x42 hb y68 ff7 fs5 fc0 sc0 ls0 ws0">Q<span class="_ _5"> </span><span class="ffa">5<span class="_ _5"> </span></span>I<span class="_ _a"></span><span class="ff6 ls23">),<span class="_ _0"></span> </span></div><div class="t m3 x26 h8 y6c ff6 fs5 fc0 sc0 ls5 ws0">and </div><div class="c x43 y6d w10 h22"><div class="t m3 x0 hb y3e ff7 fs5 fc0 sc0 ls0 ws0">L</div></div><div class="t m3 x34 hc y6d ff8 fs5 fc0 sc0 ls0 ws0">[<span class="_ _4"> </span><span class="ff9 fs6">R</span></div><div class="t m3 x3f h11 y6e ff5 fs7 fc0 sc0 ls0 ws0">n<span class="_ _6"></span><span class="ffa">3<span class="_ _2"></span></span>n</div><div class="t m3 x44 h8 y6d ff6 fs5 fc0 sc0 ls5 ws0"> is lower triangular<span class="_ _3"></span>. Then<span class="_ _2"></span> </div><div class="t m3 x45 h12 y6f ffd fs5 fc0 sc0 ls0 ws0">7<span class="_ _6"></span><span class="ff7">y<span class="_ _5"> </span><span class="ffa">2<span class="_ _5"> </span></span><span class="ls19">Hs<span class="_ _a"></span></span></span>7</div><div class="c x46 y70 w11 h23"><div class="t m3 x47 hd y71 ff6 fs7 fc0 sc0 ls0 ws0">2</div></div><div class="t m3 x48 h12 y26 ffa fs5 fc0 sc0 ls0 ws0">5<span class="_ _a"> </span><span class="ffd">7<span class="_ _6"></span><span class="ff7 ls2c">QQ</span></span></div><div class="c x49 y70 w12 h23"><div class="t m3 x4a hd y71 ff5 fs7 fc0 sc0 ls0 ws0">T</div></div><div class="t m3 x4b h12 y72 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x33 h24 y73 ff6 fs8 fc0 sc0 ls0 ws0"> </div><div class="t m3 x4c hb y26 ff7 fs5 fc0 sc0 ls0 ws0">y<span class="_ _6"></span><span class="ffa">2<span class="_ _6"></span></span><span class="ls19">Hs</span></div><div class="t m3 x4d h12 y72 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x4e h12 y26 ffd fs5 fc0 sc0 ls0 ws0">7</div><div class="c x49 y70 w12 h23"><div class="t m3 x4f hd y71 ff6 fs7 fc0 sc0 ls0 ws0">2</div></div><div class="t m3 x26 h12 y74 ff6 fs5 fc0 sc0 ls0 ws0"> <span class="_ _b"> </span><span class="ffa">1<span class="_ _a"> </span><span class="ffd">7</span></span></div><div class="t m3 x50 h12 y75 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x51 hb y74 ff7 fs5 fc0 sc0 ls0 ws0">I<span class="_ _2"></span><span class="ffa">2<span class="_ _2"></span></span><span class="ls2c">QQ</span></div><div class="c x52 y76 w13 h23"><div class="t m3 x53 hd y77 ff5 fs7 fc0 sc0 ls0 ws0">T</div></div><div class="t m3 x54 h12 y75 ffd fs5 fc0 sc0 ls2d ws0">21</div><div class="t m3 x55 h25 y78 ffb fs8 fc0 sc0 ls0 ws0"> </div><div class="t m3 x55 hb y74 ff7 fs5 fc0 sc0 ls0 ws0">y<span class="_ _c"> </span><span class="ffa">2<span class="_ _2"></span></span><span class="ls19">Hs</span></div><div class="t m3 x56 h12 y75 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x57 h12 y74 ffd fs5 fc0 sc0 ls0 ws0">7</div><div class="c x52 y76 w13 h23"><div class="t m3 x16 hd y77 ff6 fs7 fc0 sc0 ls0 ws0">2</div></div><div class="t m3 x26 h12 y79 ff6 fs5 fc0 sc0 ls0 ws0"> <span class="_ _d"> </span><span