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large-scale MIMO系统仿真,附带matlab程序及论文!
uplink large-scale MIMO systems.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://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/62523d866caf59619263edef/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/62523d866caf59619263edef/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Low-complexity<span class="_ _0"> </span>near-optimal<span class="_ _0"> </span>signal</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">detection<span class="_ _0"> </span>for<span class="_ _0"> </span>uplink<span class="_ _0"> </span>large-scale</div><div class="t m0 x1 h2 y3 ff1 fs0 fc0 sc0 ls0 ws0">MIMO<span class="_ _0"> </span>systems</div><div class="t m0 x1 h3 y4 ff2 fs1 fc0 sc0 ls0 ws0">Xinyu<span class="_ _1"> </span>Gao,<span class="_ _1"> </span>Linglong<span class="_ _1"> </span>Dai,<span class="_ _1"> </span>Yongkui<span class="_ _1"> </span>Ma<span class="_ _1"> </span>and</div><div class="t m0 x1 h3 y5 ff2 fs1 fc0 sc0 ls0 ws0">Zhaocheng<span class="_ _1"> </span>Wang</div><div class="t m0 x2 h4 y6 ff2 fs2 fc0 sc0 ls0 ws0">The<span class="_ _2"> </span>minimum<span class="_ _2"> </span>mean<span class="_ _2"> </span>square<span class="_ _2"> </span>error<span class="_ _2"> </span>(MMSE)<span class="_ _2"> </span>signal<span class="_ _2"> </span>detection<span class="_ _2"> </span>algorithm</div><div class="t m0 x2 h4 y7 ff2 fs2 fc0 sc0 ls0 ws0">is<span class="_ _3"> </span>near-optimal<span class="_ _3"> </span>for<span class="_ _3"> </span>uplink<span class="_ _3"> </span>multi-user<span class="_ _3"> </span>large-scale<span class="_ _3"> </span>multiple-input<span class="ff3">&#8211;</span>mul-</div><div class="t m0 x2 h4 y8 ff2 fs2 fc0 sc0 ls0 ws0">tiple-output<span class="_ _4"> </span>(MIMO)<span class="_ _0"> </span>systems,<span class="_ _0"> </span>but<span class="_ _4"> </span>involves<span class="_ _0"> </span>matrix<span class="_ _0"> </span>inversion<span class="_ _4"> </span>with</div><div class="t m0 x2 h4 y9 ff2 fs2 fc0 sc0 ls0 ws0">high<span class="_ _3"> </span>complexity.<span class="_ _3"> </span>It<span class="_ _1"> </span>is<span class="_ _3"> </span><span class="ff4">&#64257;</span>rstly<span class="_ _3"> </span>proved<span class="_ _3"> </span>that<span class="_ _3"> </span>the<span class="_ _1"> </span>MMSE<span class="_ _3"> </span><span class="ff4">&#64257;</span>ltering<span class="_ _1"> </span>matrix</div><div class="t m0 x2 h4 ya ff2 fs2 fc0 sc0 ls0 ws0">for<span class="_ _2"> </span>large-scale<span class="_ _2"> </span>MIMO<span class="_ _2"> </span>is<span class="_ _2"> </span>symmetric<span class="_ _2"> </span>positive<span class="_ _2"> </span>de<span class="ff4">&#64257;</span>nite,<span class="_ _2"> </span>based<span class="_ _2"> </span>on<span class="_ _2"> </span>which</div><div class="t m0 x2 h4 yb ff2 fs2 fc0 sc0 ls0 ws0">a<span class="_ _2"> </span>low-complexity<span class="_ _2"> </span>near-optimal<span class="_ _3"> </span>signal<span class="_ _2"> </span>detection<span class="_ _2"> </span>algorithm<span class="_ _3"> </span>by<span class="_ _2"> </span>exploit-</div><div class="t m0 x2 h4 yc ff2 fs2 fc0 sc0 ls0 ws0">ing<span class="_ _2"> </span>the<span class="_ _5"> </span>Richardson<span class="_ _5"> </span>method<span class="_ _2"> </span>to<span class="_ _5"> </span>avoid<span class="_ _5"> </span>the<span class="_ _2"> </span>matrix<span class="_ _5"> </span>inversion<span class="_ _2"> </span>is<span class="_ _5"> </span>proposed.</div><div class="t m0 x2 h4 yd ff2 fs2 fc0 sc0 ls0 ws0">The<span class="_ _5"> </span>complexity<span class="_ _2"> </span>can<span class="_ _5"> </span>be<span class="_ _5"> </span>reduced<span class="_ _2"> </span>from<span class="_ _5"> </span><span class="ff5">O</span>(<span class="ff6">K</span></div><div class="t m1 x3 h5 ye ff2 fs3 fc0 sc0 ls0 ws0">3</div><div class="t m0 x4 h4 yf ff2 fs2 fc0 sc0 ls1 ws0">)t<span class="_ _6"></span>o<span class="ff5 ls0">O<span class="ff2">(<span class="ff6">K</span></span></span></div><div class="t m1 x5 h5 ye ff2 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6 h4 yf ff2 fs2 fc0 sc0 ls0 ws0">),<span class="_ _5"> </span>where<span class="_ _5"> </span><span class="ff6">K<span class="_ _2"> </span></span>is<span class="_ _5"> </span>the</div><div class="t m0 x2 h4 y10 ff2 fs2 fc0 sc0 ls0 ws0">number<span class="_ _5"> </span>of<span class="_ _5"> </span>users.<span class="_ _2"> </span>The<span class="_ _5"> </span>convergence<span class="_ _5"> </span>proof<span class="_ _5"> </span>of<span class="_ _2"> </span>the<span class="_ _5"> </span>proposed<span class="_ _5"> </span>algorithm<span class="_ _2"> </span>is</div><div class="t m0 x2 h4 y11 ff2 fs2 fc0 sc0 ls0 ws0">also<span class="_ _5"> </span>provided.<span class="_ _5"> </span>Simulation<span class="_ _5"> </span>results<span class="_ _7"> </span>show<span class="_ _7"> </span>that<span class="_ _5"> </span>the<span class="_ _5"> </span>proposed<span class="_ _5"> </span>signal<span class="_ _5"> </span>detec-</div><div class="t m0 x2 h4 y12 ff2 fs2 fc0 sc0 ls0 ws0">tion<span class="_ _5"> </span>algorithm<span class="_ _5"> </span>converges<span class="_ _5"> </span>fast,<span class="_ _5"> </span>and<span class="_ _2"> </span>achieves<span class="_ _7"> </span>the<span class="_ _5"> </span>near-optimal<span class="_ _2"> </span>perform-</div><div class="t m0 x2 h4 y13 ff2 fs2 fc0 sc0 ls0 ws0">ance<span class="_ _5"> </span>of<span class="_ _2"> </span>the<span class="_ _2"> </span>classical<span class="_ _5"> </span>MMSE<span class="_ _2"> </span>algorithm.</div><div class="t m0 x1 h6 y14 ff6 fs4 fc0 sc0 ls0 ws0">Introduction:<span class="_ _0"> </span><span class="ff2">Large-scale<span class="_ _8"> </span>multiple-input<span class="ff3">&#8211;</span>multiple-output<span class="_ _8"> </span>(MIMO)</span></div><div class="t m0 x1 h6 y15 ff2 fs4 fc0 sc0 ls0 ws0">employing<span class="_ _0"> </span>hundreds<span class="_ _0"> </span>of<span class="_ _4"> </span>antennas<span class="_ _0"> </span>at<span class="_ _0"> </span>the<span class="_ _0"> </span>base<span class="_ _4"> </span>sta<span class="_ _9"></span>tion<span class="_ _0"> </span>(BS)<span class="_ _4"> </span>to<span class="_ _0"> </span>simul-</div><div class="t m0 x1 h6 y16 ff2 fs4 fc0 sc0 ls0 ws0">taneously<span class="_ _3"> </span>serve<span class="_ _3"> </span>multiple<span class="_ _1"> </span>users<span class="_ _1"> </span>is<span class="_ _3"> </span>a<span class="_ _1"> </span>promising<span class="_ _3"> </span>key<span class="_ _3"> </span>technology<span class="_ _1"> </span>for<span class="_ _1"> </span><span class="ff4">&#64257;</span>fth</div><div class="t m0 x1 h6 y17 ff2 fs4 fc0 sc0 ls0 ws0">generation<span class="_ _7"> </span>wireless<span class="_ _7"> </span>communications<span class="_ _7"> </span>[<span class="fc1">1</span>].<span class="_ _7"> </span>It<span class="_ _5"> </span>can<span class="_ _7"> </span>achieve<span class="_ _7"> </span>orders<span class="_ _7"> </span>of<span class="_ _5"> </span>magni-</div><div class="t m0 x1 h6 y18 ff2 fs4 fc0 sc0 ls0 ws0">tude<span class="_ _3"> </span>increases<span class="_ _3"> </span>in<span class="_ _3"> </span>spectrum<span class="_ _3"> </span>and<span class="_ _3"> </span>energy<span class="_ _2"> </span>ef<span class="ff4">&#64257;</span>ciency,<span class="_ _3"> </span>and<span class="_ _3"> </span>one<span class="_ _3"> </span>challenging</div><div class="t m0 x1 h6 y19 ff2 fs4 fc0 sc0 ls0 ws0">issue<span class="_ _2"> </span>to<span class="_ _5"> </span>realise<span class="_ _2"> </span>such<span class="_ _2"> </span>a<span class="_ _5"> </span>goal<span class="_ _2"> </span>is<span class="_ _2"> </span>the<span class="_ _2"> </span>low-complexity<span class="_ _5"> </span>signal<span class="_ _5"> </span>detection<span class="_ _2"> </span>algo-</div><div class="t m0 x1 h6 y1a ff2 fs4 fc0 sc0 ls0 ws0">rithm<span class="_ _7"> </span>in<span class="_ _7"> </span>the<span class="_ _7"> </span>uplink,<span class="_ _7"> </span>due<span class="_ _7"> </span>to<span class="_ _7"> </span>the<span class="_ _7"> </span>increased<span class="_ _7"> </span>dimension<span class="_ _7"> </span>of<span class="_ _7"> </span>large-scale<span class="_ _7"> </span>MIMO</div><div class="t m0 x1 h6 y1b ff2 fs4 fc0 sc0 ls0 ws0">systems<span class="_ _3"> </span>[<span class="fc1">2</span>].