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<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/6255768247503a0a93dc714a/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Performance<span class="_ _0"> </span>analysis<span class="_ _0"> </span>for<span class="_ _0"> </span>MIMO<span class="_ _0"> </span>systems<span class="_ _0"> </span>using<span class="_ _0"> </span>zero</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls1 ws0">forcing<span class="_ _0"> </span>detector<span class="_ _0"> </span>over<span class="_ _0"> </span>fading<span class="_ _0"> </span>channels</div><div class="t m0 x1 h3 y3 ff2 fs1 fc0 sc0 ls2 ws0">R.<span class="_ _1"> </span>Xu<span class="_ _1"> </span>and<span class="_ _1"> </span>F.C.M.<span class="_ _1"> </span>Lau</div><div class="t m0 x2 h4 y4 ff3 fs1 fc0 sc0 ls3 ws0">Abstract:<span class="_ _2"> </span><span class="ff4 ls4">In<span class="_ _3"> </span>the<span class="_ _3"> </span>paper,<span class="_ _3"> </span>the<span class="_ _3"> </span>performance<span class="_ _3"> </span>of<span class="_ _3"> </span>multiple-input-multiple-output<span class="_ _3"> </span>(MIMO)<span class="_ _3"> </span>systems<span class="_ _3"> </span>using</span></div><div class="t m0 x2 h4 y5 ff4 fs1 fc0 sc0 ls5 ws0">a<span class="_ _1"> </span>zero<span class="_ _1"> </span>forcing<span class="_ _1"> </span>detector<span class="_ _1"> </span>is<span class="_ _1"> </span>studied<span class="_ _1"> </span>over<span class="_ _1"> </span>Rayleigh<span class="_ _1"> </span>and<span class="_ _1"> </span>Rice<span class="_ _1"> </span>fading<span class="_ _1"> </span>channels.<span class="_ _1"> </span>Approximation<span class="_ _1"> </span>to<span class="_ _1"> </span>the</div><div class="t m0 x2 h4 y6 ff4 fs1 fc0 sc0 ls4 ws0">distribution<span class="_ _1"> </span>of<span class="_ _4"> </span>non-central<span class="_ _1"> </span>complex<span class="_ _4"> </span>Wishart<span class="_ _4"> </span>dist<span class="ls6">ribution<span class="_ _1"> </span>is<span class="_ _4"> </span>used<span class="_ _1"> </span>to<span class="_ _4"> </span>derive<span class="_ _1"> </span>the<span class="_ _4"> </span>distribution<span class="_ _1"> </span>of<span class="_ _4"> </span>the</span></div><div class="t m0 x2 h4 y7 ff4 fs1 fc0 sc0 ls6 ws0">post-detection<span class="_ _5"> </span>signal-to-noise<span class="_ _5"> </span>(SNR)<span class="_ _5"> </span>ratio.<span class="_ _5"> </span>A<span class="_ _5"> </span>clos<span class="ls4">ed-form<span class="_ _5"> </span>expression<span class="_ _5"> </span>of<span class="_ _5"> </span>bit<span class="_ _5"> </span>error<span class="_ _5"> </span>rate<span class="_ _5"> </span>is<span class="_ _5"> </span>obtained<span class="_ _5"> </span>in</span></div><div class="t m0 x2 h4 y8 ff4 fs1 fc0 sc0 ls6 ws0">terms<span class="_ _5"> </span>of<span class="_ _5"> </span>t<span class="_"> </span>he<span class="_ _5"> </span>hypergeometric<span class="_ _5"> </span>and<span class="_ _6"> </span>Gamma<span class="_ _6"> </span>functions.<span class="_ _5"> </span>The<span class="_ _6"> </span>performance<span class="_ _5"> </span>of<span class="_ _5"> </span>MIMO<span class="_ _6"> </span>s<span class="_ _7"></span>ystems<span class="_ _6"> </span>under<span class="_ _5"> </span>the</div><div class="t m0 x2 h4 y9 ff4 fs1 fc0 sc0 ls6 ws0">scenario<span class="_ _6"> </span>that<span class="_ _5"> </span>the<span class="_ _5"> </span>individual<span class="_ _5"> </span>sub-channels<span class="_ _6"> </span>are<span class="_ _5"> </span>corre<span class="ls7">lated<span class="_ _5"> </span>is<span class="_ _5"> </span>also<span class="_ _5"> </span>investigated.<span class="_ _5"> </span>It<span class="_ _5"> </span>is<span class="_ _5"> </span>shown<span class="_ _5"> </span>that,<span class="_ _5"> </span>for<span class="_ _5"> </span>the</span></div><div class="t m0 x2 h4 ya ff4 fs1 fc0 sc0 ls3 ws0">exponential<span class="_ _6"> </span>correlation<span class="_ _5"> </span>channel<span class="_ _5"> </span>model,<span class="_ _5"> </span>an<span class="_ _5"> </span>increas<span class="ls2">e<span class="_ _5"> </span>in<span class="_ _5"> </span>channel<span class="_ _5"> </span>correlation<span class="_ _5"> </span>between<span class="_ _5"> </span>the<span class="_ _5"> </span>transmission</span></div><div class="t m0 x2 h4 yb ff4 fs1 fc0 sc0 ls8 ws0">paths<span class="_ _4"> </span>results<span class="_ _4"> </span>a<span class="_ _4"> </span>decrease<span class="_ _4"> </span>in<span class="_ _4"> </span>SNR<span class="_ _4"> </span>at<span class="_ _4"> </span>the<span class="_ _8"> </span>receiver.<span class="_ _4"> </span>It<span class="_ _4"> </span>is<span class="_ _4"> </span>concluded<span class="_ _4"> </span>that<span class="_ _4"> </span>with<span class="_ _4"> </span>the<span class="_ _4"> </span>same<span class="_ _4"> </span>SNR<span class="_ _4"> </span>at<span class="_ _8"> </span>the</div><div class="t m0 x2 h4 yc ff4 fs1 fc0 sc0 ls9 ws0">receiver,<span class="_ _6"> </span>the<span class="_ _5"> </span>performance<span class="_ _5"> </span>of<span class="_ _6"> </span>MIMO<span class="_ _5"> </span>systems<span class="_ _5"> </span>over<span class="_ _6"> </span><span class="lsa">a<span class="_ _5"> </span>Rice<span class="_ _6"> </span>f<span class="_ _7"></span>ading<span class="_ _6"> </span>channel<span class="_ _5"> </span>degrades<span class="_ _5"> </span>compared<span class="_ _5"> </span>to<span class="_ _6"> </span>that</span></div><div class="t m0 x2 h4 yd ff4 fs1 fc0 sc0 ls6 ws0">over<span class="_ _1"> </span>an<span class="_ _1"> </span>independent<span class="_ _1"> </span>and<span class="_ _1"> </span>identical<span class="_ _1"> </span>distri<span class="lsb">buted<span class="_ _1"> </span>(i.i.d.)<span class="_ _1"> </span>Raylei<span class="lsc">gh<span class="_ _1"> </span>fading<span class="_ _1"> </span>channel.