class="ffa">5<span class="_ _6"></span><span class="ffd">7<span class="_ _6"></span><span class="ff7">Q</span></span></span></div><div class="c x49 y7a w14 h23"><div class="t m3 x58 hd y71 ff5 fs7 fc0 sc0 ls0 ws0">T</div></div><div class="t m3 x59 h12 y79 ff7 fs5 fc0 sc0 ls0 ws0">y<span class="_ _5"> </span><span class="ffa">2<span class="_ _5"> </span></span><span class="ls2e">Ls<span class="_ _6"></span></span><span class="ffd">7</span></div><div class="c x49 y7a w14 h23"><div class="t m3 x5a hd y71 ff6 fs7 fc0 sc0 ls0 ws0">2</div></div><div class="t m3 x26 h12 y7b ff6 fs5 fc0 sc0 ls0 ws0"> <span class="_ _b"> </span><span class="ffa">1<span class="_ _a"> </span><span class="ffd">7</span></span></div><div class="t m3 x50 h12 y7c ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x51 hb y7b ff7 fs5 fc0 sc0 ls0 ws0">I<span class="_ _5"> </span><span class="ffa">2<span class="_ _5"> </span></span><span class="ls2c">QQ</span></div><div class="c x5b y7d w15 h23"><div class="t m3 x5c hd y71 ff5 fs7 fc0 sc0 ls0 ws0">T</div></div><div class="t m3 x5d h12 y7c ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x4d h25 y7e ffb fs8 fc0 sc0 ls0 ws0"> </div><div class="t m3 x55 h12 y7b ff7 fs5 fc0 sc0 ls0 ws0">y<span class="_ _6"></span><span class="ffd">7</span></div><div class="c x5b y7d w15 h23"><div class="t m3 x5e hd y71 ff6 fs7 fc0 sc0 ls0 ws0">2</div></div><div class="t m3 x5f h8 y7b ff6 fs5 fc0 sc0 ls0 ws0">,</div><div class="t m3 x26 h8 y7f ff6 fs5 fc0 sc0 lsc ws26">where the last term does not depend on </div><div class="c x60 y80 w5 h13"><div class="t m3 x0 hb y25 ff7 fs5 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x61 h8 y81 ff6 fs5 fc0 sc0 ls0 ws0">. </div><div class="t m3 x26 h8 y82 ff6 fs5 fc0 sc0 ls5 ws0">It follows that we can reformulate (2) as </div><div class="t m3 x26 h8 y39 ff6 fs5 fc0 sc0 ls2f ws27">Equivalent problem: min</div><div class="c x26 y83 w16 h26"><div class="t m3 x16 h1d y3b ff7 fs7 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x31 h1e y84 ff8 fs7 fc0 sc0 ls0 ws0">[<span class="_ _2"> </span><span class="fff">S<span class="_"> </span><span class="ff6 fs8"> </span></span></div><div class="t m3 x37 h1f y85 ff5 fs9 fc0 sc0 ls0 ws0">n</div><div class="t m3 x54 h12 y39 ffd fs5 fc0 sc0 ls0 ws0">7<span class="_ _c"></span><span class="ff7">y</span></div><div class="t m3 x5d h27 y86 ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 x4e h12 y39 ffa fs5 fc0 sc0 ls0 ws0">2<span class="_ _5"> </span><span class="ff7 ls2e">Ls<span class="_ _6"></span></span><span class="ffd">7<span class="_ _a"></span><span class="ff6 ls30">, </span></span></div><div class="t m3 x26 h8 y87 ff6 fs5 fc0 sc0 ls5 ws0">where </div><div class="c x62 y88 w17 h28"><div class="t m3 x63 h8 y2b ff5 fs5 fc0 sc0 ls0 ws0">y</div></div><div class="t m3 x64 h27 y89 ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 x50 h16 y8a fff fs5 fc0 sc0 ls0 ws0">!