<span class="_ _1"> </span>The<span class="_ _1"> </span>optimal<span class="_ _a"> </span>signal<span class="_ _1"> </span>detection<span class="_ _1"> </span>algorithm<span class="_ _a"> </span>is<span class="_ _3"> </span>the<span class="_ _1"> </span>maximum-</div><div class="t m0 x1 h6 y1c ff2 fs4 fc0 sc0 ls0 ws0">likelihood<span class="_ _a"> </span>algorithm,<span class="_ _a"> </span>but<span class="_ _0"> </span>its<span class="_ _a"> </span>complexity<span class="_ _a"> </span>increases<span class="_ _a"> </span>exponentially<span class="_ _a"> </span>with</div><div class="t m0 x1 h6 y1d ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _3"> </span>number<span class="_ _1"> </span>of<span class="_ _1"> </span>transmit<span class="_ _3"> </span>antennas,<span class="_ _1"> </span>making<span class="_ _1"> </span>it<span class="_ _1"> </span>impossible<span class="_ _1"> </span>for<span class="_ _1"> </span>large-scale</div><div class="t m0 x1 h6 y1e ff2 fs4 fc0 sc0 ls0 ws0">MIMO<span class="_ _a"> </span>systems.<span class="_ _b"> </span>The<span class="_ _a"> </span><span class="ff4">&#64257;</span>xed-complexity<span class="_ _b"> </span>sphere<span class="_ _a"> </span>decoding<span class="_ _b"> </span>[<span class="fc1">3</span>]<span class="_ _b"> </span>and<span class="_ _b"> </span>tabu</div><div class="t m0 x1 h6 y1f ff2 fs4 fc0 sc0 ls0 ws0">search<span class="_ _b"> </span>[<span class="fc1">4</span>]<span class="_ _b"> </span>algorithms<span class="_ _0"> </span>have<span class="_ _b"> </span>been<span class="_ _b"> </span>proposed<span class="_ _b"> </span>with<span class="_ _0"> </span>reduced<span class="_ _b"> </span>complexity,</div><div class="t m0 x1 h6 y20 ff2 fs4 fc0 sc0 ls0 ws0">but<span class="_ _c"> </span>their<span class="_ _c"> </span>complexity<span class="_ _c"> </span>is<span class="_ _c"> </span>unafordable<span class="_ _c"> </span>when<span class="_ _c"> </span>the<span class="_ _c"> </span>dimension<span class="_ _c"> </span>of<span class="_ _c"> </span>the</div><div class="t m0 x1 h6 y21 ff2 fs4 fc0 sc0 ls0 ws0">large-scale<span class="_ _3"> </span>MIMO<span class="_ _3"> </span>sy<span class="_ _9"></span>stem<span class="_ _3"> </span>is<span class="_ _3"> </span>large<span class="_ _3"> </span>or<span class="_ _3"> </span>the<span class="_ _2"> </span>modulation<span class="_ _3"> </span>order<span class="_ _3"> </span>is<span class="_ _3"> </span>high<span class="_ _3"> </span>[<span class="fc1">5</span>].</div><div class="t m0 x1 h6 y22 ff2 fs4 fc0 sc0 ls0 ws0">Low-complexity<span class="_ _1"> </span>linear<span class="_ _a"> </span>detection<span class="_ _b"> </span>algorithms<span class="_ _a"> </span>such<span class="_ _a"> </span>as<span class="_ _a"> </span>zero-forcing<span class="_ _1"> </span>and</div><div class="t m0 x1 h6 y23 ff2 fs4 fc0 sc0 ls0 ws0">minimum<span class="_ _1"> </span>mean<span class="_ _1"> </span>square<span class="_ _1"> </span>error<span class="_ _1"> </span>(MMSE)<span class="_ _a"> </span>with<span class="_ _1"> </span>near-optimal<span class="_ _1"> </span>performance</div><div class="t m0 x1 h6 y24 ff2 fs4 fc0 sc0 ls0 ws0">have<span class="_ _2"> </span>been<span class="_ _3"> </span>investigated<span class="_ _2"> </span>[<span class="fc1">2</span>],<span class="_ _3"> </span>but<span class="_ _3"> </span>these<span class="_ _2"> </span>algorithms<span class="_ _3"> </span>have<span class="_ _3"> </span>to<span class="_ _2"> </span>use<span class="_ _3"> </span>unfavour-</div><div class="t m0 x1 h6 y25 ff2 fs4 fc0 sc0 ls0 ws0">able<span class="_ _b"> </span>matrix<span class="_ _b"> </span>inversion,<span class="_ _b"> </span>whose<span class="_ _b"> </span>high<span class="_ _b"> </span>complexity<span class="_ _b"> </span>is<span class="_ _0"> </span>still<span class="_ _a"> </span>not<span class="_ _0"> </span>acceptable</div><div class="t m0 x1 h6 y26 ff2 fs4 fc0 sc0 ls0 ws0">for<span class="_ _2"> </span>large-scale<span class="_ _3"> </span>MIMO<span class="_ _3"> </span>sy<span class="_ _9"></span>stems.<span class="_ _2"> </span>Recently,<span class="_ _3"> </span>the<span class="_ _2"> </span>Neumann<span class="_ _3"> </span>series<span class="_ _2"> </span>approxi-</div><div class="t m0 x1 h6 y27 ff2 fs4 fc0 sc0 ls0 ws0">mation<span class="_ _1"> </span>algorithm<span class="_ _a"> </span>has<span class="_ _1"> </span>been<span class="_ _a"> </span>proposed<span class="_ _1"> </span>to<span class="_ _a"> </span>approximate<span class="_ _1"> </span>matrix<span class="_ _1"> </span>inversion</div><div class="t m0 x1 h6 y28 ff2 fs4 fc0 sc0 ls0 ws0">[<span class="fc1">6</span>],<span class="_ _3"> </span>which<span class="_ _3"> </span>converts<span class="_ _2"> </span>the<span class="_ _3"> </span>matrix<span class="_ _3"> </span>inversion<span class="_ _3"> </span>into<span class="_ _3"> </span>a<span class="_ _2"> </span>series<span class="_ _3"> </span>of<span class="_ _3"> </span>matrix<span class="ff3">&#8211;</span>vector</div><div class="t m0 x1 h6 y29 ff2 fs4 fc0 sc0 ls0 ws0">multiplications.<span class="_ _1"> </span>However,<span class="_ _1"> </span>only<span class="_ _1"> </span>marginal<span class="_ _a"> </span>complexity<span class="_ _1"> </span>reduction<span class="_ _1"> </span>can<span class="_ _1"> </span>be</div><div class="t m0 x1 h6 y2a ff2 fs4 fc0 sc0 ls0 ws0">achieved.</div><div class="t m0 x7 h6 y2b ff2 fs4 fc0 sc0 ls0 ws0">In<span class="_ _4"> </span>this<span class="_ _4"> </span>Letter,<span class="_ _4"> </span>we<span class="_ _4"> </span>propose<span class="_ _4"> </span>a<span class="_ _4"> </span>low-complexity<span class="_ _4"> </span>near-optimal<span class="_ _4"> </span>signal</div><div class="t m0 x1 h6 y2c ff2 fs4 fc0 sc0 ls0 ws0">detection<span class="_ _3"> </span>algorithm<span class="_ _1"> </span>by<span class="_ _3"> </span>exploiting<span class="_ _1"> </span>the<span class="_ _3"> </span>Richardson<span class="_ _1"> </span>method<span class="_ _1"> </span>[<span class="fc1">7</span>]<span class="_ _3"> </span>to<span class="_ _1"> </span>avoid</div><div class="t m0 x1 h6 y2d ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _1"> </span>complicated<span class="_ _a"> </span>matrix<span class="_ _1"> </span>inversion.<span class="_ _a"> </span>We<span class="_ _1"> </span><span class="ff4">&#64257;</span>rstly<span class="_ _a"> </span>prove<span class="_ _3"> </span>a<span class="_ _a"> </span>special<span class="_ _a"> </span>property</div><div class="t m0 x1 h6 y2e ff2 fs4 fc0 sc0 ls0 ws0">of<span class="_ _3"> </span>large-scale<span class="_ _1"> </span>MIMO<span class="_ _1"> </span>systems<span class="_ _3"> </span>that<span class="_ _3"> </span>the<span class="_ _3"> </span>MMSE<span class="_ _1"> </span><span class="ff4">&#64257;</span>ltering<span class="_ _1"> </span>matrix<span class="_ _3"> </span>is<span class="_ _1"> </span>sym-</div><div class="t m0 x1 h6 y2f ff2 fs4 fc0 sc0 ls0 ws0">metric<span class="_ _0"> </span>positive<span class="_ _4"> </span>de<span class="ff4">&#64257;</span>nite,<span class="_ _0"> </span>based<span class="_ _4"> </span>on<span class="_ _b"> </span>which<span class="_ _4"> </span>we<span class="_ _0"> </span>propose<span class="_ _0"> </span>to<span class="_ _4"> </span>ex<span class="_ _9"></span>ploit<span class="_ _4"> </span>the</div><div class="t m0 x1 h6 y30 ff2 fs4 fc0 sc0 ls0 ws0">Richardson<span class="_ _a"> </span>method<span class="_ _b"> </span>to<span class="_ _b"> </span>avoid<span class="_ _a"> </span>the<span class="_ _b"> </span>complicated<span class="_ _a"> </span>matrix<span class="_ _b"> </span>inversion.