</span></span></div><div class="t m0 x1 h5 ye ff1 fs1 fc0 sc0 lsd ws0">1<span class="_ _9"> </span>Introduction</div><div class="t m0 x1 h4 yf ff4 fs1 fc0 sc0 ls2 ws0">Multiple-input-multiple-out<span class="ls3">put<span class="_ _2"> </span>(<span class="_"> </span>MIMO)<span class="_ _2"> </span>systems<span class="_ _a"> </span>have<span class="_ _a"> </span>at-</span></div><div class="t m0 x1 h4 y10 ff4 fs1 fc0 sc0 ls3 ws0">tracted<span class="_ _6"> </span>a<span class="_ _5"> </span>lot<span class="_ _6"> </span>of<span class="_ _5"> </span>research<span class="_ _6"> </span>a<span class="_ _7"></span>ttention<span class="_ _6"> </span>in<span class="_ _5"> </span>the<span class="_ _6"> </span>past<span class="_ _5"> </span>decade<span class="_ _6"> </span>because</div><div class="t m0 x1 h4 y11 ff4 fs1 fc0 sc0 lse ws0">they<span class="_ _b"> </span>possess<span class="_ _b"> </span>the<span class="_ _b"> </span>ability<span class="_ _b"> </span>to<span class="_ _b"> </span>provide<span class="_ _b"> </span>high<span class="_ _b"> </span>speed<span class="_ _b"> </span>wireless</div><div class="t m0 x1 h4 y12 ff4 fs1 fc0 sc0 lse ws0">communications<span class="_ _4"> </span>over<span class="_ _4"> </span>a<span class="_ _4"> </span>multipath<span class="_ _4"> </span>environment<span class="_ _4"> </span><span class="fc1 lsf">[1,<span class="_ _4"> </span>2]</span><span class="ls10">.T<span class="_ _c"></span>h<span class="_ _c"></span>e</span></div><div class="t m0 x1 h4 y13 ff4 fs1 fc0 sc0 ls5 ws0">performance<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _4"> </span>MIMO<span class="_ _8"> </span>systems<span class="_ _4"> </span>is<span class="_ _4"> </span>very<span class="_ _8"> </span>much<span class="_ _4"> </span>affected</div><div class="t m0 x1 h4 y14 ff4 fs1 fc0 sc0 lsc ws0">by<span class="_ _5"> </span>the<span class="_ _3"> </span>characteristics<span class="_ _5"> </span>of<span class="_ _5"> </span>the<span class="_ _3"> </span>radio<span class="_ _5"> </span>propagation<span class="_ _3"> </span>channel.<span class="_ _5"> </span>For</div><div class="t m0 x1 h4 y15 ff4 fs1 fc0 sc0 ls6 ws0">example,<span class="_ _a"> </span>the<span class="_ _a"> </span>correlation<span class="_ _a"> </span>between<span class="_ _a"> </span>the<span class="_ _a"> </span>sub-channels<span class="_ _a"> </span>in<span class="_ _a"> </span>the</div><div class="t m0 x1 h4 y16 ff4 fs1 fc0 sc0 ls6 ws0">propagation<span class="_ _b"> </span>channel<span class="_ _b"> </span>limits<span class="_ _b"> </span>the<span class="_ _b"> </span>capacity<span class="_ _b"> </span>of<span class="_ _b"> </span>the<span class="_ _b"> </span>MIMO</div><div class="t m0 x1 h4 y17 ff4 fs1 fc0 sc0 ls11 ws0">architecture.<span class="_ _1"> </span>In<span class="_ _1"> </span><span class="fc1 ls12">[3<span class="_"> </span>]</span><span class="ls6">,<span class="_ _3"> </span>the<span class="_ _1"> </span>effect<span class="_ _3"> </span>of<span class="_ _1"> </span>a<span class="_ _3"> </span>fading<span class="_ _3"> </span>correlation<span class="_ _1"> </span>on<span class="_ _3"> </span>the</span></div><div class="t m0 x1 h4 y18 ff4 fs1 fc0 sc0 ls13 ws0">capacity<span class="_ _5"> </span>of<span class="_ _3"> </span>the<span class="_ _5"> </span>MIMO<span class="_ _5"> </span>architecture<span class="_ _3"> </span>has<span class="_ _5"> </span>been<span class="_ _5"> </span>investigated<span class="_ _5"> </span>in</div><div class="t m0 x1 h4 y19 ff4 fs1 fc0 sc0 lse ws0">detail<span class="_ _a"> </span>based<span class="_ _a"> </span>on<span class="_ _a"> </span>an<span class="_ _a"> </span>exponential<span class="_ _a"> </span>correlation<span class="_ _8"> </span>matrix.<span class="_ _a"> </span>It<span class="_ _a"> </span>has</div><div class="t m0 x1 h4 y1a ff4 fs1 fc0 sc0 lsc ws0">been<span class="_ _8"> </span>shown<span class="_ _a"> </span>that<span class="_ _4"> </span>a<span class="_ _7"></span>n<span class="_ _8"> </span>increase<span class="_ _a"> </span>i<span class="ls3">n<span class="_ _8"> </span>the<span class="_ _8"> </span>correlation<span class="_ _8"> </span>will<span class="_ _a"> </span>result</span></div><div class="t m0 x1 h4 y1b ff4 fs1 fc0 sc0 ls14 ws0">in<span class="_ _b"> </span>a<span class="_ _d"> </span>decrease<span class="_ _d"> </span>in<span class="_ _b"> </span>capacity.<span class="_ _d"> </span>In<span class="_ _b"> </span>the<span class="_ _b"> </span>worst<span class="_ _d"> </span>case<span class="_ _b"> </span>when<span class="_ _d"> </span>the</div><div class="t m0 x1 h4 y1c ff4 fs1 fc0 sc0 ls6 ws0">correlation<span class="_ _3"> </span>coefficient<span class="_ _3"> </span>equals<span class="_ _3"> </span>to<span class="_ _1"> </span>unity,<span class="_ _3"> </span>no<span class="_ _3"> </span>advantage<span class="_ _3"> </span>will<span class="_ _1"> </span>be</div><div class="t m0 x1 h4 y1d ff4 fs1 fc0 sc0 ls6 ws0">provided<span class="_ _0"> </span>by<span class="_ _0"> </span>the<span class="_ _0"> </span>MIMO<span class="_ _0"> </span>architecture.<span class="_ _0"> </span>The<span class="_ _0"> </span>exponential</div><div class="t m0 x1 h4 y1e ff4 fs1 fc0 sc0 ls9 ws0">correlation<span class="_ _d"> </span>matrix,<span class="_ _b"> </span>which<span class="_ _d"> </span>has<span class="_ _b"> </span>been<span class="_ _d"> </span>successfully<span class="_ _d"> </span>used<span class="_ _b"> </span>in</div><div class="t m0 x1 h4 y1f ff4 fs1 fc0 sc0 ls5 ws0">many<span class="_ _1"> </span>communication<span class="_ _1"> </span>problem<span class="ls15">s,<span class="_ _4"> </span>will<span class="_ _1"> </span>be<span class="_ _1"> </span>used<span class="_ _1"> </span>to<span class="_ _1"> </span>model<span class="_ _4"> </span>the</span></div><div class="t m0 x1 h4 y20 ff4 fs1 fc0 sc0 ls3 ws0">correlation<span class="_ _e"> </span>between<span class="_ _e"> </span>sub-<span class="ls6">channels<span class="_ _e"> </span>in<span class="_ _e"> </span>this<span class="_ _e"> </span>paper.<span class="_ _e"> </span>An</span></div><div class="t m0 x1 h4 y21 ff4 fs1 fc0 sc0 ls3 ws0">analytical<span class="_ _1"> </span>formula<span class="_ _1"> </span>will<span class="_ _1"> </span>then<span class="_ _3"> </span>be<span class="_ _1"> </span>derived<span class="_ _1"> </span>for<span class="_ _1"> </span>the<span class="_ _1"> </span>degradation</div><div class="t m0 x1 h4 y22 ff4 fs1 fc0 sc0 ls2 ws0">of<span class="_ _2"> </span>signal-to-noise<span class="_ _d"> </span>(SNR)<span class="_ _2"> </span>ratio<span class="_ _2"> </span>based<span class="_ _d"> </span>on<span class="_ _2"> </span>the<span class="_ _2"> </span>exponential</div><div class="t m0 x1 h4 y23 ff4 fs1 fc0 sc0 lse ws0">correlation<span class="_ _1"> </span>coefficients.