<span class="_ _4"> </span><span class="ff7">Q</span></div><div class="t m3 x31 hd y8b ff5 fs7 fc0 sc0 ls0 ws0">T</div><div class="t m3 x65 h8 y8a ff6 fs5 fc0 sc0 ls0 ws0"> </div><div class="c x62 y88 w17 h28"><div class="t m3 x66 hb y2b ff7 fs5 fc0 sc0 ls0 ws0">y</div></div><div class="t m3 x42 h8 y8a ff6 fs5 fc0 sc0 ls5 ws28"> (3)</div><div class="t m3 x26 h8 y8c ff6 fs5 fc0 sc0 ls5 ws0">or<span class="_ _3"></span>, in yet another equivalent form, as<span class="_ _2"></span> </div><div class="t m3 x67 h8 y8d ff6 fs5 fc0 sc0 ls8 ws0">min </div><div class="t m3 x68 h29 y8e ffd fs7 fc0 sc0 ls0 ws0">5</div><div class="t m3 x69 hd y8f ff5 fs7 fc0 sc0 ls0 ws0">s</div><div class="t m3 x6a h1f y90 ff6 fs9 fc0 sc0 ls0 ws0">1</div><div class="t m3 x6b h1e y91 ff6 fs7 fc0 sc0 ls0 ws0">,<span class="_ _6"></span><span class="fff">c<span class="_ _7"></span><span class="ff6 ls31">, <span class="ff5 ls0">s</span></span></span></div><div class="t m3 x6c h1f y90 ff5 fs9 fc0 sc0 ls0 ws0">n</div><div class="t m3 x6d h29 y8e ffd fs7 fc0 sc0 ls0 ws0">6</div><div class="t m3 x6e hd y8f ff6 fs7 fc0 sc0 ls0 ws0"> </div><div class="t m3 x6f hd y92 ff5 fs7 fc0 sc0 ls0 ws0">s</div><div class="t m3 x70 h1f y93 ff5 fs9 fc0 sc0 ls0 ws0">k</div><div class="t m3 x71 hd y94 ff11 fs7 fc0 sc0 ls0 ws0">P<span class="ff5">S</span></div><div class="t m3 x72 hd y95 ff6 fs7 fc0 sc0 ls31 ws0"> </div><div class="t m3 x6e h12 y96 ffd fs5 fc0 sc0 ls0 ws0">5</div><div class="t m3 x73 h8 y7 ff5 fs5 fc0 sc0 ls0 ws0">f</div><div class="t m3 x74 hd y97 ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 x75 h12 y96 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x76 h8 y7 ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 x77 hd y97 ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 x78 h12 y96 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x79 h1b y7 ffa fs5 fc0 sc0 ls0 ws0">1<span class="_ _5"> </span><span class="ff5">f</span></div><div class="t m3 x7a hd y97 ff6 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m3 x7b h12 y96 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x7c h8 y7 ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 x7d hd y97 ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 x7e h8 y7 ff6 fs5 fc0 sc0 ls8 ws0">, <span class="ff5 ls0">s</span></div><div class="t m3 x7f hd y97 ff6 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m3 x80 h12 y96 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x81 h1b y98 ff6 fs5 fc0 sc0 ls0 ws0"> <span class="_ _e"> </span><span class="ffa">1</span></div><div class="t m3 x78 h16 y99 fff fs5 fc0 sc0 ls0 ws0">c</div><div class="t m3 x82 h1b y98 ffa fs5 fc0 sc0 ls0 ws0">1<span class="_ _5"> </span><span class="ff5">f</span></div><div class="c x76 y9a w18 h2a"><div class="t m3 x83 hd y3b ff5 fs7 fc0 sc0 ls0 ws0">n</div></div><div class="t m3 x84 h12 y9b ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x85 h8 y9c ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 x86 hd y9d ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 x7f h16 y9c ff6 fs5 fc0 sc0 ls0 ws0">,<span class="_ _4"> </span><span class="fff">c</span><span class="ls8">, </span><span class="ff5">s</span></div><div class="c x76 y9a w18 h2a"><div class="t m3 x87 hd y3b ff5 fs7 fc0 sc0 ls0 ws0">n</div></div><div class="t m3 x88 h12 y9b ffd fs5 fc0 sc0 ls32 ws0">26</div><div class="t m3 x89 h8 y9c ff6 fs5 fc0 sc0 ls0 ws0">,</div><div class="t m3 x81 h8 y6 ff6 fs5 fc0 sc0 ls5 ws0">where</div><div class="t m3 x81 h8 y9e ff6 fs5 fc0 sc0 ls0 ws0"> </div><div class="t m3 x8a h8 y9f ff6 fsb fc0 sc0 ls0 ws0"> <span class="ff5 fs5">f</span></div><div class="t m3 x8b hd ya0 ff5 fs7 fc0 sc0 ls0 ws0">k</div><div class="t m3 x6b h12 ya1 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x8c h8 y9f ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 x8d hd ya0 ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 x8e h16 y9f ff6 fs5 fc0 sc0 ls0 ws0">,<span class="_ _4"> </span><span class="fff">c</span><span class="ls8">, </span><span class="ff5">s</span></div><div class="t m3 x8f hd ya0 ff5 fs7 fc0 sc0 ls0 ws0">k</div><div class="t m3 x90 h12 ya1 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x2 h16 y9f fff fs5 fc0 sc0 ls0 ws0">!</div><div class="t m3 x91 h12 ya2 ffd fs5 fc0 sc0 ls0 ws0">a</div><div class="t m3 x7a h8 y9f ff5 fs5 fc0 sc0 ls0 ws0">y</div><div class="t m3 x7a h27 ya3 ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 x92 hd ya0 ff5 fs7 fc0 sc0 ls0 ws0">k</div><div class="t m3 x93 h1b y9f ffa fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x94 h2b ya4 ff12 fs5 fc0 sc0 ls0 ws0">a</div><div class="c x8a ya5 w19 h2c"><div class="t m3 x95 hd ya6 ff5 fs7 fc0 sc0 ls0 ws0">k</div><div class="t m3 x1e hd ya7 ff5 fs7 fc0 sc0 ls0 ws0">l</div></div><div class="t m3 x7f h11 ya8 ffa fs7 fc0 sc0 ls0 ws0">5<span class="_ _2"></span><span class="ff6">1</span></div><div class="t m3 x4 h8 y9f ff5 fs5 fc0 sc0 ls0 ws0">L</div><div class="t m3 x96 hd ya0 ff5 fs7 fc0 sc0 ls0 ws0">k<span class="ff6 ls31">, </span>l<span class="ff6"> </span></div><div class="t m3 x97 h8 y9f ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 x98 hd ya0 ff5 fs7 fc0 sc0 ls0 ws0">l</div><div class="t m3 x99 h12 ya2 ffd fs5 fc0 sc0 ls0 ws0">b</div><div class="t m3 x9a hd ya9 ff6 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m3 x9b h8 y9f ff6 fs5 fc0 sc0 ls33 ws29">. (4)</div><div class="t m3 x9c h8 yaa ff6 fs5 fc0 sc0 ls34 ws5">Problem (4) can be visualized as a </div><div class="t m3 x81 h1b yab ff6 fs5 fc0 sc0 ls35 ws2a">decision tree with <span class="_ _2"></span><span class="ff5 ls0 ws0">n<span class="_ _5"> </span><span class="ffa">1<span class="_ _5"> </span></span></span><span class="ls36 ws2b">1 layers, </span></div><div class="c x9d yac w1a h17"><div class="t m3 x0 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 x9e h2d yab ff13 fs5 fc0 sc0 ls0 ws0">S</div><div class="c x9d yac w1a h17"><div class="t m3 x22 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 x9f h8 yab ff6 fs5 fc0 sc0 ls0 ws0"> </div><div class="t m3 x81 h8 yad ff6 fs5 fc0 sc0 ls0 