<span class="_ _a"> </span>Then</div><div class="t m0 x1 h6 y31 ff2 fs4 fc0 sc0 ls0 ws0">we<span class="_ _b"> </span>prove<span class="_ _a"> </span>the<span class="_ _b"> </span>convergence<span class="_ _b"> </span>of<span class="_ _0"> </span>the<span class="_ _b"> </span>proposed<span class="_ _b"> </span>algorithm<span class="_ _0"> </span>for<span class="_ _b"> </span>any<span class="_ _b"> </span>initial</div><div class="t m0 x1 h6 y32 ff2 fs4 fc0 sc0 ls0 ws0">solution<span class="_ _d"> </span>when<span class="_ _d"> </span>the<span class="_ _d"> </span>relaxation<span class="_ _4"> </span>parameter<span class="_ _d"> </span>is<span class="_ _d"> </span>appropriate.<span class="_ _4"> </span>Finally,<span class="_ _d"> </span>we</div><div class="t m0 x1 h6 y33 ff2 fs4 fc0 sc0 ls0 ws0">verify<span class="_ _5"> </span>through<span class="_ _3"> </span>simulations<span class="_ _5"> </span>that<span class="_ _2"> </span>the<span class="_ _2"> </span>proposed<span class="_ _2"> </span>signal<span class="_ _2"> </span>detection<span class="_ _3"> </span>algorithm</div><div class="t m0 x1 h6 y34 ff2 fs4 fc0 sc0 ls0 ws0">can<span class="_ _1"> </span>ef<span class="ff4">&#64257;</span>ciently<span class="_ _1"> </span>solve<span class="_ _1"> </span>the<span class="_ _1"> </span>matrix<span class="_ _1"> </span>inversion<span class="_ _1"> </span>problem<span class="_ _1"> </span>in<span class="_ _a"> </span>an<span class="_ _1"> </span>iterative<span class="_ _3"> </span>way</div><div class="t m0 x1 h6 y35 ff2 fs4 fc0 sc0 ls0 ws0">until<span class="_ _0"> </span>the<span class="_ _4"> </span>desired<span class="_ _b"> </span>accuracy<span class="_ _0"> </span>is<span class="_ _4"> </span>attained,<span class="_ _b"> </span>and<span class="_ _4"> </span>achieve<span class="_ _b"> </span>the<span class="_ _4"> </span>near-optimal</div><div class="t m0 x1 h6 y36 ff2 fs4 fc0 sc0 ls0 ws0">performance<span class="_ _2"> </span>of<span class="_ _3"> </span>the<span class="_ _2"> </span>MMSE<span class="_ _3"> </span>algorithm<span class="_ _2"> </span>with<span class="_ _3"> </span>exact<span class="_ _2"> </span>matrix<span class="_ _2"> </span>inversion.</div><div class="t m0 x1 h6 y37 ff6 fs4 fc0 sc0 ls0 ws0">Large-scale<span class="_ _a"> </span>MIMO<span class="_ _b"> </span>system<span class="_ _a"> </span>model:<span class="_ _0"> </span><span class="ff2">We<span class="_ _a"> </span>consider<span class="_ _a"> </span>an<span class="_ _b"> </span>uplink<span class="_ _a"> </span>multi-user</span></div><div class="t m0 x1 h6 y38 ff2 fs4 fc0 sc0 ls0 ws0">large-scale<span class="_ _2"> </span>MIMO<span class="_ _2"> </span>syst<span class="_ _9"></span>em<span class="_ _2"> </span>which<span class="_ _2"> </span>employs<span class="_ _5"> </span><span class="ff6">N<span class="_ _2"> </span></span>antennas<span class="_ _2"> </span>at<span class="_ _2"> </span>the<span class="_ _2"> </span>BS<span class="_ _2"> </span>to<span class="_ _2"> </span>sim-</div><div class="t m0 x1 h6 y39 ff2 fs4 fc0 sc0 ls0 ws0">ultaneously<span class="_ _3"> </span>serve<span class="_ _3"> </span><span class="ff6">K<span class="_ _3"> </span></span>single-antenna<span class="_ _3"> </span>users.<span class="_ _3"> </span>Usually<span class="_ _3"> </span>we<span class="_ _3"> </span>have<span class="_ _3"> </span><span class="ff6">N<span class="_ _e"> </span></span>&gt;<span class="_ _e"> </span><span class="ff6">K</span>,<span class="_ _3"> </span>e.g.</div><div class="t m0 x1 h6 y3a ff6 fs4 fc0 sc0 ls0 ws0">N<span class="_ _e"> </span><span class="ff2">=<span class="_ _e"> </span>128<span class="_ _1"> </span>and<span class="_ _3"> </span></span>K<span class="_ _e"> </span><span class="ff2">=<span class="_ _7"> </span>16<span class="_ _3"> </span>have<span class="_ _3"> </span>been<span class="_ _1"> </span>considered<span class="_ _3"> </span>[<span class="fc1">1</span>,<span class="_ _3"> </span><span class="fc1">2</span>].<span class="_ _1"> </span>For<span class="_ _3"> </span>signal<span class="_ _1"> </span>detection,</span></div><div class="t m0 x1 h6 y3b ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _1"> </span>complex-valued<span class="_ _1"> </span>system<span class="_ _3"> </span>model<span class="_ _1"> </span>can<span class="_ _1"> </span>be<span class="_ _1"> </span>directly<span class="_ _1"> </span>converted<span class="_ _1"> </span>to<span class="_ _1"> </span>a<span class="_ _1"> </span>cor-</div><div class="t m0 x1 h6 y3c ff2 fs4 fc0 sc0 ls0 ws0">responding<span class="_ _3"> </span>real-valued<span class="_ _1"> </span>system<span class="_ _1"> </span>model,<span class="_ _1"> </span>then<span class="_ _1"> </span>the<span class="_ _1"> </span>estimate<span class="_ _1"> </span>of<span class="_ _1"> </span>the<span class="_ _1"> </span>2<span class="ff6">K<span class="_ _7"> </span></span><span class="ls2">&#215;1</span></div><div class="t m0 x1 h6 y3d ff2 fs4 fc0 sc0 ls0 ws0">transmitted<span class="_ _0"> </span>signal<span class="_ _4"> </span>vector</div><div class="t m0 x8 h7 y3e ff7 fs4 fc0 sc0 ls0 ws0">&#710;</div><div class="t m0 x8 h6 y3d ff8 fs4 fc0 sc0 ls0 ws0">s<span class="_ _0"> </span><span class="ff2">coming<span class="_ _4"> </span>from<span class="_ _0"> </span><span class="ff6">K<span class="_ _4"> </span></span>differe<span class="_ _9"></span>nce<span class="_ _4"> </span>users<span class="_ _0"> </span>can<span class="_ _4"> </span>be</span></div><div class="t m0 x1 h6 y3f ff2 fs4 fc0 sc0 ls0 ws0">achieved<span class="_ _2"> </span>by<span class="_ _2"> </span>the<span class="_ _3"> </span>classical<span class="_ _2"> </span>MMSE<span class="_ _3"> </span>algorithm<span class="_ _2"> </span>as<span class="_ _3"> </span>[<span class="fc1">2</span>]</div><div class="t m0 x9 h7 y40 ff7 fs4 fc0 sc0 ls0 ws0">&#710;</div><div class="t m0 x9 h6 y41 ff8 fs4 fc0 sc0 ls0 ws0">s<span class="_ _5"> </span><span class="ff5">=<span class="_ _2"> </span><span class="ff2">(</span></span>H</div><div class="t m0 xa h8 y42 ff9 fs5 fc0 sc0 ls0 ws0">H</div><div class="t m0 xb h9 y41 ff8 fs4 fc0 sc0 ls0 ws0">H<span class="_ _7"> </span><span class="ff5">+</span></div><div class="t m2 xc ha y41 ffa fs6 fc0 sc0 ls0 ws0">s</div><div class="t m0 xd hb y42 ff2 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 xe hc y41 ff8 fs4 fc0 sc0 ls0 ws0">I</div><div class="t m0 xf hb y43 ff2 fs5 fc0 sc0 ls0 ws0">2<span class="ff6">K</span></div><div class="t m0 x10 h6 y41 ff2 fs4 fc0 sc0 ls0 ws0">)</div><div class="t m0 x11 hb y44 ff5 fs5 fc0 sc0 ls0 ws0">&#8722;<span class="ff2">1</span></div><div class="t m0 x12 hc y41 ff8 fs4 fc0 sc0 ls0 ws0">H</div><div class="t m0 x13 h8 y42 ff9 fs5 fc0 sc0 ls0 ws0">H</div><div class="t m0 x14 h9 y41 ff8 fs4 fc0 sc0 ls0 ws0">y<span class="_ _5"> </span><span class="ff5">=<span class="_ _2"> </span></span>W</div><div class="t m0 x15 hb y42 ff5 fs5 fc0 sc0 ls0 ws0">&#8722;<span class="ff2">1</span></div><div class="t m0 x16 h7 y45 ff7 fs4 fc0 sc0 ls0 ws0">&#710;</div><div class="t m0 x16 h6 y46 ff8 fs4 fc0 sc0 ls0 ws0">y<span class="_ _f"> </span><span class="ff2">(1)</span></div><div class="t m0 x1 h6 y47 ff2 fs4 fc0 sc0 ls0 ws0">where<span class="_ _2"> </span><span class="ffb">H<span class="_ _2"> </span></span>is<span class="_ _2"> </span>the<span class="_ _2"> </span>2<span class="ff6">N<span class="_ _7"> </span></span><span class="ls2">&#215;2<span class="_ _10"></span><span class="ff6 ls0">K<span class="_ _2"> </span><span class="ff2">MIMO<span class="_ _2"> </span>channel<span class="_ _3"> </span>matrix,<span class="_ _5"> </span>which<span class="_ _3"> </span>can<span class="_ _2"> </span>be<span class="_ _2"> </span>obtained</span></span></span></div><div class="t m0 x1 h6 y48 ff2 fs4 fc0 sc0 ls0 ws0">through<span class="_ _5"> </span>frequency-domain<span class="_ _2"> </span>and/or<span class="_ _2"> </span>time-domain<span class="_ _2"> </span>training<span class="_ _2"> </span>pilots<span class="_ _2"> </span>[<span class="fc1">8</span>],<span class="_ _2"> </span><span