<span class="_ _4"> </span>Finally,<span class="_ _1"> </span>we<span class="_ _4"> </span>will<span class="_ _1"> </span>show<span class="_ _4"> </span>that,<span class="_ _1"> </span>for<span class="_ _4"> </span>this</div><div class="t m0 x1 h4 y24 ff4 fs1 fc0 sc0 lse ws0">model,<span class="_ _3"> </span>an<span class="_ _3"> </span>increase<span class="_ _3"> </span>in<span class="_ _3"> </span>correlation<span class="_ _5"> </span>is<span class="_ _3"> </span>equivalent<span class="_ _3"> </span>to<span class="_ _3"> </span>a<span class="_ _3"> </span>decrease</div><div class="t m0 x1 h4 y25 ff4 fs1 fc0 sc0 lse ws0">in<span class="_ _1"> </span>the<span class="_ _1"> </span>SNR<span class="_ _1"> </span>under<span class="_ _1"> </span>some<span class="_ _1"> </span>realistic<span class="_ _1"> </span>conditions.</div><div class="t m0 x3 h4 y26 ff4 fs1 fc0 sc0 lse ws0">Although<span class="_ _d"> </span>Rayleigh<span class="_ _d"> </span>and<span class="_ _2"> </span>Rice<span class="_ _d"> </span>fading<span class="_ _d"> </span>channels<span class="_ _d"> </span>are<span class="_ _d"> </span>the</div><div class="t m0 x1 h4 y27 ff4 fs1 fc0 sc0 lsc ws0">most<span class="_ _a"> </span>popular<span class="_ _2"> </span>models<span class="_ _a"> </span>for<span class="_ _2"> </span>wireless<span class="_ _a"> </span>channels,<span class="_ _a"> </span>much<span class="_ _2"> </span>of<span class="_ _a"> </span>the</div><div class="t m0 x1 h4 y28 ff4 fs1 fc0 sc0 ls16 ws0">research<span class="_ _8"> </span>on<span class="_ _4"> </span>the<span class="_ _8"> </span>performance<span class="_ _8"> </span>of<span class="_ _8"> </span>MIMO<span class="_ _8"> </span>systems<span class="_ _4"> </span>has<span class="_ _8"> </span>been</div><div class="t m0 x1 h4 y29 ff4 fs1 fc0 sc0 ls17 ws0">f<span class="_ _f"></span>o<span class="_ _f"></span>c<span class="_ _f"></span>u<span class="_ _f"></span>s<span class="_ _f"></span>e<span class="_ _f"></span>do<span class="_ _f"></span>nR<span class="_ _f"></span>a<span class="_ _f"></span>y<span class="_ _10"></span>l<span class="_ _f"></span>e<span class="_ _10"></span>i<span class="_ _f"></span>g<span class="_ _f"></span>hf<span class="_ _f"></span>a<span class="_ _f"></span>d<span class="_ _f"></span>i<span class="_ _f"></span>n<span class="_ _f"></span>gc<span class="_ _f"></span>h<span class="_ _10"></span>a<span class="_ _f"></span>n<span class="_ _f"></span>n<span class="_ _f"></span>e<span class="_ _f"></span>l<span class="_ _f"></span>s<span class="fc1 ls18">[4–6]</span><span class="ls19">.O<span class="_ _10"></span>n<span class="_ _f"></span>eo<span class="_ _f"></span>ft<span class="_ _f"></span>h<span class="_ _f"></span>ef<span class="_ _10"></span>e<span class="_ _f"></span>w</span></div><div class="t m0 x1 h4 y2a ff4 fs1 fc0 sc0 ls6 ws0">MIMO<span class="_ _6"> </span>systems<span class="_ _5"> </span>that<span class="_ _6"> </span>has<span class="_ _6"> </span>been<span class="_ _6"> </span>evaluated<span class="_ _5"> </span>under<span class="_ _6"> </span>a<span class="_ _6"> </span>Rice<span class="_ _5"> </span>fading</div><div class="t m0 x1 h4 y2b ff4 fs1 fc0 sc0 ls8 ws0">channel<span class="_ _4"> </span>is<span class="_ _1"> </span>the<span class="_ _4"> </span>Alamouti<span class="_ _4"> </span>scheme<span class="_ _1"> </span><span class="fc1 lsf">[7,<span class="_ _4"> </span>8]</span><span class="ls1a">.I<span class="_ _10"></span>n<span class="fc1 lsf">[9]</span><span class="ls1b">,<span class="_ _4"> </span>the<span class="_ _4"> </span>bit<span class="_ _1"> </span>error</span></span></div><div class="t m0 x1 h4 y2c ff4 fs1 fc0 sc0 ls5 ws0">rate<span class="_ _b"> </span>(BER)<span class="_ _d"> </span>performance<span class="_ _b"> </span>of<span class="_ _d"> </span>MIMO<span class="_ _b"> </span>systems<span class="_ _b"> </span>using<span class="_ _d"> </span>zero</div><div class="t m0 x1 h4 y2d ff4 fs1 fc0 sc0 ls4 ws0">forcing<span class="_ _a"> </span>(ZF)<span class="_ _a"> </span>detectors<span class="_ _a"> </span>in<span class="_ _a"> </span>both<span class="_ _a"> </span>Rayleigh<span class="_ _a"> </span>and<span class="_ _a"> </span>Rice<span class="_ _2"> </span>fading</div><div class="t m0 x1 h4 y2e ff4 fs1 fc0 sc0 lse ws0">channels<span class="_ _5"> </span>was<span class="_ _5"> </span>evaluated<span class="_ _5"> </span>by<span class="_ _5"> </span>simul<span class="lsc">ations.<span class="_ _5"> </span>Results<span class="_ _5"> </span>showed<span class="_ _5"> </span>that</span></div><div class="t m0 x4 h4 y2f ff4 fs1 fc0 sc0 lse ws0">provided<span class="_ _8"> </span>that<span class="_ _a"> </span>the<span class="_ _4"> </span>signal<span class="_ _a"> </span>power<span class="_ _4"> </span>from<span class="_ _a"> </span>the<span class="_ _8"> </span>non-line-of-sight</div><div class="t m0 x4 h4 y30 ff4 fs1 fc0 sc0 ls2 ws0">components<span class="_ _b"> </span>remains<span class="_ _d"> </span>the<span class="_ _b"> </span>same,<span class="_ _b"> </span>the<span class="_ _b"> </span>performance<span class="_ _d"> </span>of<span class="_ _b"> </span>the</div><div class="t m0 x4 h4 y31 ff4 fs1 fc0 sc0 ls3 ws0">MIMO<span class="_ _2"> </span>system<span class="_ _2"> </span>in<span class="_ _2"> </span>Rayleigh<span class="_ _2"> </span>and<span class="_ _d"> </span>Rice<span class="_ _2"> </span>fading<span class="_ _2"> </span>channels<span class="_ _2"> </span>is</div><div class="t m0 x4 h4 y32 ff4 fs1 fc0 sc0 ls2 ws0">almost<span class="_ _8"> </span>the<span class="_ _4"> </span>same.<span class="_ _a"> </span>In<span class="_ _4"> </span>this<span class="_ _8"> </span>paper,<span class="_ _8"> </span>we<span class="_ _8"> </span>investigate<span class="_ _8"> </span>the<span class="_ _8"> </span>perfor-</div><div class="t m0 x4 h4 y33 ff4 fs1 fc0 sc0 ls16 ws0">mance<span class="_ _1"> </span>of<span class="_ _4"> </span>the<span class="_ _1"> </span>MIMO<span class="_ _1"> </span>systems<span class="_ _4"> </span><span class="fc1 ls6">[10]<span class="_ _1"> </span><span class="fc0">over<span class="_ _1"> </span>Rayleigh<span class="_ _4"> </span>and<span class="_ _1"> </span>Rice</span></span></div><div class="t m0 x4 h4 y34 ff4 fs1 fc0 sc0 ls1c ws0">fading<span class="_ _5"> </span>channels.<span class="_ _5"> </span>Ass<span class="_ _7"></span><span class="ls1b">uming<span class="_ _5"> </span>ZF<span class="_ _5"> </span>detectors<span class="_ _3"> </span><span class="fc1 ls6">[11]<span class="_ _5"> </span></span><span class="ls1d">are<span class="_ _5"> </span>used<span class="_ _5"> </span>at<span class="_ _5"> </span>the</span></span></div><div class="t m0 x4 h4 y35 ff4 fs1 fc0 sc0 ls4 ws0">receiver,<span class="_ _b"> </span>a<span class="_ _d"> </span>closed-form<span class="_ _b"> </span>expression<span class="_ _b"> </span>of<span class="_ _d"> </span>the<span class="_ _b"> </span>bit<span class="_ _b"> </span>error<span class="_ _d"> </span>rate</div><div class="t m0 x4 h4 y36 ff4 fs1 fc0 sc0 lse ws0">(BER)<span class="_ _11"> </span>is<span class="_ _11"> </span>derived<span class="_ _11"> </span>in<span class="_ _11"> </span>terms<span class="_ _11"> </span>of<span class="_ _11"> </span>the<span class="_ _11"> </span>hypergeometric<span class="_ _11"> </span>and</div><div class="t m0 x4 h4 y37 ff4 fs1 fc0 sc0 lse ws0">Gamma<span class="_ _d"> </span>functions.