ws5">branches emanating from each nonleaf </div><div class="t m3 x81 h8 yae ff6 fs5 fc0 sc0 ls37 ws5">node, and </div><div class="c xa0 yaf w1b h17"><div class="t m3 x0 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 x75 h2d yb0 ff13 fs5 fc0 sc0 ls0 ws0">S</div><div class="c xa0 yaf w1b h17"><div class="t m3 x22 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 xa1 hd yb1 ff5 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m3 xa2 h8 yb0 ff6 fs5 fc0 sc0 ls37 ws5"> leaf nodes. See Figure 1. </div><div class="t m3 x81 h8 yb2 ff6 fs5 fc0 sc0 ls5 ws2c">T<span class="_ _3"></span>o any branch, we associate a hypotheti-</div><div class="t m3 x81 h8 yb3 ff6 fs5 fc0 sc0 ls5 ws2d">cal decision on <span class="_ _2"></span><span class="ff5 ls0 ws0">s</span></div><div class="c xa3 yb4 w6 h14"><div class="t m3 x1f hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 xa4 h8 yb5 ff6 fs5 fc0 sc0 ls5 ws2d">, and the branch metric </div><div class="t m3 x81 h8 yb6 ff5 fs5 fc0 sc0 ls0 ws0">f</div><div class="c x81 yb7 w1c h2a"><div class="t m3 xa5 hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 x69 h12 yb8 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x67 h8 yb9 ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 xa6 hd yba ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 x71 h16 yb9 ff6 fs5 fc0 sc0 ls0 ws0">,<span class="_ _4"> </span><span class="fff">c</span><span class="ls8">, </span><span class="ff5">s</span></div><div class="c x81 yb7 w1c h2a"><div class="t m3 x5c hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 x8f h12 yb8 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 xa1 h8 yb9 ff6 fs5 fc0 sc0 ls5 ws2e">. Also, to any node (except </div><div class="t m3 x81 h8 ybb ff6 fs5 fc0 sc0 ls38 ws5">the root), we associate the cumulative </div><div class="t m3 x81 h8 ybc ff6 fs5 fc0 sc0 ls39 ws0">metric <span class="_ _f"> </span><span class="ff5 ls0">f</span></div><div class="t m3 x76 hd ybd ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 xa1 h12 ybe ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x78 h8 ybf ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 xa3 hd ybd ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 xa7 h12 ybe ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 x82 h1b ybf ffa fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 x92 h16 yc0 fff fs5 fc0 sc0 ls0 ws0">c</div><div class="t m3 xa8 h1b ybf ffa fs5 fc0 sc0 ls0 ws0">1<span class="_ _5"> </span><span class="ff5">f</span></div><div class="c xa9 yc1 w1d h2a"><div class="t m3 xaa hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 xab h12 ybe ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 xac h8 ybf ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 xad hd ybd ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 xae h16 ybf ff6 fs5 fc0 sc0 ls0 ws0">,<span class="_ _4"> </span><span class="fff">c</span><span class="ls8">, </span><span class="ff5">s</span></div><div class="c xa9 yc1 w1d h2a"><div