class="ffc">&#963;</span></div><div class="t m3 x17 hd y49 ff2 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m0 x18 h6 y4a ff2 fs4 fc0 sc0 ls0 ws0">is</div><div class="t m0 x1 h6 y4b ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _7"> </span>additive<span class="_ _7"> </span>white<span class="_ _7"> </span>Gaussian<span class="_ _5"> </span>noise<span class="_ _7"> </span>power,<span class="_ _7"> </span><span class="ffb">I</span></div><div class="t m3 x19 hd y4c ff2 fs7 fc0 sc0 ls0 ws0">2<span class="ff6">K</span></div><div class="t m0 x1a h6 y4d ff2 fs4 fc0 sc0 ls0 ws0">is<span class="_ _7"> </span>an<span class="_ _7"> </span>identity<span class="_ _5"> </span>matrix<span class="_ _7"> </span>of<span class="_ _7"> </span>size</div><div class="t m0 x1 h6 y4e ff2 fs4 fc0 sc0 ls0 ws0">2<span class="ff6">K<span class="_ _e"> </span></span><span class="ls2">&#215;2<span class="_ _10"></span><span class="ff6 ls0">K<span class="ff2">,<span class="_ _2"> </span><span class="ffb">y<span class="_ _2"> </span></span>is<span class="_ _2"> </span>the<span class="_ _2"> </span>2</span>N<span class="_ _7"> </span><span class="ff2">&#215;<span class="_ _e"> </span>1<span class="_ _2"> </span>received<span class="_ _2"> </span>signal<span class="_ _2"> </span>vector<span class="_ _2"> </span>at<span class="_ _5"> </span>the<span class="_ _3"> </span>BS,</span></span></span></div><div class="t m0 x1b he y4f ffd fs4 fc0 sc0 ls0 ws0">&#58881;</div><div class="t m0 x1c h9 y4e ff8 fs4 fc0 sc0 ls0 ws0">y<span class="_ _5"> </span><span class="ff5">=<span class="_ _2"> </span></span>H</div><div class="t m0 x1d h8 y50 ff9 fs5 fc0 sc0 ls0 ws0">H</div><div class="t m0 x1e h6 y51 ff8 fs4 fc0 sc0 ls0 ws0">y<span class="_ _2"> </span><span class="ff2">can</span></div><div class="t m0 x1 h6 y52 ff2 fs4 fc0 sc0 ls0 ws0">be<span class="_ _1"> </span>interpreted<span class="_ _1"> </span>as<span class="_ _1"> </span>the<span class="_ _1"> </span>matched-<span class="ff4">&#64257;</span>lter<span class="_ _1"> </span>output<span class="_ _1"> </span>of<span class="_ _a"> </span><span class="ffb">y<span class="_ _3"> </span></span>and<span class="_ _a"> </span><span class="ff4">&#64257;</span>nally<span class="_ _3"> </span><span class="ffb">W<span class="_ _7"> </span></span>=<span class="_ _e"> </span><span class="ffb">H</span></div><div class="t m3 x1f hd y53 ff2 fs7 fc0 sc0 ls0 ws0">H</div><div class="t m0 x20 hf y54 ffb fs4 fc0 sc0 ls0 ws0">H</div><div class="t m0 x1 h6 y55 ff2 fs4 fc0 sc0 ls0 ws0">+<span class="_ _e"> </span><span class="ffc">&#963;</span></div><div class="t m3 x21 hd y56 ff2 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m0 x22 hf y57 ffb fs4 fc0 sc0 ls0 ws0">I</div><div class="t m3 x23 hd y58 ff2 fs7 fc0 sc0 ls0 ws0">2<span class="ff6">K</span></div><div class="t m0 x24 h6 y57 ff2 fs4 fc0 sc0 ls0 ws0">denotes<span class="_ _5"> </span>the<span class="_ _5"> </span>MMSE<span class="_ _5"> </span><span class="ff4">&#64257;</span>ltering<span class="_ _2"> </span>matrix.<span class="_ _5"> </span>Note<span class="_ _5"> </span>that<span class="_ _5"> </span>the<span class="_ _5"> </span>direct<span class="_ _5"> </span>compu-</div><div class="t m0 x1 h6 y59 ff2 fs4 fc0 sc0 ls0 ws0">tation<span class="_ _7"> </span>of<span class="_ _7"> </span>the<span class="_ _7"> </span>matrix<span class="_ _7"> </span>inversion<span class="_ _7"> </span><span class="ffb">W</span></div><div class="t m3 x25 hd y5a ffe fs7 fc0 sc0 ls0 ws0">&#8722;<span class="_ _11"></span><span class="ff2">1</span></div><div class="t m0 x26 h6 y5b ff2 fs4 fc0 sc0 ls0 ws0">requires<span class="_ _e"> </span>relatively<span class="_ _7"> </span>high<span class="_ _7"> </span>complexity<span class="_ _e"> </span>of</div><div class="t m0 x1 h6 y5c ff5 fs4 fc0 sc0 ls0 ws0">O<span class="ff2">(<span class="ff6">K</span></span></div><div class="t m0 x27 hb y5d ff2 fs5 fc0 sc0 ls0 ws0">3</div><div class="t m0 x28 h6 y5e ff2 fs4 fc0 sc0 ls0 ws0">).</div><div class="t m0 x29 h6 y5f ff6 fs4 fc0 sc0 ls0 ws0">Proposed<span class="_ _a"> </span>signal<span class="_ _b"> </span>detection<span class="_ _b"> </span>based<span class="_ _b"> </span>on<span class="_ _b"> </span>Richardson<span class="_ _b"> </span>method:<span class="_ _4"> </span><span class="ff2">Unlike<span class="_ _a"> </span>the</span></div><div class="t m0 x29 h6 y60 ff2 fs4 fc0 sc0 ls0 ws0">conventional<span class="_ _4"> </span>(small-scale)<span class="_ _4"> </span>MIMO<span class="_ _4"> </span>syst<span class="_ _9"></span>ems<span class="_ _4"> </span>with<span class="_ _4"> </span>a<span class="_ _4"> </span>small<span class="_ _4"> </span>number<span class="_ _4"> </span>of</div><div class="t m0 x29 h6 y61 ff2 fs4 fc0 sc0 ls0 ws0">antennas,<span class="_ _b"> </span>large-scale<span class="_ _b"> </span>MIMO<span class="_ _b"> </span>systems<span class="_ _a"> </span>have<span class="_ _a"> </span>a<span class="_ _b"> </span>special<span class="_ _b"> </span>property<span class="_ _b"> </span>in<span class="_ _b"> </span>that</div><div class="t m0 x29 h6 y62 ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _d"> </span>MMSE<span class="_ _12"> </span><span class="ff4">&#64257;</span>ltering<span class="_ _d"> </span>matrix<span class="_ _12"> </span><span class="ffb">W<span class="_ _d"> </span></span>determined<span class="_ _12"> </span>by<span class="_ _d"> </span>the<span class="_ _12"> </span>MIMO<span class="_ _d"> </span>channel</div><div class="t m0 x29 h6 y63 ff2 fs4 fc0 sc0 ls0 ws0">matrix<span class="_ _5"> </span><span class="ffb">H<span class="_ _2"> </span></span>is<span class="_ _2"> </span>symmetric<span class="_ _5"> </span>positive<span class="_ _5"> </span>de<span class="_"> </span><span class="ff4">&#64257;</span>nite,<span class="_ _2"> </span>which<span class="_ _2"> </span>can<span class="_ _2"> </span>be<span class="_ _5"> </span>proved<span class="_ _5"> </span>as<span class="_ _2"> </span>below.</div><div class="t m0 x29 h10 y64 ff6 fs4 fc0 sc0 ls0 ws0">Lemma<span class="_ _4"> </span>1:<span class="_ _4"> </span>For<span class="_ _4"> </span>signal<span class="_ _d"> </span>detection<span class="_ _4"> </span>of<span class="_ _4"> </span>large-scale<span class="_ _d"> </span>MIMO<span class="_ _4"> </span>systems,<span class="_ _4"> </span>the</div><div class="t m0 x29 h6 y65 ff6 fs4 fc0 sc0 ls0 ws0">MMSE<span class="_ _2"> </span><span class="fff">&#64257;</span>ltering<span class="_ _3"> </span>matrix<span class="_ _2"> </span><span class="ffb">W<span class="_ _2"> </span></span>is<span class="_ _3"> </span>symmetric<span class="_ _2"> </span>positive<span class="_ _3"> </span>de<span class="fff">&#64257;</span>nite<span class="ff2">.</span></div><div class="t m0 x29 h6 y66 ff6 fs4 fc0 sc0 ls0 ws0">Proof:<span class="_ _0"> </span><span class="ff2">The<span class="_ _4"> </span>column<span class="_ _4"> </span>vectors<span class="_ _0"> </span>of<span class="_ _4"> </span>the<span class="_ _0"> </span>channel<span class="_ _4"> </span>matrix<span class="_ _4"> </span><span class="ffb">H<span class="_ _0"> </span></span>in<span class="_ _4"> </span>large-scale</span></div><div class="t m0 x29 h6 y67 ff2 fs4 fc0 sc0 ls0 ws0">MIMO<span class="_ _1"> </span>systems<span class="_ _3"> </span>are<span class="_ _1"> </span>asymptotically<span class="_ _1"> </span>orthogonal<span class="_ _1"> </span>(i.e.<span class="_ _a"> </span>rank(<span class="ffb">H</span><span class="ls3">)=2<span class="_ _10"></span><span class="ff6 ls0">K<span class="_ _11"></span><span class="ff2 ls4">)[<span class="_ _13"></span><span class="fc1 ls0">2<span class="fc0">].</span></span></span></span></span></div><div class="t m0 x29 h6 y68 ff2 fs4 fc0 sc0 ls0 ws0">Then<span class="_ _3"> </span>we<span class="_ _3"> </span>have<span class="_ _3"> </span>the<span class="_ _1"> </span>equation<span class="_ _3"> </span><span class="ffb">Hr<span class="_ _e"> </span></span>=<span class="_ _7"> </span>0<span class="_ _3"> </span>when<span class="_ _1"> </span>and<span class="_ _3"> </span>only<span class="_ _1"> </span>when<span class="_ _3"> </span><span class="ffb">r<span class="_ _1"> </span></span>is<span class="_ _3"> </span>a<span class="_ _1"> </span>2<span class="ff6">K<span class="_ _e"> </span></span><span class="ls2">&#215;1</span></div><div class="t m0 x29 h6 y69 ff2 fs4 fc0 sc0 ls0 ws0">zero<span class="_ _2"> </span>vector.