<span class="_ _2"> </span>The<span class="_ _d"> </span>performance<span class="_ _d"> </span>degradatio<span class="_"> </span>n<span class="_ _d"> </span>of<span class="_ _2"> </span>the</div><div class="t m0 x4 h4 y38 ff4 fs1 fc0 sc0 ls1c ws0">system<span class="_ _4"> </span>over<span class="_ _4"> </span>a<span class="_ _4"> </span>Rice<span class="_ _4"> </span>fading<span class="_ _8"> </span>channel<span class="_ _4"> </span>compared<span class="_ _4"> </span>to<span class="_ _4"> </span>that<span class="_ _4"> </span>over</div><div class="t m0 x4 h4 y39 ff4 fs1 fc0 sc0 ls5 ws0">an<span class="_ _2"> </span>independent<span class="_ _2"> </span>and<span class="_ _2"> </span>identical<span class="_ _2"> </span>di<span class="ls1e">stributed<span class="_ _a"> </span>(i.i.d.)<span class="_ _2"> </span>Rayleigh</span></div><div class="t m0 x4 h4 y3a ff4 fs1 fc0 sc0 ls2 ws0">fading<span class="_ _a"> </span>channel<span class="_ _a"> </span>is<span class="_ _a"> </span>also<span class="_ _a"> </span>evaluated<span class="_ _a"> </span>assuming<span class="_ _a"> </span>identical<span class="_ _a"> </span>SNR</div><div class="t m0 x4 h4 y3b ff4 fs1 fc0 sc0 ls2 ws0">at<span class="_ _1"> </span>the<span class="_ _1"> </span>receiver.</div><div class="t m0 x5 h4 y3c ff4 fs1 fc0 sc0 ls2 ws0">Before<span class="_ _1"> </span>proceeding<span class="_ _1"> </span>further,<span class="_ _1"> </span>we<span class="_ _4"> </span>first<span class="_ _1"> </span>describe<span class="_ _1"> </span>the<span class="_ _4"> </span>notation</div><div class="t m0 x4 h4 y3d ff4 fs1 fc0 sc0 lse ws0">used<span class="_ _3"> </span>in<span class="_ _3"> </span>this<span class="_ _3"> </span>paper.<span class="_ _3"> </span>All<span class="_ _5"> </span>vectors<span class="_ _3"> </span>and<span class="_ _3"> </span>matrices<span class="_ _3"> </span>are<span class="_ _3"> </span>in<span class="_ _3"> </span>boldface.</div><div class="t m0 x4 h6 y3e ff5 fs1 fc0 sc0 ls1f ws0">j<span class="ff6">x</span>j<span class="_ _1"> </span><span class="ff4 ls4">represents<span class="_ _3"> </span>the<span class="_ _1"> </span>norm<span class="_ _1"> </span>value<span class="_ _3"> </span>of<span class="_ _1"> </span>complex<span class="_ _1"> </span>number<span class="_ _1"> </span></span><span class="ff7">x<span class="_ _5"> </span><span class="ff4 ls8">and<span class="_ _1"> </span></span>x</span></div><div class="t m1 x6 h7 y3f ff4 fs2 fc0 sc0 ls1f ws0">*</div><div class="t m0 x4 h8 y40 ff4 fs1 fc0 sc0 ls3 ws0">represents<span class="_ _a"> </span>the<span class="_ _a"> </span>complex<span class="_ _2"> </span>conjugate<span class="_ _a"> </span>of<span class="_ _a"> </span><span class="ff7 ls1f">x<span class="_ _12"></span><span class="ff4">.<span class="_ _a"> </span><span class="ff8">I</span></span></span></div><div class="t m1 x7 h9 y41 ff7 fs2 fc0 sc0 ls1f ws0">m</div><div class="t m0 x8 h4 y40 ff4 fs1 fc0 sc0 lsf ws0">is<span class="_ _a"> </span>the<span class="_ _2"> </span>identity</div><div class="t m0 x4 h6 y42 ff4 fs1 fc0 sc0 ls20 ws0">matrix<span class="_ _8"> </span>of<span class="_ _a"> </span>size<span class="_ _a"> </span><span class="ff7 ls1f">m<span class="_ _13"></span><span class="ff5"><span class="_ _14"></span></span>m<span class="ff4">.<span class="_ _4"> </span><span class="ff8">X</span></span></span></div><div class="t m1 x9 h9 y43 ff7 fs2 fc0 sc0 ls1f ws0">H</div><div class="t m0 xa h4 y44 ff4 fs1 fc0 sc0 ls21 ws0">denotes<span class="_ _a"> </span>the<span class="_ _8"> </span>complex<span class="_ _a"> </span>conjugate</div><div class="t m0 x4 h4 y45 ff4 fs1 fc0 sc0 ls2 ws0">transpose<span class="_ _5"> </span>of<span class="_ _3"> </span><span class="ff8 ls1f">X<span class="_ _6"> </span></span><span class="ls8">and<span class="_ _3"> </span><span class="ff8 ls1f">X</span></span></div><div class="t m1 xb h9 y46 ff7 fs2 fc0 sc0 ls1f ws0">T</div><div class="t m0 xc h4 y47 ff4 fs1 fc0 sc0 ls21 ws0">is<span class="_ _5"> </span>the<span class="_ _3"> </span>transpose<span class="_ _3"> </span>of<span class="_ _5"> </span><span class="ff8 ls1f">X</span><span class="ls22">.[<span class="_ _15"></span><span class="ff8 ls1f">X<span class="_ _12"></span><span class="ff4">]</span></span></span></div><div class="t m1 xd h9 y48 ff7 fs2 fc0 sc0 ls23 ws0">ij</div><div class="t m0 xe h4 y47 ff4 fs1 fc0 sc0 ls24 ws0">represents</div><div class="t m0 x4 h6 y49 ff4 fs1 fc0 sc0 ls25 ws0">the<span class="_ _4"> </span><span class="ff5 ls1f">ð<span class="ff6">i<span class="ff9">;<span class="_ _14"> </span></span>j</span>Þ</span><span class="ls6">th<span class="_ _8"> </span>element<span class="_ _8"> </span>of<span class="_ _8"> </span><span class="ff8 ls1f">X<span class="_ _4"> </span></span><span class="ls5">while<span class="_ _8"> </span><span class="ff8 ls1f">X</span></span></span></div><div class="t m1 xf ha y4a ff5 fs2 fc0 sc0 ls1f ws0"><span class="ff4">1</span></div><div class="t m0 x10 h4 y4b ff4 fs1 fc0 sc0 ls1b ws0">represents<span class="_ _4"> </span>the<span class="_ _8"> </span>inverse</div><div class="t m0 x4 h6 y4c ff4 fs1 fc0 sc0 lsf ws0">of<span class="_ _4"> </span>square<span class="_ _8"> </span>matrix<span class="_ _8"> </span><span class="ff8 ls1f">X<span class="_ _12"></span><span class="ff4 ls26">.E<span class="_ _c"></span>[<span class="_ _16"></span><span class="ff5 ls1f"><span class="_ _14"></span><span class="ff4 ls2">]<span class="_ _4"> </span>is<span class="_ _8"> </span>the<span class="_ _4"> </span>expectation<span class="_ _8"> </span>operator.<span class="_ _4"> </span>The</span></span></span></span></div><div class="t m0 x4 h4 y4d ff4 fs1 fc0 sc0 ls8 ws0">operator<span class="_ _11"> </span>det<span class="_ _14"> </span>(<span class="ff8 ls1f">X<span class="_ _13"> </span></span><span class="ls4">)<span class="_ _11"> </span>stands<span class="_ _11"> </span>for<span class="_ _11"> </span>the<span class="_ _11"> </span>determinant<span class="_ _b"> </span>of<span class="_ _11"> </span>square</span></div><div class="t m0 x4 h4 y4e ff4 fs1 fc0 sc0 ls27 ws0">matrix<span class="_ _1"> </span><span class="ff8 ls1f">X<span class="_ _12"></span><span class="ff4">.