class="t m3 xc hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 xaf h12 ybe ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 xb0 h8 ybf ff6 fs5 fc0 sc0 ls0 ws0">, </div><div class="t m3 x81 h8 yc2 ff6 fs5 fc0 sc0 ls5 ws2f">which is just the sum of all branch met-</div><div class="t m3 x81 h8 yc3 ff6 fs5 fc0 sc0 ls5 ws30">rics accumulated when traveling to that </div><div class="t m3 x81 h8 yc4 ff6 fs5 fc0 sc0 ls5 ws31">node from the root. Finally<span class="_ _0"></span>, to each node, </div><div class="t m3 x81 h8 yc5 ff6 fs5 fc0 sc0 ls3a ws5">we associate the symbols </div><div class="t m3 xb1 h12 yc6 ffd fs5 fc0 sc0 ls0 ws0">5</div><div class="t m3 xb2 h8 yc5 ff5 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 xb3 hd yc7 ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 xb4 h16 yc8 ff6 fs5 fc0 sc0 ls0 ws0">,<span class="_ _4"> </span><span class="fff">c</span><span class="ls8">, </span><span class="ff5">s</span></div><div class="c xb1 yc9 w1e h2a"><div class="t m3 xb5 hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 xb6 h12 yca ffd fs5 fc0 sc0 ls0 ws0">6</div><div class="t m3 x9e h8 yc8 ff6 fs5 fc0 sc0 ls3a ws5"> it </div><div class="t m3 x81 h8 ycb ff6 fs5 fc0 sc0 ls5 ws0">takes to reach there from the root. </div><div class="t m3 x9c h8 ycc ff6 fs5 fc0 sc0 ls3b ws32">Clearly<span class="_ _0"></span>, a naive but valid way of solving<span class="_ _2"></span> </div><div class="t m3 x81 h8 ycd ff6 fs5 fc0 sc0 ls3b ws33">(4) would be to traverse the entire tree to </div><div class="t m3 x81 h8 yce ff6 fs5 fc0 sc0 ls3b ws34">find the leaf node with the smallest cumu-</div><div class="t m3 x81 h8 ycf ff6 fs5 fc0 sc0 ls3b ws35">lative metric. However<span class="_ _3"></span>, such a brute-force<span class="_ _2"></span> </div><div class="t m3 x81 h8 yd0 ff6 fs5 fc0 sc0 ls3b ws36">search is extremely inefficient, since there </div><div class="t m3 x81 h8 yd1 ff6 fs5 fc0 sc0 ls2c ws0">are </div><div class="c x70 yd2 w1b h17"><div class="t m3 x0 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 x71 h2d yd3 ff13 fs5 fc0 sc0 ls0 ws0">S</div><div class="c x70 yd2 w1b h17"><div class="t m3 x22 h8 y40 ff6 fs5 fc0 sc0 ls0 ws0">|</div></div><div class="t m3 xb7 hd yd4 ff5 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m3 x6d h8 yd3 ff6 fs5 fc0 sc0 ls3c ws4"> leaf nodes to examine. We will </div><div class="t m3 x81 h8 yd5 ff6 fs5 fc0 sc0 ls3d ws20">now review some efficient, popular<span class="_ _3"></span>, but </div><div class="t m3 x81 h8 yd6 ff6 fs5 fc0 sc0 ls3b ws0">approximate solutions to (4). </div><div class="t m3 x81 h9 yd7 ff14 fs5 fc0 sc0 ls5 ws0">ZERO-FORCING (ZF) DETECTOR</div><div class="t m3 x81 h8 yd8 ff6 fs5 fc0 sc0 ls5 ws37">The ZF detector first solves (2), neglect-</div><div class="t m3 x81 h8 yd9 ff6 fs5 fc0 sc0 ls5 ws0">ing the constraint </div><div class="c xb8 yda we ha"><div class="t m3 x0 hb y25 ff7 fs5 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 xb9 hc ydb ff8 fs5 fc0 sc0 ls0 