<span class="_ _2"> </span>Thus,<span class="_ _3"> </span>for<span class="_ _2"> </span>an<span class="_ _3"> </span>arbitrary<span class="_ _2"> </span>nonzero<span class="_ _2"> </span>2<span class="ff6">K<span class="_ _e"> </span></span>&#215;<span class="_ _e"> </span>1<span class="_ _3"> </span>vector<span class="_ _2"> </span><span class="ffb">r</span>,<span class="_ _3"> </span>we<span class="_ _2"> </span>have</div><div class="t m0 x2a h9 y6a ff8 fs4 fc0 sc0 ls0 ws0">Hr<span class="_ _14"></span><span class="ff5 ls5">()</span></div><div class="t m0 x2b h8 y6b ff9 fs5 fc0 sc0 ls0 ws0">H</div><div class="t m0 x2c h9 y6c ff8 fs4 fc0 sc0 ls0 ws0">Hr<span class="_ _5"> </span><span class="ff5">=<span class="_ _2"> </span></span>r</div><div class="t m0 x2d h8 y6b ff9 fs5 fc0 sc0 ls0 ws0">H</div><div class="t m0 x2e h6 y6c ff2 fs4 fc0 sc0 ls0 ws0">(<span class="ff8">H</span></div><div class="t m0 x2f h8 y6b ff9 fs5 fc0 sc0 ls0 ws0">H</div><div class="t m0 x30 h6 y6c ff8 fs4 fc0 sc0 ls0 ws0">H<span class="ff2">)</span>r<span class="_ _2"> </span><span class="ff10">.<span class="_ _2"> </span><span class="ff2">0<span class="_ _15"> </span>(2)</span></span></div><div class="t m0 x29 h6 y6d ff2 fs4 fc0 sc0 ls0 ws0">which<span class="_ _3"> </span>indicates<span class="_ _3"> </span>that<span class="_ _3"> </span>the<span class="_ _3"> </span>Gram<span class="_ _3"> </span>matrix<span class="_ _3"> </span><span class="ffb">G<span class="_ _e"> </span></span>=<span class="_ _e"> </span><span class="ffb">H</span></div><div class="t m3 x31 hd y6e ff2 fs7 fc0 sc0 ls0 ws0">H</div><div class="t m0 x32 h6 y6f ffb fs4 fc0 sc0 ls0 ws0">H<span class="_ _3"> </span><span class="ff2">is<span class="_ _3"> </span>positive<span class="_ _3"> </span>de<span class="ff4">&#64257;</span>nite.<span class="_ _3"> </span>In</span></div><div class="t m0 x29 h6 y70 ff2 fs4 fc0 sc0 ls0 ws0">addition,<span class="_ _b"> </span>as<span class="_ _a"> </span>we<span class="_ _b"> </span>have<span class="_ _a"> </span><span class="ffb">G</span></div><div class="t m3 x33 hd y71 ff2 fs7 fc0 sc0 ls0 ws0">H</div><div class="t m0 x2b h6 y72 ff2 fs4 fc0 sc0 ls2 ws0">=(<span class="_ _10"></span><span class="ffb ls0">H</span></div><div class="t m3 x34 hd y71 ff2 fs7 fc0 sc0 ls0 ws0">H</div><div class="t m0 x35 h6 y72 ffb fs4 fc0 sc0 ls0 ws0">H<span class="ff2">)</span></div><div class="t m3 x2d hd y71 ff2 fs7 fc0 sc0 ls0 ws0">H</div><div class="t m0 x36 h6 y72 ff2 fs4 fc0 sc0 ls0 ws0">=<span class="_ _e"> </span><span class="ffb">G</span>,<span class="_ _b"> </span><span class="ffb">G<span class="_ _b"> </span></span>is<span class="_ _a"> </span>also<span class="_ _b"> </span>symmetric.<span class="_ _b"> </span>Thus,</div><div class="t m0 x29 h6 y73 ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _2"> </span>Gram<span class="_ _3"> </span>matrix<span class="_ _2"> </span><span class="ffb">G<span class="_ _3"> </span></span>is<span class="_ _2"> </span>symmetric<span class="_ _3"> </span>positive<span class="_ _2"> </span>de<span class="ff4">&#64257;</span>nite.<span class="_ _3"> </span>Finally,<span class="_ _5"> </span>as<span class="_ _3"> </span>the<span class="_ _2"> </span>noise</div><div class="t m0 x29 h6 y74 ff2 fs4 fc0 sc0 ls0 ws0">power<span class="_ _2"> </span><span class="ffc">&#963;</span></div><div class="t m3 x37 hd y75 ff2 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m0 x38 h6 y76 ff2 fs4 fc0 sc0 ls0 ws0">is<span class="_ _3"> </span>positive,<span class="_ _2"> </span>we<span class="_ _3"> </span>can<span class="_ _3"> </span>conclude<span class="_ _3"> </span>that<span class="_ _3"> </span>the<span class="_ _2"> </span>MMSE<span class="_ _3"> </span><span class="ff4">&#64257;</span>ltering<span class="_ _3"> </span>matrix</div><div class="t m0 x29 h6 y77 ffb fs4 fc0 sc0 ls0 ws0">W<span class="_ _e"> </span><span class="ff2">=<span class="_ _e"> </span></span>H</div><div class="t m3 x39 hd y78 ff2 fs7 fc0 sc0 ls0 ws0">H</div><div class="t m0 x3a h6 y79 ffb fs4 fc0 sc0 ls0 ws0">H<span class="_ _e"> </span><span class="ff2">+<span class="_ _e"> </span><span class="ffc">&#963;</span></span></div><div class="t m3 x3b hd y78 ff2 fs7 fc0 sc0 ls0 ws0">2</div><div class="t m0 x3c hf y79 ffb fs4 fc0 sc0 ls0 ws0">I</div><div class="t m3 x3d hd y7a ff2 fs7 fc0 sc0 ls0 ws0">2<span class="ff6">K</span></div><div class="t m0 x3e h6 y79 ff2 fs4 fc0 sc0 ls0 ws0">is<span class="_ _2"> </span>also<span class="_ _3"> </span>a<span class="_ _2"> </span>symmetric<span class="_ _3"> </span>positive<span class="_ _2"> </span>de<span class="ff4">&#64257;</span>nite<span class="_ _2"> </span>matrix.<span class="_ _16"> </span><span class="ff11">&#9633;</span></div><div class="t m0 x29 h6 y7b ff2 fs4 fc0 sc0 ls0 ws0">The<span class="_ _1"> </span>special<span class="_ _1"> </span>property<span class="_ _3"> </span>that<span class="_ _1"> </span>the<span class="_ _1"> </span>MMSE<span class="_ _1"> </span><span class="ff4">&#64257;</span>ltering<span class="_ _1"> </span>matrix<span class="_ _1"> </span><span class="ffb">W<span class="_ _3"> </span></span>in<span class="_ _1"> </span>large-scale</div><div class="t m0 x29 h6 y7c ff2 fs4 fc0 sc0 ls0 ws0">MIMO<span class="_ _0"> </span>syst<span class="_ _9"></span>ems<span class="_ _0"> </span>is<span class="_ _0"> </span>symmetric<span class="_ _b"> </span>positive<span class="_ _0"> </span>de<span class="ff4">&#64257;</span>nite<span class="_ _0"> </span>inspires<span class="_ _0"> </span>us<span class="_ _b"> </span>to<span class="_ _0"> </span>exploit</div><div class="t m0 x29 h6 y7d ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _1"> </span>Richardson<span class="_ _1"> </span>method<span class="_ _a"> </span>[<span class="fc1">7</span><span class="ls6">]t<span class="_ _13"></span>oe<span class="_ _17"></span>f<span class="_ _17"></span><span class="ff4 ls0">&#64257;<span class="ff2">ciently<span class="_ _1"> </span>solve<span class="_ _1"> </span>(1)<span class="_ _1"> </span>in<span class="_ _a"> </span>an<span class="_ _1"> </span>inversionless</span></span></span></div><div class="t m0 x29 h6 y7e ff2 fs4 fc0 sc0 ls0 ws0">way.<span class="_ _3"> </span>The<span class="_ _1"> </span>Richardson<span class="_ _1"> </span>method<span class="_ _1"> </span>is<span class="_ _3"> </span>used<span class="_ _1"> </span>to<span class="_ _1"> </span>solve<span class="_ _1"> </span>the<span class="_ _1"> </span><span class="ff6">N</span>-dimension<span class="_ _1"> </span>linear</div><div class="t m0 x29 h6 y7f ff2 fs4 fc0 sc0 ls0 ws0">equation<span class="_ _4"> </span><span class="ffb">Ax<span class="_ _e"> </span></span>=<span class="_ _e"> </span><span class="ffb">b</span>,<span class="_ _4"> </span>where<span class="_ _4"> </span><span class="ffb">A<span class="_ _4"> </span></span>is<span class="_ _4"> </span>the<span class="_ _4"> </span><span class="ff6">N<span class="_ _e"> </span></span>&#215;<span class="_ _e"> </span><span class="ff6">N<span class="_ _4"> </span></span>symmetric<span class="_ _4"> </span>positive<span class="_ _0"> </span>de<span class="ff4">&#64257;</span>nite</div><div class="t m0 x29 h6 y80 ff2 fs4 fc0 sc0 ls0 ws0">matrix,<span class="_ _3"> </span><span class="ffb">x<span class="_ _1"> </span></span>is<span class="_ _1"> </span>the<span class="_ _1"> </span><span class="ff6">N<span class="_ _e"> </span></span>&#215;<span class="_ _e"> </span>1<span class="_ _1"> </span>solution<span class="_ _1"> </span>vector<span class="_ _3"> </span>and<span class="_ _1"> </span><span class="ffb">b<span class="_ _1"> </span></span>is<span class="_ _1"> </span>the<span class="_ _3"> </span><span class="ff6">N<span class="_ _7"> </span></span>&#215;<span class="_ _e"> </span>1<span class="_ _1"> </span>measurement</div><div class="t m0 x29 h6 y81 ff2 fs4 fc0 sc0 ls0 ws0">vector.