</span></span></div><div class="t m0 x4 h5 y4f ff1 fs1 fc0 sc0 ls28 ws0">2<span class="_ _9"> </span>System<span class="_ _1"> </span>overview</div><div class="t m0 x4 h8 y50 ff4 fs1 fc0 sc0 ls13 ws0">We<span class="_ _4"> </span>consider<span class="_ _4"> </span>a<span class="_ _4"> </span>MIMO<span class="_ _4"> </span>systems<span class="_ _4"> </span>(<span class="ff7 ls1f">t<span class="_ _12"></span><span class="ff4">,<span class="_ _14"> </span><span class="ff7">r<span class="_ _12"></span><span class="ff4 ls29">)w<span class="_ _c"></span>h<span class="_ _c"></span>e<span class="_ _c"></span>r<span class="_ _c"></span>e<span class="ff7 ls1f">t<span class="_ _4"> </span></span><span class="ls2a">and<span class="_ _4"> </span><span class="ff7 ls1f">r<span class="_ _1"> </span></span><span class="lsc">denote</span></span></span></span></span></span></div><div class="t m0 x4 h4 y51 ff4 fs1 fc0 sc0 ls13 ws0">the<span class="_ _4"> </span>number<span class="_ _1"> </span>of<span class="_ _4"> </span>transmit<span class="_ _1"> </span>and<span class="_ _4"> </span>receive<span class="_ _4"> </span>antennas,<span class="_ _1"> </span>respectively.</div><div class="t m0 x4 hb y52 ff4 fs1 fc0 sc0 ls2b ws0">We<span class="_ _1"> </span>always<span class="_ _3"> </span>assume<span class="_ _1"> </span><span class="ff7 ls1f">r<span class="_ _13"> </span><span class="ffa">Z<span class="_ _14"> </span></span>t<span class="_ _12"></span><span class="ff4 ls2">.<span class="_ _1"> </span>The<span class="_ _3"> </span>incoming<span class="_ _1"> </span>data<span class="_ _1"> </span>stream<span class="_ _1"> </span>is<span class="_ _3"> </span>first</span></span></div><div class="t m0 x4 h8 y53 ff4 fs1 fc0 sc0 ls2c ws0">split<span class="_ _3"> </span>into<span class="_ _3"> </span><span class="ff7 ls1f">t<span class="_ _3"> </span></span><span class="ls6">data<span class="_ _3"> </span>sub-streams<span class="_ _3"> </span>with<span class="_ _3"> </span>the<span class="_ _3"> </span>same<span class="_ _1"> </span>symbol<span class="_ _3"> </span>rate<span class="_ _3"> </span>1/<span class="ff7 ls1f">T</span></span></div><div class="t m0 x4 h8 y54 ff4 fs1 fc0 sc0 ls1f ws0">(<span class="_ _12"></span><span class="ff7">T<span class="_ _3"> </span><span class="ff4 ls3">is<span class="_ _1"> </span>the<span class="_ _1"> </span>symbol<span class="_ _3"> </span>period),<span class="_ _1"> </span>which<span class="_ _1"> </span>are<span class="_ _1"> </span>independently<span class="_ _1"> </span>encoded</span></span></div><div class="t m0 x4 h4 y55 ff4 fs1 fc0 sc0 ls3 ws0">and<span class="_ _b"> </span>modulated.<span class="_ _b"> </span>We<span class="_ _d"> </span>assume<span class="_ _b"> </span>that<span class="_ _b"> </span>the<span class="_ _d"> </span>same<span class="_ _b"> </span>modulation</div><div class="t m0 x4 h8 y56 ff4 fs1 fc0 sc0 ls5 ws0">scheme<span class="_ _2"> </span><span class="ff7 ls1f">V<span class="_ _2"> </span></span><span class="ls2">is<span class="_ _2"> </span>used<span class="_ _d"> </span>for<span class="_ _2"> </span>each<span class="_ _2"> </span>data<span class="_ _d"> </span>s<span class="_"> </span>ub-stream<span class="_ _2"> </span>and<span class="_ _d"> </span>all<span class="_ _2"> </span>the</span></div><div class="t m0 x4 h8 y57 ff4 fs1 fc0 sc0 ls2d ws0">transmitters<span class="_ _d"> </span>(1<span class="_ _b"> </span>to<span class="_ _b"> </span><span class="ff7 ls1f">t<span class="_ _12"></span><span class="ff4 ls2e">)<span class="_ _b"> </span>are<span class="_ _b"> </span>operating<span class="_ _b"> </span>at<span class="_ _b"> </span>the<span class="_ _b"> </span>same<span class="_ _b"> </span>carrier</span></span></div><div class="t m0 x4 h4 y58 ff4 fs1 fc0 sc0 ls2e ws0">frequency.<span class="_ _1"> </span>Also,<span class="_ _1"> </span>data<span class="_ _4"> </span>transmi<span class="ls2f">ssion<span class="_ _1"> </span>is<span class="_ _4"> </span>organised<span class="_ _1"> </span>into<span class="_ _1"> </span>bursts</span></div><div class="t m0 x4 h8 y59 ff4 fs1 fc0 sc0 ls6 ws0">of<span class="_ _11"> </span><span class="ff7 ls1f">L<span class="_ _b"> </span></span><span class="lsc">symbols<span class="_ _11"> </span>and<span class="_ _b"> </span>the<span class="_ _11"> </span>modulated<span class="_ _11"> </span>signals<span class="_ _11"> </span>are<span class="_ _11"> </span>radiated</span></div><div class="t m0 x4 h4 y5a ff4 fs1 fc0 sc0 ls6 ws0">through<span class="_ _1"> </span>the<span class="_ _1"> </span>respective<span class="_ _1"> </span>antennas.</div><div class="t m0 x5 h4 y5b ff4 fs1 fc0 sc0 ls2e ws0">Consider<span class="_ _17"> </span>the<span class="_ _17"> </span>structure<span class="_ _17"> </span>of<span class="_ _17"> </span>the<span class="_ _17"> </span>equivalent<span class="_ _17"> </span>baseband</div><div class="t m0 x4 h4 y5c ff4 fs1 fc0 sc0 ls13 ws0">systems<span class="_ _18"> </span>of<span class="_ _18"> </span>a<span class="_ _18"> </span>single-user<span class="_ _18"> </span>link.<span class="_ _18"> </span>We<span class="_ _18"> </span>assume<span class="_ _18"> </span>th<span class="_"> </span>at<span class="_ _18"> </span>the</div><div class="t m0 x4 h4 y5d ff4 fs1 fc0 sc0 ls9 ws0">transmitted<span class="_ _2"> </span>signals<span class="_ _2"> </span><span class="ls1c">encounter<span class="_ _2"> </span>flat<span class="_ _2"> </span>fading<span class="_ _2"> </span>before<span class="_ _2"> </span>reaching</span></div><div class="t m0 x4 h4 y5e ff4 fs1 fc0 sc0 ls2 ws0">the<span class="_ _1"> </span>receiving<span class="_ _3"> </span>antennas.<span class="_ _1"> </span>Denote<span class="_ _1"> </span>the<span class="_ _3"> </span>impulse<span class="_ _1"> </span>response<span class="_ _1"> </span>of<span class="_ _1"> </span>the</div><div class="t m0 x4 h6 y5f ff4 fs1 fc0 sc0 ls1d ws0">channel<span class="_ _5"> </span>between<span class="_ _3"> </span>the<span class="_ _3"> </span><span class="ff7 ls1f">j<span class="_ _13"> </span></span><span class="ls30">th<span class="_ _3"> </span><span class="ff5 ls1f">ð<span class="_ _14"></span><span class="ff6">j<span class="_ _a"> </span></span>¼<span class="_ _a"> </span><span class="ff4">1<span class="ff9">;<span class="_ _14"> </span></span>2<span class="ff9">;<span class="_ _14"> </span><span class="ffb ls31">...