ws0">[</div><div class="c xb8 yda we ha"><div class="t m3 x39 h16 y25 fff fs5 fc0 sc0 ls0 ws0">S</div></div><div class="t m3 xba h20 ydb ff6 fsa fc0 sc0 ls0 ws0"> </div><div class="c xb8 yda we ha"><div class="t m3 x3b hd y27 ff5 fs7 fc0 sc0 ls0 ws0">n</div></div><div class="t m3 xbb h8 ydb ff6 fs5 fc0 sc0 ls3e ws0"> </div><div class="t m3 x9c hb ydc ff7 fs5 fc0 sc0 ls0 ws0">s</div><div class="t m3 x9c h27 ydd ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 xbc h16 ydc fff fs5 fc0 sc0 ls0 ws0">!<span class="_ _4"> </span><span class="ff6 ls3f ws38">arg min </span></div><div class="c xbd yde w1f h2e"><div class="t m3 xbe hd y3b ff5 fs7 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 x77 h2f ydf ff8 fs7 fc0 sc0 ls0 ws0">[<span class="_ _2"> </span><span class="ff9 fsc">R</span></div><div class="t m3 x79 h1f ye0 ff5 fs9 fc0 sc0 ls0 ws0">n</div><div class="t m3 x3 h12 y7b ffd fs5 fc0 sc0 ls0 ws0">7<span class="_ _6"></span><span class="ff7">y<span class="_ _5"> </span><span class="ffa">2<span class="_ _5"> </span></span><span class="ls19">Hs<span class="_ _a"></span></span></span>7</div><div class="t m3 xbc h1b ye1 ffa fs5 fc0 sc0 ls0 ws0">5<span class="_ _5"> </span><span class="ff6 ls3f ws38">arg min </span></div><div class="c x8d ye2 w20 h30"><div class="t m3 xbf hd y3b ff5 fs7 fc0 sc0 ls0 ws0">s</div></div><div class="t m3 xc0 h2f ye3 ff8 fs7 fc0 sc0 ls0 ws0">[<span class="_ _2"> </span><span class="ff9 fsc">R</span></div><div class="t m3 xa3 h1f ye4 ff5 fs9 fc0 sc0 ls0 ws0">n</div><div class="t m3 xc1 h12 ye5 ffd fs5 fc0 sc0 ls0 ws0">7<span class="_ _c"></span><span class="ff7">y</span></div><div class="t m3 xc2 h31 ye6 ffc fs7 fc0 sc0 ls0 ws0">,</div><div class="t m3 xc3 h12 ye5 ffa fs5 fc0 sc0 ls0 ws0">2<span class="_ _5"> </span><span class="ff7 ls2e">Ls<span class="_ _6"></span></span><span class="ffd">7<span class="_ _4"> </span></span>5<span class="_ _5"> </span><span class="ff7">L</span></div><div class="t m3 xad h11 ye6 ffa fs7 fc0 sc0 ls0 ws0">2<span class="_ _2"></span><span class="ff6">1</span></div><div class="t m3 xc4 hb ye5 ff7 fs5 fc0 sc0 ls0 ws0">y</div><div class="t m3 xc4 h31 ye6 ffc fs7 fc0 sc0 ls0 ws0">,</div><div class="t m3 xc5 h8 ye5 ff6 fs5 fc0 sc0 ls5 ws39">. (5) </div><div class="t m3 x81 h8 ye7 ff6 fs5 fc0 sc0 lsd ws3a">Of course, </div><div class="c x74 ye7 w21 h32"><div class="t m3 x0 hb y3e ff7 fs5 fc0 sc0 ls0 ws0">L</div></div><div class="t m3 xc6 h11 ye8 ffa fs7 fc0 sc0 ls0 ws0">2<span class="_ _2"></span><span class="ff6">1</span></div><div class="t m3 x78 h8 ye7 ff6 fs5 fc0 sc0 lsd ws3a"> does not need to be explic-</div><div class="t m3 x81 h8 ye9 ff6 fs5 fc0 sc0 ls40 ws5">itly computed. For example, one can do </div><div class="t m3 x81 h8 yea ff6 fs5 fc0 sc0 ls41 ws5">Gaussian elimination: take <span class="_ _c"> </span><span class="ff5 ls0 ws0">s</span></div><div class="t m3 xb3 h27 yeb ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 xad hd yec ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 