<span class="_ _2"> </span>The<span class="_ _3"> </span>Richardson<span class="_ _2"> </span>iteration<span class="_ _2"> </span>can<span class="_ _3"> </span>be<span class="_ _2"> </span>described<span class="_ _3"> </span>as</div><div class="t m0 x3f hc y82 ff8 fs4 fc0 sc0 ls0 ws0">x</div><div class="t m0 x40 hb y83 ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i<span class="ff5">+</span></span><span class="ls7">1)</span></div><div class="t m0 x41 h9 y84 ff5 fs4 fc0 sc0 ls0 ws0">=<span class="_ _5"> </span><span class="ff8">x</span></div><div class="t m0 x42 hb y83 ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i</span>)</div><div class="t m0 x43 h6 y84 ff5 fs4 fc0 sc0 ls0 ws0">+<span class="_ _7"> </span><span class="ff6">w<span class="ff2">(<span class="ff8">b<span class="_ _7"> </span></span></span></span>&#8722;<span class="_ _7"> </span><span class="ff8">Ax</span></div><div class="t m0 x44 hb y83 ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i</span>)</div><div class="t m0 x45 h6 y84 ff2 fs4 fc0 sc0 ls0 ws0">)<span class="ff9">,<span class="_ _b"> </span><span class="ff6">i<span class="_ _5"> </span><span class="ff5">=<span class="_ _2"> </span></span></span></span>0<span class="ff9">,<span class="_ _b"> </span></span>1<span class="ff9">,<span class="_ _b"> </span></span>2<span class="ff9">,<span class="_ _18"> </span><span class="ff7 ls8">...<span class="_ _19"> </span></span></span>(3)</div><div class="t m0 x29 h6 y85 ff2 fs4 fc0 sc0 ls0 ws0">where<span class="_ _7"> </span>the<span class="_ _7"> </span>superscript<span class="_ _7"> </span><span class="ff6">i<span class="_ _7"> </span></span>denotes<span class="_ _7"> </span>the<span class="_ _7"> </span>number<span class="_ _7"> </span>of<span class="_ _7"> </span>iterations<span class="_ _e"> </span>and<span class="_ _7"> </span><span class="ff6">w<span class="_ _5"> </span></span>represents</div><div class="t m0 x29 h6 y86 ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _5"> </span>relaxation<span class="_ _2"> </span>parameter.<span class="_ _5"> </span>Since<span class="_ _2"> </span><span class="ffb">W<span class="_ _5"> </span></span>in<span class="_ _2"> </span>(1)<span class="_ _2"> </span>is<span class="_ _2"> </span>also<span class="_ _5"> </span>symmetric<span class="_ _2"> </span>positive<span class="_ _5"> </span>def-</div><div class="t m0 x29 h6 y87 ff2 fs4 fc0 sc0 ls0 ws0">inite<span class="_ _7"> </span>as<span class="_ _5"> </span>proved<span class="_ _e"> </span>above,<span class="_ _7"> </span>we<span class="_ _7"> </span>can<span class="_ _7"> </span>exploit<span class="_ _7"> </span>the<span class="_ _5"> </span>Richardson<span class="_ _7"> </span>method<span class="_ _5"> </span>to<span class="_ _7"> </span>estimate</div><div class="t m0 x29 h6 y88 ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _2"> </span>transmitted<span class="_ _3"> </span>signal<span class="_ _2"> </span>vector</div><div class="t m0 x46 h7 y89 ff7 fs4 fc0 sc0 ls0 ws0">&#710;</div><div class="t m0 x46 h6 y88 ff8 fs4 fc0 sc0 ls0 ws0">s<span class="_ _2"> </span><span class="ff2">without<span class="_ _3"> </span>matrix<span class="_ _2"> </span>inversion<span class="_ _2"> </span>as<span class="_ _3"> </span>follows:</span></div><div class="t m0 x47 hc y8a ff8 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x3f hb y8b ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i<span class="ff5">+</span></span><span class="ls7">1)</span></div><div class="t m0 x48 h9 y8c ff5 fs4 fc0 sc0 ls0 ws0">=<span class="_ _5"> </span><span class="ff8">s</span></div><div class="t m0 x49 hb y8b ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i</span>)</div><div class="t m0 x4a h6 y8c ff5 fs4 fc0 sc0 ls0 ws0">+<span class="_ _e"> </span><span class="ff6">w<span class="_"> </span><span class="ff2">(</span></span></div><div class="t m0 x4b he y8d ffd fs4 fc0 sc0 ls0 ws0">&#58881;</div><div class="t m0 x4c h9 y8c ff8 fs4 fc0 sc0 ls0 ws0">y<span class="_ _7"> </span><span class="ff5">&#8722;<span class="_ _7"> </span></span>Ws</div><div class="t m0 x4d hb y8b ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i</span>)</div><div class="t m0 x4e h6 y8c ff2 fs4 fc0 sc0 ls0 ws0">)<span class="ff9">,<span class="_ _1a"> </span><span class="ff6">i<span class="_ _5"> </span><span class="ff5">=<span class="_ _2"> </span></span></span></span>0<span class="ff9">,<span class="_ _b"> </span></span>1<span class="ff9">,<span class="_ _b"> </span></span>2<span class="ff9">,<span class="_ _12"> </span><span class="ff7 ls9">...<span class="_ _1b"> </span></span></span>(4)</div><div class="t m0 x29 h6 y8e ff2 fs4 fc0 sc0 ls0 ws0">where<span class="_ _5"> </span>the<span class="_ _3"> </span>initial<span class="_ _5"> </span>solution<span class="_ _2"> </span><span class="ffb">s</span></div><div class="t m3 x4f hd y8f ff2 fs7 fc0 sc0 lsa ws0">(0)</div><div class="t m0 x50 h6 y90 ff2 fs4 fc0 sc0 ls0 ws0">can<span class="_ _2"> </span>be<span class="_ _2"> </span>usually<span class="_ _5"> </span>set<span class="_ _2"> </span>as<span class="_ _3"> </span>a<span class="_ _5"> </span>2<span class="ff6">K<span class="_ _e"> </span></span>&#215;<span class="_ _7"> </span>1<span class="_ _2"> </span>zero<span class="_ _5"> </span>vector</div><div class="t m0 x29 h6 y91 ff2 fs4 fc0 sc0 ls0 ws0">without<span class="_ _7"> </span>loss<span class="_ _5"> </span>of<span class="_ _5"> </span>generality<span class="_ _7"> </span>as<span class="_ _7"> </span>no<span class="_ _5"> </span><span class="ff6">a<span class="_ _7"> </span>priori<span class="_ _5"> </span></span>information<span class="_ _7"> </span>of<span class="_ _5"> </span>the<span class="_ _7"> </span><span class="ff4">&#64257;</span>nal<span class="_ _5"> </span>solution</div><div class="t m0 x29 h6 y92 ff2 fs4 fc0 sc0 ls0 ws0">is<span class="_ _2"> </span>available<span class="_ _2"> </span>[<span class="fc1">7</span>].<span class="_ _2"> </span>Such<span class="_ _2"> </span>an<span class="_ _2"> </span>initial<span class="_ _3"> </span>solution<span class="_ _2"> </span>will<span class="_ _2"> </span>not<span class="_ _2"> </span>affect<span class="_ _2"> </span>the<span class="_ _2"> </span>convergence</div><div class="t m0 x29 h6 y93 ff2 fs4 fc0 sc0 ls0 ws0">of<span class="_ _2"> </span>the<span class="_ _2"> </span>Richardson<span class="_ _2"> </span>method,<span class="_ _3"> </span>since<span class="_ _5"> </span>the<span class="_ _3"> </span>symmetric<span class="_ _5"> </span>positive<span class="_ _2"> </span>de<span class="ff4">&#64257;</span>nite<span class="_ _2"> </span>matrix</div><div class="t m0 x29 h6 y94 ffb fs4 fc0 sc0 ls0 ws0">W<span class="_ _5"> </span><span class="ff2">guarantees<span class="_ _2"> </span>the<span class="_ _5"> </span>convergence<span class="_ _2"> </span>of<span class="_ _5"> </span>the<span class="_ _2"> </span>Richardson<span class="_ _2"> </span>method<span class="_ _2"> </span>for<span class="_ _5"> </span>any<span class="_ _2"> </span>initial</span></div><div class="t m0 x29 h6 y95 ff2 fs4 fc0 sc0 ls0 ws0">solution<span class="_ _2"> </span>as<span class="_ _5"> </span>we<span class="_ _5"> </span>will<span class="_ _2"> </span>prove<span class="_ _5"> </span>in<span class="_ _2"> </span>the<span class="_ _5"> </span>following<span class="_ _2"> </span>Lemma<span class="_ _2"> </span>2.<span class="_ _5"> </span>Consequently,<span class="_ _2"> </span>the</div><div class="t m0 x29 h6 y96 ff4 fs4 fc0 sc0 ls0 ws0">&#64257;<span class="ff2">nal<span class="_ _b"> </span>accurac<span class="_ _9"></span>y<span class="_ _b"> </span>will<span class="_ _b"> </span>also<span class="_ _b"> </span>not<span class="_ _b"> </span>be<span class="_ _b"> </span>affected<span class="_ _a"> </span>by<span class="_ _b"> </span>the<span class="_ _b"> </span>initial<span class="_ _b"> </span>solution<span class="_ _b"> </span>if<span class="_ _b"> </span>the</span></div><div class="t m0 x29 h6 y97 ff2 fs4 fc0 sc0 ls0 ws0">number<span class="_ _a"> </span>of<span class="_ _1"> </span>iterations<span class="_ _a"> </span><span class="ff6">i<span class="_ _a"> </span></span>is<span class="_ _1"> </span>large<span class="_ _a"> </span>(e.g.