<span class="_ _7"></span></span>;<span class="_ _14"> </span><span class="ff6">t</span></span></span>Þ<span class="_ _3"> </span></span><span class="ls3">transmit<span class="_ _3"> </span>antenna</span></span></div><div class="t m0 x1 hc y60 ff4 fs3 fc0 sc0 ls32 ws0">E-mail:<span class="_ _6"> </span>xurt@eie.polyu.edu.hk</div><div class="t m0 x1 hc y61 ff4 fs3 fc0 sc0 ls33 ws0">The<span class="_ _b"> </span>authors<span class="_ _d"> </span>are<span class="_ _b"> </span>with<span class="_ _b"> </span>the<span class="_ _d"> </span>Department<span class="_ _b"> </span>of<span class="_ _d"> </span>Electronic<span class="_ _b"> </span>and<span class="_ _b"> </span>Information</div><div class="t m0 x1 hc y62 ff4 fs3 fc0 sc0 ls34 ws0">Engineering,<span class="_ _6"> </span>The<span class="_ _5"> </span>Hong<span class="_ _5"> </span>Kong<span class="_ _5"> </span>Polytechnic<span class="_ _6"> </span>University,<span class="_ _5"> </span>Hong<span class="_ _5"> </span>Kong,<span class="_ _5"> </span>China</div><div class="t m0 x1 hc y63 ffc fs3 fc0 sc0 ls1f ws0">r<span class="_ _6"> </span><span class="ff4 ls35">IEE,<span class="_ _5"> </span>2006</span></div><div class="t m0 x1 hd y64 ff7 fs3 fc0 sc0 ls36 ws0">IEE<span class="_ _6"> </span>Proceedings<span class="_ _5"> </span><span class="ff4 ls37">online<span class="_ _5"> </span>no.<span class="_ _5"> </span>20045281</span></div><div class="t m0 x1 hc y65 ff4 fs3 fc0 sc0 ls35 ws0">doi:10.1049/ip-com:20045281</div><div class="t m0 x1 hc y66 ff4 fs3 fc0 sc0 ls37 ws0">Paper<span class="_ _14"> </span>first<span class="_ _14"> </span>recei<span class="_"> </span>ved<span class="_ _14"> </span>6th<span class="_ _14"> </span>November<span class="_ _14"> </span>2004<span class="_ _14"> </span>and<span class="_ _14"> </span>in<span class="_ _19"> </span>final<span class="_ _14"> </span>revise<span class="_"> </span>d<span class="_ _14"> </span>form<span class="_ _14"> </span>28th<span class="_ _14"> </span>April<span class="_ _14"> </span>2005</div><div class="t m0 x1 he y67 ff4 fs4 fc0 sc0 ls38 ws0">74<span class="_ _1a"> </span><span class="ff7 fs3 ls37">IEE<span class="_ _5"> </span>Proc.-Commun.,<span class="_ _6"> </span>Vol.<span class="_ _5"> </span>153,<span class="_ _5"> </span>No.<span class="_ _5"> </span>1,<span class="_ _5"> </span>February<span class="_ _6"> </span>2006</span></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div>
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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6255768247503a0a93dc714a/bg2.jpg"><div class="t m0 x1 h6 y68 ff4 fs1 fc0 sc0 ls9 ws0">and<span class="_ _a"> </span>the<span class="_ _a"> </span><span class="ff7 ls1f">i<span class="_ _12"></span><span class="ff4 ls30">th<span class="_ _8"> </span><span class="ff5 ls1f">ð<span class="ff6">i<span class="_ _a"> </span></span>¼<span class="_ _a"> </span><span class="ff4">1<span class="ff9">;<span class="_ _14"> </span></span>2<span class="ff9">;<span class="_ _14"> </span><span class="ffb ls31">...<span class="_ _7"></span></span>;<span class="_ _14"> </span><span class="ff6">r<span class="_ _7"></span></span></span></span>Þ<span class="_ _a"> </span></span><span class="lsc">receive<span class="_ _a"> </span>antenna<span class="_ _a"> </span>by<span class="_ _8"> </span><span class="ff6 ls1f">h</span></span></span></span></div><div class="t m0 x11 hf y69 ff6 fs5 fc0 sc0 ls1f ws0">i<span class="ff9">;</span>j</div><div class="t m0 x12 h4 y68 ff4 fs1 fc0 sc0 ls39 ws0">.T<span class="_ _1b"></span>h<span class="_ _1b"></span>e</div><div class="t m0 x1 h4 y6a ff4 fs1 fc0 sc0 ls1b ws0">received<span class="_ _4"> </span>si<span class="_ _7"></span>gnal<span class="_ _4"> </span>can<span class="_ _8"> </span>be<span class="_ _8"> </span>presented<span class="_ _8"> </span>in<span class="_ _4"> </span>the<span class="_ _8"> </span>following<span class="_ _8"> </span>complex</div><div class="t m0 x1 h4 y6b ff4 fs1 fc0 sc0 lsc ws0">baseband<span class="_ _1"> </span>vector<span class="_ _1"> </span>form<span class="_ _1"> </span><span class="fc1 ls27">[12]</span></div><div class="t m0 x13 h6 y6c ff8 fs1 fc0 sc0 ls1f ws0">y<span class="_ _a"> </span><span class="ff5">¼<span class="_ _a"> </span></span><span class="ls2a">Hd<span class="_ _1"> </span></span><span class="ff5">þ<span class="_ _19"></span></span>n<span class="_ _1c"> </span><span class="ff5">ð<span class="ff4">1</span>Þ</span></div><div class="t m0 x1 h6 y6d ff4 fs1 fc0 sc0 ls13 ws0">where<span class="_ _2"> </span><span class="ff8 ls1f">y<span class="_ _a"> </span><span class="ff5 ls3a">¼½<span class="_ _1b"></span><span class="ff6 ls1f">y</span></span></span></div><div class="t m0 x14 h10 y6e ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x15 h11 y6f ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _14"> </span><span class="ffb ls31">...<span class="_ _7"></span></span>;<span class="_ _14"> </span><span class="ff6">y</span></div><div class="t m0 x16 h12 y6e ff6 fs5 fc0 sc0 ls1f ws0">r</div><div class="t m0 x17 h6 y6f ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 x18 h12 y70 ff6 fs5 fc0 sc0 ls1f ws0">T</div><div class="t m0 x19 h4 y6f ff4 fs1 fc0 sc0 ls1b ws0">is<span class="_ _2"> </span>the<span class="_ _d"> </span>received<span class="_ _2"> </span>signal<span class="_ _2"> </span>vector,<span class="_ _d"> </span><span class="ff8 ls1f">H</span></div><div class="t m0 x1 h6 y71 ff4 fs1 fc0 sc0 ls12 ws0">is<span class="_ _d"> </span>an<span class="_ _b"> </span><span class="ff7 ls1f">r<span class="_ _13"> </span><span class="ff5"><span class="_ _14"></span></span>t<span class="_ _d"> </span></span><span class="lsc">channel<span class="_ _d"> </span>matrix<span class="_ _b"> </span>containing<span class="_ _d"> </span>the<span class="_ _b"> </span>elements<span class="_ _d"> </span><span class="ff6 ls1f">h</span></span></div><div class="t m0 x1a hf y72 ff6 fs5 fc0 sc0 ls1f ws0">i<span class="ff9">;</span>j</div><div class="t m0 x1b h4 y73 ff4 fs1 fc0 sc0 ls1f ws0">,</div><div class="t m0 x1 h6 y74 ff8 fs1 fc0 sc0 ls1f ws0">d<span class="_ _2"> </span><span class="ff5 ls3a">¼½<span class="_ _1b"></span><span class="ff6 ls1f">d</span></span></div><div class="t m0 x1c h10 y75 ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x1d h4 y76 ffb fs1 fc0 sc0 ls3b ws0">...