x97 h1b y3d ffa fs5 fc0 sc0 ls0 ws0">5<span class="_ _10"> </span><span class="ff5">y</span></div><div class="t m3 xc5 h27 yed ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 x89 hd yec ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m3 xc7 h8 y3d ffb fs5 fc0 sc0 ls0 ws0">/<span class="_ _2"></span><span class="ff5">L</span></div><div class="t m3 x9d hd yec ff6 fs7 fc0 sc0 ls42 ws0">1,1</div><div class="t m3 xb0 h8 y3d ff6 fs5 fc0 sc0 ls0 ws0">, </div><div class="t m3 x81 h8 yee ff6 fs5 fc0 sc0 lsd ws0">then<span class="_ _c"> </span><span class="ff5 ls0">s</span></div><div class="t m3 x71 h27 yef ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 xc8 hd yf0 ff6 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m3 x6c h1b y42 ffa fs5 fc0 sc0 ls0 ws0">5</div><div class="t m3 xc9 h12 yf1 ffd fs5 fc0 sc0 ls0 ws0">1</div><div class="t m3 xca h8 y42 ff5 fs5 fc0 sc0 ls0 ws0">y</div><div class="t m3 xa0 h27 yf2 ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 x76 hd yf0 ff6 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m3 xa1 h1b y42 ffa fs5 fc0 sc0 ls0 ws0">2<span class="_ _4"> </span><span class="ff5">s</span></div><div class="t m3 xa7 h27 yf2 ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="t m3 xc2 hd yf0 ff6 fs7 fc0 sc0 ls43 ws0">1 </div><div class="t m3 x7a h8 y42 ff5 fs5 fc0 sc0 ls0 ws0">L</div><div class="t m3 x92 hd yf0 ff6 fs7 fc0 sc0 ls42 ws0">2,1</div><div class="t m3 x7e h12 yf1 ffd fs5 fc0 sc0 ls0 ws0">2</div><div class="t m3 xa8 h8 y42 ff6 fs5 fc0 sc0 ls0 ws0">/<span class="_ _2"></span><span class="ff5">L</span></div><div class="t m3 xbb hd yf0 ff6 fs7 fc0 sc0 ls42 ws0">2,2</div><div class="t m3 xb2 h8 y42 ff6 fs5 fc0 sc0 lsd ws3b">, and so forth. </div><div class="t m3 x81 h8 yf3 ff6 fs5 fc0 sc0 ls44 ws20">ZF then approximates (2) by projecting </div><div class="t m3 x81 h8 yf4 ff6 fs5 fc0 sc0 lsd ws0">each <span class="_ _c"> </span><span class="ff5 ls0">s</span></div><div class="t m3 xcb h27 yf5 ff10 fs5 fc0 sc0 ls0 ws0">|</div><div class="c xcb yf6 w22 h33"><div class="t m3 x41 hd y3b ff5 fs7 fc0 sc0 ls0 ws0">k</div></div><div class="t m3 x6c h8 y8c ff6 fs5 fc0 sc0 lsd ws0"> onto the constellation </div><div class="c xb4 y8c w7 h15"><div class="t m3 x0 h16 y3e fff fs5 fc0 sc0 ls0 ws0">S</div></div><div class="t m3 x97 h8 y8c ff6 fs5 fc0 sc0 ls0 ws0"> </div><div class="t m3 x6 h34 yf7 ff1 fs5 fc0 sc0 ls5 ws0">Erik G. Larsson</div><div class="t m3 xcc h35 yf8 ff3 fsd fc0 sc0 ls45 ws0">MIMO Detection Methods: How They W<span class="_ _7"></span>ork</div><div class="t m0 xcc h36 yf9 ff1 fs3 fc0 sc0 ls0 ws0">1053-5888/09/$25.00&#169;2009IEEE</div><div class="t m3 xcd h37 yfa ff15 fs0 fc4 sc0 ls0 ws0">Authorized licensed use limited to: SUN YAT-SEN UNIVERSITY. Downloaded on April 14,2021 at 07:32:15 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.693122,0.000000,0.000000,1.693122,0.000000,0.000000]}'></div></div></body></html>
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