<span class="_ _a"> </span><span class="ff6">i<span class="_ _e"> </span></span>=<span class="_ _7"> </span>5),<span class="_ _a"> </span>as<span class="_ _a"> </span>will<span class="_ _1"> </span>be<span class="_ _a"> </span>veri<span class="ff4">&#64257;</span>ed<span class="_ _a"> </span>later<span class="_ _1"> </span>in</div><div class="t m0 x29 h6 y98 ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _2"> </span>simulation<span class="_ _3"> </span>results.<span class="_ _2"> </span>Since<span class="_ _3"> </span>the<span class="_ _2"> </span>relaxation<span class="_ _2"> </span>parameter<span class="_ _3"> </span><span class="ff6">w<span class="_ _2"> </span></span>in<span class="_ _3"> </span>(4)<span class="_ _2"> </span>plays<span class="_ _2"> </span>an</div><div class="t m0 x29 h6 y99 ff2 fs4 fc0 sc0 ls0 ws0">important<span class="_ _1"> </span>role<span class="_ _1"> </span>in<span class="_ _1"> </span>convergence,<span class="_ _a"> </span>next<span class="_ _3"> </span>we<span class="_ _1"> </span>prove<span class="_ _1"> </span>the<span class="_ _1"> </span>convergence<span class="_ _1"> </span>of<span class="_ _1"> </span>the</div><div class="t m0 x29 h6 y9a ff2 fs4 fc0 sc0 ls0 ws0">Richardson<span class="_ _a"> </span>method<span class="_ _a"> </span>for<span class="_ _1"> </span>any<span class="_ _a"> </span>initial<span class="_ _a"> </span>solution<span class="_ _a"> </span>when<span class="_ _1"> </span>the<span class="_ _a"> </span>relaxation<span class="_ _a"> </span>para-</div><div class="t m0 x29 h6 y9b ff2 fs4 fc0 sc0 ls0 ws0">meter<span class="_ _3"> </span>is<span class="_ _2"> </span>appropriately<span class="_ _2"> </span>selected.</div><div class="t m0 x29 h10 y9c ff6 fs4 fc0 sc0 ls0 ws0">Lemma<span class="_ _3"> </span>2:<span class="_ _3"> </span>For<span class="_ _3"> </span>the<span class="_ _3"> </span>N-dimension<span class="_ _3"> </span>linear<span class="_ _3"> </span>equation<span class="_ _3"> </span><span class="ffb">Ax<span class="_ _e"> </span></span>=<span class="_ _e"> </span><span class="ffb">b</span>,<span class="_ _3"> </span>the<span class="_ _3"> </span>necessary</div><div class="t m0 x29 h10 y9d ff6 fs4 fc0 sc0 ls0 ws0">and<span class="_ _3"> </span>suf<span class="fff">&#64257;</span>cient<span class="_ _3"> </span>conditions<span class="_ _3"> </span>for<span class="_ _2"> </span>convergence<span class="_ _3"> </span>of<span class="_ _3"> </span>the<span class="_ _3"> </span>Richardson<span class="_ _3"> </span>method<span class="_ _3"> </span>is</div><div class="t m0 x29 h10 y9e ff6 fs4 fc0 sc0 ls0 ws0">that<span class="_ _3"> </span>the<span class="_ _3"> </span>relaxation<span class="_ _3"> </span>parameter<span class="_ _3"> </span>w<span class="_ _3"> </span>satis<span class="fff">&#64257;</span>es<span class="_ _3"> </span>0<span class="_ _e"> </span>&lt;<span class="_ _e"> </span>w<span class="_ _7"> </span>&lt;<span class="_ _e"> </span>2/<span class="ffc">&#955;</span></div><div class="t m3 x51 h11 y9f ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m0 x52 h10 ya0 ff6 fs4 fc0 sc0 ls0 ws0">,<span class="_ _3"> </span>where<span class="_ _3"> </span><span class="ffc">&#955;</span></div><div class="t m3 x53 h11 y9f ff6 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m0 x54 h10 ya0 ff6 fs4 fc0 sc0 ls0 ws0">is<span class="_ _3"> </span>the</div><div class="t m0 x29 h10 ya1 ff6 fs4 fc0 sc0 ls0 ws0">largest<span class="_ _2"> </span>eigenvalue<span class="_ _3"> </span>of<span class="_ _2"> </span>symmetric<span class="_ _3"> </span>positive<span class="_ _2"> </span>de<span class="fff">&#64257;</span>nite<span class="_ _2"> </span>matrix<span class="_ _3"> </span><span class="ffb">A</span>.</div><div class="t m0 x29 h6 ya2 ff6 fs4 fc0 sc0 ls0 ws0">Proof:<span class="_ _b"> </span><span class="ff2 lsb">We<span class="_ _4"> </span>de</span><span class="ff4">&#64257;<span class="ff2">ne<span class="_ _0"> </span><span class="ffb">D<span class="_ _7"> </span></span>=<span class="_ _e"> </span><span class="ffb">I</span></span></span></div><div class="t m3 x55 h11 ya3 ff6 fs7 fc0 sc0 ls0 ws0">N</div><div class="t m0 x56 h6 ya4 ffe fs4 fc0 sc0 ls0 ws0">&#8722;<span class="_ _e"> </span><span class="ff6">w<span class="ffb">A<span class="_ _0"> </span><span class="ff2">and<span class="_ _0"> </span></span>c<span class="_ _e"> </span><span class="ff2">=<span class="_ _e"> </span></span></span>w<span class="ffb">b<span class="ff2">,<span class="_ _0"> </span>where<span class="_ _0"> </span></span>D<span class="_ _b"> </span><span class="ff2">is<span class="_ _0"> </span>the<span class="_ _0"> </span>iteration</span></span></span></div><div class="t m0 x29 h6 ya5 ff2 fs4 fc0 sc0 ls0 ws0">matrix.<span class="_ _2"> </span>Then<span class="_ _3"> </span>the<span class="_ _2"> </span>Richardson<span class="_ _3"> </span>iteration<span class="_ _2"> </span>(3)<span class="_ _2"> </span>can<span class="_ _3"> </span>be<span class="_ _2"> </span>rewritten<span class="_ _2"> </span>as</div><div class="t m0 x48 hc ya6 ff8 fs4 fc0 sc0 ls0 ws0">x</div><div class="t m0 x41 hb ya7 ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i<span class="ff5">+</span></span><span class="ls7">1)</span></div><div class="t m0 x4a h9 ya8 ff5 fs4 fc0 sc0 ls0 ws0">=<span class="_ _5"> </span><span class="ff8">Dx</span></div><div class="t m0 x57 hb ya7 ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i</span>)</div><div class="t m0 x58 h6 ya8 ff5 fs4 fc0 sc0 ls0 ws0">+<span class="_ _e"> </span><span class="ff8">c<span class="_ _11"></span><span class="ff9">,<span class="_ _a"> </span><span class="ff6">i<span class="_ _2"> </span></span></span></span>=<span class="_ _2"> </span><span class="ff2">0<span class="ff9">,<span class="_ _b"> </span></span>1<span class="ff9">,<span class="_ _b"> </span></span>2<span class="ff9">,<span class="_ _12"> </span><span class="ff7 ls8">...<span class="_ _1c"> </span></span></span>(5)</span></div><div class="t m0 x29 h6 ya9 ff2 fs4 fc0 sc0 ls0 ws0">We<span class="_ _1d"> </span>call<span class="_ _1d"> </span>the<span class="_ _1e"> </span>iteration<span class="_ _1d"> </span>procedure<span class="_ _1d"> </span>convergent<span class="_ _1d"> </span>if<span class="_ _1f"> </span><span class="ff9">lim</span></div><div class="t m0 x59 h12 yaa ff6 fs5 fc0 sc0 ls0 ws0">i<span class="ff5">&#58882;<span class="ff12">1</span></span></div><div class="t m0 x5a hc yab ff8 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x5b hb yac ff2 fs5 fc0 sc0 ls0 ws0">(<span class="ff6">i</span>)</div><div class="t m0 x5c h9 yab ff5 fs4 fc0 sc0 ls0 ws0">=</div><div class="t m0 x5d h7 yad ff7 fs4 fc0 sc0 ls0 ws0">&#710;</div><div class="t m0 x5d h6 yab ff8 fs4 fc0 sc0 ls0 ws0">s<span class="_ _1d"> </span><span class="ff2">and</span></div><div class="t m0 x29 h7 yae ff7 fs4 fc0 sc0 ls0 ws0">&#710;</div><div class="t m0 x29 h9 yaf ff8 fs4 fc0 sc0 ls0 ws0">s<span class="_ _5"> </span><span class="ff5">=<span class="_ _2"> </span></span>B</div><div class="t m0 x5e hc yae ff8 fs4 fc0 sc0 ls0 ws0">&#710;</div><div class="t m0 x5e h6 yaf ff8 fs4 fc0 sc0 ls0 ws0">s<span class="_ _7"> </span><span class="ff5">+<span class="_ _7"> </span></span>c<span class="_ _2"> </span><span class="ff2">for<span class="_ _3"> </span>any<span class="_ _2"> </span>initial<span class="_ _3"> </span>solution<span class="_ _2"> </span><span class="ffb">s</span></span></div><div class="t m3 x4d hd yb0 ff2 fs7 fc0 sc0 lsa ws0">(0)</div><div class="t m0 x5f h6 yb1 ff2 fs4 fc0 sc0 ls0 ws0">.