<span class="_ _7"></span><span class="ff6 ls1f">d</span></div><div class="t m0 x1e h12 y75 ff6 fs5 fc0 sc0 ls1f ws0">t</div><div class="t m0 x1f h6 y76 ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 x14 h12 y77 ff6 fs5 fc0 sc0 ls1f ws0">T</div><div class="t m0 x20 h4 y76 ff4 fs1 fc0 sc0 ls5 ws0">is<span class="_ _0"> </span>the<span class="_ _1d"> </span>transmitted<span class="_ _0"> </span>symbol<span class="_ _1d"> </span>vector,<span class="_ _0"> </span>and</div><div class="t m0 x1 h6 y78 ff8 fs1 fc0 sc0 ls1f ws0">n<span class="_ _a"> </span><span class="ff5 ls3a">¼½<span class="_ _1b"></span><span class="ff6 ls1f">n</span></span></div><div class="t m0 x1c h10 y79 ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x21 h11 y7a ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _14"> </span><span class="ffb ls31">...<span class="_ _7"></span></span>;<span class="_ _14"> </span><span class="ff6">n</span></div><div class="t m0 x22 h12 y79 ff6 fs5 fc0 sc0 ls1f ws0">r</div><div class="t m0 x15 h6 y7a ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 x23 h12 y7b ff6 fs5 fc0 sc0 ls1f ws0">T</div><div class="t m0 x24 h4 y7a ff4 fs1 fc0 sc0 lsa ws0">is<span class="_ _a"> </span>the<span class="_ _a"> </span>noise<span class="_ _2"> </span>vector.<span class="_ _a"> </span>The<span class="_ _a"> </span>elements<span class="_ _a"> </span>of<span class="_ _a"> </span><span class="ff8 ls1f">H</span></div><div class="t m0 x1 h4 y7c ff4 fs1 fc0 sc0 ls3c ws0">are<span class="_ _2"> </span>assum<span class="_"> </span>ed<span class="_ _a"> </span>as<span class="_ _2"> </span>zero<span class="_ _2"> </span>mean<span class="_ _a"> </span>circularly<span class="_ _a"> </span>symmetric<span class="_ _2"> </span>complex</div><div class="t m0 x1 h4 y7d ff4 fs1 fc0 sc0 ls6 ws0">Gaussian<span class="_ _1"> </span>random<span class="_ _1"> </span>variables<span class="_ _1"> </span><span class="fc1 ls12">[8]</span><span class="ls1f">.</span></div><div class="t m0 x3 h4 y7e ff4 fs1 fc0 sc0 ls9 ws0">The<span class="_ _2"> </span>received<span class="_ _2"> </span>signal<span class="_ _2"> </span><span class="ff8 ls1f">y<span class="_ _a"> </span></span><span class="lsc">will<span class="_ _d"> </span>be<span class="_ _2"> </span>processed<span class="_ _2"> </span>by<span class="_ _2"> </span>a<span class="_ _2"> </span>detector</span></div><div class="t m0 x1 h4 y7f ff4 fs1 fc0 sc0 ls6 ws0">whose<span class="_ _5"> </span>aim<span class="_ _6"> </span>is<span class="_ _5"> </span>to<span class="_ _5"> </span>determine<span class="_ _5"> </span>the<span class="_ _5"> </span>received<span class="_ _5"> </span>data<span class="_ _5"> </span>symbols.<span class="_ _5"> </span>In<span class="_ _5"> </span>our</div><div class="t m0 x1 h4 y80 ff4 fs1 fc0 sc0 ls13 ws0">analysis<span class="_ _1"> </span>it<span class="_ _4"> </span>is<span class="_ _1"> </span>assumed<span class="_ _1"> </span>that<span class="_ _4"> </span>the<span class="_ _1"> </span>channel<span class="_ _1"> </span>matrix<span class="_ _1"> </span><span class="ff8 ls1f">H<span class="_ _1"> </span></span><span class="ls3d">is<span class="_ _4"> </span>known</span></div><div class="t m0 x1 h4 y81 ff4 fs1 fc0 sc0 ls6 ws0">at<span class="_ _2"> </span>the<span class="_ _2"> </span>receiver.<span class="_ _2"> </span>We<span class="_ _2"> </span>assume<span class="_ _a"> </span>that<span class="_ _2"> </span>the<span class="_ _2"> </span>channel<span class="_ _2"> </span>is<span class="_ _2"> </span>random,</div><div class="t m0 x1 h4 y82 ff4 fs1 fc0 sc0 ls1b ws0">quasistatic,<span class="_ _18"> </span>frequency<span class="_ _e"> </span>inde<span class="lsc">pendent<span class="_ _18"> </span>and<span class="_ _e"> </span>the<span class="_ _18"> </span>signal<span class="_ _e"> </span>is</span></div><div class="t m0 x1 h4 y83 ff4 fs1 fc0 sc0 ls14 ws0">corrupted<span class="_ _1"> </span>by<span class="_ _1"> </span>complex<span class="_ _1"> </span>AWGN.<span class="_ _1"> </span>Moreover,<span class="_ _1"> </span>the<span class="_ _4"> </span>elements<span class="_ _1"> </span>of</div><div class="t m0 x1 h4 y84 ff8 fs1 fc0 sc0 ls1f ws0">n<span class="_ _6"> </span><span class="ff4 ls6">are<span class="_ _5"> </span>taken<span class="_ _5"> </span>to<span class="_ _3"> </span>be<span class="_ _5"> </span>i.i.d.<span class="_ _5"> </span>Gaussi<span class="ls16">an<span class="_ _5"> </span>random<span class="_ _5"> </span>variables<span class="_ _5"> </span>with<span class="_ _5"> </span>zero</span></span></div><div class="t m0 x1 h4 y85 ff4 fs1 fc0 sc0 ls3e ws0">mean<span class="_ _1"> </span>and<span class="_ _1"> </span>variance<span class="_ _3"> </span><span class="ffd ls1f">s</span></div><div class="t m1 x25 h7 y86 ff4 fs2 fc0 sc0 ls1f ws0">2</div><div class="t m0 x16 h4 y87 ff4 fs1 fc0 sc0 ls1f ws0">.</div><div class="t m0 x1 h5 y88 ff1 fs1 fc0 sc0 ls3f ws0">3<span class="_ _9"> </span>ZF<span class="_ _1"> </span>detec<span class="_ _7"></span>tor<span class="_ _1"> </span>performa<span class="_ _7"></span>nce</div><div class="t m0 x1 h4 y89 ff4 fs1 fc0 sc0 ls8 ws0">In<span class="_ _11"> </span>this<span class="_ _11"> </span>Section,<span class="_ _11"> </span>we<span class="_ _11"> </span>study<span class="_ _11"> </span>the<span class="_ _11"> </span>performance<span class="_ _11"> </span>of<span class="_ _11"> </span>MIMO</div><div class="t m0 x1 h4 y8a ff4 fs1 fc0 sc0 ls5 ws0">systems<span class="_ _4"> </span>using<span class="_ _4"> </span>ZF<span class="_ _8"> </span>detector<span class="_ _8"> </span>ov<span class="ls13">er<span class="_ _4"> </span>Rayleigh<span class="_ _4"> </span>and<span class="_ _8"> </span>Rice<span class="_ _4"> </span>fading</span></div><div class="t m0 x1 h4 y8b ff4 fs1 fc0 sc0 ls5 ws0">channel.<span class="_ _2"> </span>In<span class="_ _2"> </span>the<span class="_ _d"> </span>r<span class="_"> </span>eceiver,<span class="_ _2"> </span>the<span class="_ _2"> </span>post-detection<span class="_ _2"> </span>SNR<span class="_ _d"> </span>with<span class="_ _2"> </span>a</div><div class="t m0 x1 h4 y8c ff4 fs1 fc0 sc0 lse ws0">linear<span class="_ _1"> </span>ZF<span class="_ _1"> </span>equalizer<span class="_ _1"> </span>is<span class="_ _1"> </span>expressed<span class="_ _1"> </span>as<span class="_ _1"> </span><span class="fc1 ls12">[8]</span></div><div class="t m0 x2 h13 y8d ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x14 h12 y8e ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x26 h6 y8f ff5 fs1 fc0 sc0 ls1f ws0">¼</div><div class="t m0 x18 h13 y90 ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x27 h12 y91 ff6 fs5 fc0 sc0 ls1f ws0">o</div><div class="t m0 x28 h6 y92 ff5 fs1 fc0 sc0 ls1f ws0">½<span class="ff8">H</span></div><div class="t m0 x29 h12 y93 ff6 fs5 fc0 sc0 ls1f ws0">H</div><div class="t m0 x2a h6 y92 ff8 fs1 fc0 sc0 ls1f ws0">H<span class="_ _1e"></span><span class="ff5"></span></div><div class="t m0 x2b h14 y94 ff5 fs5 fc0 sc0 ls1f ws0"><span class="ff4">1</span></div><div class="t m0 x2b h12 y95 ff6 fs5 fc0 sc0 ls40 ws0">kk</div><div class="t m0 x2c h6 y8f ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _1f"> </span><span class="ff6">k<span class="_ _2"> </span><span class="ff5">¼<span class="_ _a"> </span><span class="ff4">1</span></span></span>;<span class="_ _14"> </span><span class="ffb ls3b">...