</div><div class="t m0 x29 h6 yb2 ff2 fs4 fc0 sc0 ls0 ws0">The<span class="_ _1"> </span>spectral<span class="_ _3"> </span>radius<span class="_ _1"> </span>of<span class="_ _1"> </span>iteration<span class="_ _1"> </span>matrix<span class="_ _1"> </span><span class="ff8">D<span class="_ _5"> </span><span class="ff13">[<span class="_ _2"> </span><span class="ff14">R</span></span></span></div><div class="t m0 x60 h12 yb3 ff6 fs5 fc0 sc0 ls0 ws0">N<span class="_ _11"></span><span class="ff5">&#215;</span>N</div><div class="t m0 x61 h6 yb4 ff2 fs4 fc0 sc0 ls0 ws0">is<span class="_ _3"> </span>the<span class="_ _1"> </span>non-negative</div><div class="t m0 x29 h6 yb5 ff2 fs4 fc0 sc0 ls0 ws0">number</div><div class="t m2 x37 ha yb6 ffa fs6 fc0 sc0 ls0 ws0">r</div><div class="t m0 x62 h6 yb6 ff2 fs4 fc0 sc0 ls0 ws0">(<span class="ff8">D</span>)<span class="_ _5"> </span><span class="ff5">=<span class="_ _d"> </span><span class="ff9">max</span></span></div><div class="t m0 x63 hb yb7 ff2 fs5 fc0 sc0 ls0 ws0">1<span class="ff5">&#8804;<span class="ff6">n</span>&#8804;<span class="ff6">N</span></span></div><div class="t m2 x2a ha yb6 ffa fs6 fc0 sc0 ls0 ws0">m</div><div class="t m0 x55 h13 yb8 ff6 fs5 fc0 sc0 ls0 ws0">n</div><div class="t m0 x64 h6 yb6 ff2 fs4 fc0 sc0 ls0 ws0">(<span class="ff8">D</span>)</div><div class="t m0 x65 he yb9 ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x65 he yba ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x4c he yb9 ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x4c he yba ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x50 h6 ybb ff2 fs4 fc0 sc0 ls0 ws0">,<span class="_ _2"> </span>where<span class="_ _3"> </span><span class="ffc">&#956;</span></div><div class="t m3 x45 h11 ybc ff6 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m0 x66 h6 yb6 ff2 fs4 fc0 sc0 ls0 ws0">(<span class="ffb">D</span>)<span class="_ _2"> </span>denotes<span class="_ _3"> </span>the<span class="_ _2"> </span><span class="ff6">n</span>th<span class="_ _3"> </span>eigenvalue</div><div class="t m0 x29 h6 ybd ff2 fs4 fc0 sc0 ls0 ws0">of<span class="_ _3"> </span><span class="ffb">D<span class="_ _11"></span></span>,<span class="_ _3"> </span>and<span class="_ _1"> </span>the<span class="_ _3"> </span>necessary<span class="_ _1"> </span>and<span class="_ _1"> </span>suf<span class="ff4">&#64257;</span>cient<span class="_ _1"> </span>conditions<span class="_ _3"> </span>for<span class="_ _1"> </span>the<span class="_ _1"> </span>convergence</div><div class="t m0 x29 h6 ybe ff2 fs4 fc0 sc0 ls0 ws0">of<span class="_ _2"> </span>(5)<span class="_ _3"> </span>is<span class="_ _2"> </span>that<span class="_ _2"> </span>the<span class="_ _3"> </span>spectral<span class="_ _2"> </span>radius<span class="_ _3"> </span>should<span class="_ _2"> </span>satisfy<span class="_ _2"> </span>[7,<span class="_ _3"> </span>Theorem<span class="_ _2"> </span>7.2.2]</div><div class="t m2 x67 ha ybf ffa fs6 fc0 sc0 ls0 ws0">r</div><div class="t m0 x68 h6 ybf ff2 fs4 fc0 sc0 ls0 ws0">(<span class="ff8">D</span>)<span class="_ _2"> </span><span class="ff5">=<span class="_ _4"> </span><span class="ff9">max</span></span></div><div class="t m0 x69 hb yc0 ff2 fs5 fc0 sc0 ls0 ws0">1<span class="ff5">&#8804;<span class="ff6">n</span>&#8804;<span class="ff6">N</span></span></div><div class="t m2 x2e ha ybf ffa fs6 fc0 sc0 ls0 ws0">m</div><div class="t m0 x6a h13 yc1 ff6 fs5 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2f h6 ybf ff2 fs4 fc0 sc0 ls0 ws0">(<span class="ff8">D</span>)</div><div class="t m0 x4d he yc2 ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x4d he yc3 ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x6b he yc2 ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x6b he yc3 ffd fs4 fc0 sc0 ls0 ws0">&#58883;</div><div class="t m0 x6c h6 ybf ff10 fs4 fc0 sc0 ls0 ws0">,<span class="_ _5"> </span><span class="ff2">1<span class="_ _20"> </span>(6)</span></div><div class="t m0 x29 h6 yc4 ff2 fs4 fc0 sc0 ls0 ws0">Without<span class="_ _3"> </span>loss<span class="_ _3"> </span>of<span class="_ _3"> </span>generality,<span class="_ _3"> </span>we<span class="_ _3"> </span>use</div><div class="t m2 x6d ha y51 ffa fs6 fc0 sc0 ls0 ws0">l</div><div class="t m0 x6e hb yc5 ff2 fs5 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6a h9 y51 ff5 fs4 fc0 sc0 ls0 ws0">&#8805;</div><div class="t m2 x6f ha y51 ffa fs6 fc0 sc0 ls0 ws0">l</div><div class="t m0 x70 hb yc5 ff2 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6c h9 y51 ff5 fs4 fc0 sc0 lsc ws0">&#8805;<span class="_ _21"> </span>&#183;&#183;&#183;<span class="_ _21"> </span>&#8805;</div><div class="t m2 x51 ha y51 ffa fs6 fc0 sc0 ls0 ws0">l</div><div class="t m0 x71 h13 yc5 ff6 fs5 fc0 sc0 ls0 ws0">N</div><div class="t m0 x72 h6 y51 ff10 fs4 fc0 sc0 ls0 ws0">.<span class="_ _5"> </span><span class="ff2">0<span class="_ _1"> </span>to<span class="_ _3"> </span>denote</span></div><div class="t m0 x29 h6 y52 ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _2"> </span><span class="ff6">N<span class="_ _3"> </span></span>eigenvalues<span class="_ _3"> </span>of<span class="_ _2"> </span>symmetric<span class="_ _2"> </span>positive<span class="_ _3"> </span>de<span class="ff4">&#64257;</span>nite<span class="_ _2"> </span>matrix<span class="_ _3"> </span><span class="ffb">A</span>,<span class="_ _2"> </span>where<span class="_ _3"> </span><span class="ffc">&#955;</span></div><div class="t m3 x73 hd yc6 ff2 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m0 x74 h6 y54 ff2 fs4 fc0 sc0 ls0 ws0">is</div><div class="t m0 x29 h6 y55 ff2 fs4 fc0 sc0 ls0 ws0">the<span class="_ _3"> </span>largest<span class="_ _1"> </span>one.<span class="_ _3"> </span>Since<span class="_ _1"> </span><span class="ffb">D<span class="_ _e"> </span></span>=<span class="_ _e"> </span><span class="ffb">I<span class="_ _7"> </span><span class="ffe">&#8722;<span class="_ _e"> </span><span class="ff6">w</span></span>A</span>,<span class="_ _1"> </span>we<span class="_ _3"> </span>have<span class="_ _3"> </span><span class="ffc">&#956;</span></div><div class="t m3 x75 h11 y58 ff6 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m0 x76 h6 y57 ff2 fs4 fc0 sc0 ls0 ws0">(<span class="ffb">D</span><span class="ls3">)=1</span><span class="ffe">&#8722;<span class="_ _e"> </span><span class="ff6">w<span class="ffc">&#955;</span></span></span></div><div class="t m3 x5b h11 y58 ff6 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m0 x77 h6 y57 ff2 fs4 fc0 sc0 ls0 ws0">,<span class="_ _3"> </span>where<span class="_ _1"> </span><span class="ffc">&#955;</span></div><div class="t m3 x78 h11 y58 ff6 fs7 fc0 sc0 ls0 ws0">n</div><div class="t m0 x29 h6 y5b ff2 fs4 fc0 sc0 ls0 ws0">is<span class="_ _3"> </span>the<span class="_ _2"> </span><span class="ff6">n</span>th<span class="_ _3"> </span>eigenvalue<span class="_ _3"> </span>of<span class="_ _3"> </span><span class="ffb">A</span>,<span class="_ _2"> </span>which<span class="_ _3"> </span>can<span class="_ _3"> </span>be<span class="_ _3"> </span>substituted<span class="_ _2"> </span>into<span class="_ _3"> </span>(6),<span class="_ _3"> </span>and<span class="_ _3"> </span>then</div><div class="t m0 x29 h6 y5c ff2 fs4 fc0 sc0 ls0 ws0">we<span class="_ _2"> </span>have<span class="_ _2"> </span>0<span class="_ _e"> </span>&lt;<span class="_ _7"> </span><span class="ff6">w<span class="_ _e"> </span></span><span class="ls2">&lt;2<span class="_ _10"></span>/<span class="_ _10"></span><span class="ffc ls0">&#955;</span></span></div><div class="t m3 x79 hd yc7 ff2 fs7 fc0 sc0 ls0 ws0">1</div><div class="t m0 x7a h6 y5e ff2 fs4 fc0 sc0 ls0 ws0">.<span class="_ _22"> </span><span class="ff11">&#9633;</span></div><div class="t m0 x1 h14 yc8 ff15 fs8 fc0 sc0 ls0 ws0">ELECTRONICS<span class="_ _a"> </span>LETTERS<span class="_ _23"> </span>28th<span class="_ _a"> </span>August<span class="_ _b"> </span>2014<span class="_ _23"> </span>Vol.<span class="_ _a"> </span>50<span class="_ _23"> </span>No.<span class="_ _b"> </span>18<span class="_ _23"> </span>pp.<span class="_ _a"> </span>1326<span class="ff16">&#8211;</span>1328</div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div> </body> </html>
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