<span class="_ _7"></span></span>;<span class="_ _14"> </span><span class="ff6">t<span class="_ _20"> </span><span class="ff5">ð<span class="ff4">2</span>Þ</span></span></div><div class="t m0 x1 h4 y96 ff4 fs1 fc0 sc0 ls13 ws0">where<span class="_ _a"> </span><span class="ffd ls1f">g</span></div><div class="t m1 x2d h9 y97 ff7 fs2 fc0 sc0 ls1f ws0">o</div><div class="t m0 x2e h4 y98 ff4 fs1 fc0 sc0 ls9 ws0">is<span class="_ _a"> </span>the<span class="_ _a"> </span>normalised<span class="_ _8"> </span>received<span class="_ _a"> </span>SNR<span class="_ _a"> </span>at<span class="_ _a"> </span>each<span class="_ _a"> </span>receive</div><div class="t m0 x1 h4 y99 ff4 fs1 fc0 sc0 lsc ws0">antenna<span class="_ _1"> </span>and<span class="_ _1"> </span>is<span class="_ _1"> </span>defined<span class="_ _1"> </span>as</div><div class="t m0 x2f h13 y9a ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x16 h12 y9b ff6 fs5 fc0 sc0 ls1f ws0">o</div><div class="t m0 x30 h6 y9c ff5 fs1 fc0 sc0 ls1f ws0">¼<span class="_ _a"> </span><span class="ff6">E</span>½j<span class="ff6">d</span></div><div class="t m0 x31 h12 y9d ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x32 h6 y9c ff5 fs1 fc0 sc0 ls1f ws0">j</div><div class="t m0 x33 h10 y9e ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 x34 h6 y9c ff5 fs1 fc0 sc0 ls1f ws0"><span class="ff9">=<span class="ffd">s</span></span></div><div class="t m0 x35 h10 y9f ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 x36 h6 y9c ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4">3</span>Þ</div><div class="t m0 x1 h8 ya0 ff4 fs1 fc0 sc0 ls8 ws0">and<span class="_ _5"> </span><span class="ff7 ls1f">d</span></div><div class="t m1 x37 h9 ya1 ff7 fs2 fc0 sc0 ls1f ws0">k</div><div class="t m0 x38 h8 ya2 ff4 fs1 fc0 sc0 ls1c ws0">denotes<span class="_ _3"> </span>the<span class="_ _5"> </span>symbols<span class="_ _3"> </span>from<span class="_ _3"> </span>the<span class="_ _3"> </span><span class="ff7 ls1f">k<span class="_ _13"> </span></span><span class="ls27">th<span class="_ _3"> </span>transmit<span class="_ _5"> </span>antenna.</span></div><div class="t m0 x1 h6 ya3 ff4 fs1 fc0 sc0 ls1d ws0">Define<span class="_ _8"> </span><span class="ff8 ls1f">Z<span class="_ _d"> </span><span class="ff5">¼<span class="_ _a"> </span></span>H</span></div><div class="t m0 x39 h12 ya4 ff6 fs5 fc0 sc0 ls1f ws0">H</div><div class="t m0 x20 h4 ya5 ff8 fs1 fc0 sc0 ls1f ws0">H<span class="_ _2"> </span><span class="ff4 ls6">and<span class="_ _a"> </span>take<span class="_ _4"> </span>the<span class="_ _a"> </span>row<span class="_ _8"> </span>vectors<span class="_ _8"> </span>of<span class="_ _a"> </span></span>H<span class="_ _4"> </span><span class="ff4 ls7">to<span class="_ _8"> </span>be</span></div><div class="t m0 x1 h4 ya6 ff4 fs1 fc0 sc0 ls5 ws0">independent<span class="_ _d"> </span>under<span class="_ _2"> </span>the<span class="_ _d"> </span>assumption<span class="_ _d"> </span>that<span class="_ _2"> </span>there<span class="_ _d"> </span>is<span class="_ _d"> </span>a<span class="_ _d"> </span>large</div><div class="t m0 x1 h4 ya7 ff4 fs1 fc0 sc0 lse ws0">physical<span class="_ _a"> </span>separation<span class="_ _a"> </span>between<span class="_ _a"> </span>the<span class="_ _2"> </span>receive<span class="_ _a"> </span>antennas.<span class="_ _a"> </span>Let<span class="_ _a"> </span><span class="ff8 ls1f">H</span></div><div class="t m1 x3a h9 ya8 ff7 fs2 fc0 sc0 ls1f ws0">i</div><div class="t m0 x1 h8 ya9 ff4 fs1 fc0 sc0 ls2 ws0">denote<span class="_ _a"> </span>the<span class="_ _8"> </span><span class="ff7 ls1f">i</span><span class="lsf">th<span class="_ _a"> </span>row<span class="_ _a"> </span>vector<span class="_ _8"> </span>of<span class="_ _a"> </span><span class="ff8 ls1f">H</span><span class="ls41">.T<span class="_ _1b"></span>h<span class="_ _1b"></span>e<span class="_ _1b"></span>n<span class="ff8 ls1f">H</span></span></span></div><div class="t m1 x3b h9 yaa ff7 fs2 fc0 sc0 ls1f ws0">i</div><div class="t m0 x3c h4 ya9 ff4 fs1 fc0 sc0 ls14 ws0">has<span class="_ _a"> </span>a<span class="_ _8"> </span>complex</div><div class="t m0 x1 h4 yab ff4 fs1 fc0 sc0 ls42 ws0">multivariate<span class="_ _1f"> </span>normal<span class="_ _1f"> </span>distribution<span class="_ _1f"> </span><span class="fc1 ls6">[13]<span class="_ _1f"> </span><span class="fc0">denoted<span class="_ _21"> </span>by</span></span></div><div class="t m0 x1 h15 yac ff8 fs1 fc0 sc0 ls1f ws0">H</div><div class="t m0 x3d h12 yad ff6 fs5 fc0 sc0 ls1f ws0">i</div><div class="t m0 x3e h6 yae ff5 fs1 fc0 sc0 ls1f ws0"><span class="_ _5"> </span><span class="ff6">N</span></div><div class="t m0 x2d h12 yaf ff6 fs5 fc0 sc0 ls1f ws0">c</div><div class="t m0 x3f h12 yb0 ff6 fs5 fc0 sc0 ls1f ws0">t</div><div class="t m0 x40 h6 yae ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ffe">l</span></div><div class="t m0 x41 h12 yb1 ff6 fs5 fc0 sc0 ls1f ws0">i</div><div class="t m0 x42 h11 yae ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _14"> </span><span class="fff">R</span></div><div class="t m0 x14 h12 yad ff6 fs5 fc0 sc0 ls1f ws0">t</div><div class="t m0 x43 h6 yae ff5 fs1 fc0 sc0 ls1f ws0">Þ<span class="ff4 ls2c">,<span class="_ _1d"> </span>with<span class="_ _22"> </span>mean<span class="_ _1d"> </span>vector<span class="_ _1d"> </span></span><span class="ffe">l</span></div><div class="t m1 x44 h9 yb2 ff7 fs2 fc0 sc0 ls1f ws0">i</div><div class="t m0 x45 h4 yae ff4 fs1 fc0 sc0 ls8 ws0">and<span class="_ _1d"> </span>covariance</div><div class="t m0 x1 h4 yb3 ff4 fs1 fc0 sc0 ls27 ws0">matrix<span class="_ _0"> </span><span class="fff ls1f">R</span></div><div class="t m1 x2e h9 yb4 ff7 fs2 fc0 sc0 ls1f ws0">i</div><div class="t m0 x46 h4 yb5 ff4 fs1 fc0 sc0 ls2 ws0">.<span class="_ _1d"> </span>Suppose<span class="_ _0"> </span>a<span class="_ _7"></span>ll<span class="_ _1d"> </span>the<span class="_ _0"> </span>r<span class="_ _7"></span>ow<span class="_ _1d"> </span>vectors<span class="_ _1d"> </span><span class="ff8 ls1f">H</span></div><div class="t m1 x47 h9 yb4 ff7 fs2 fc0 sc0 ls1f ws0">i</div><div class="t m0 x48 h4 yb5 ff4 fs1 fc0 sc0 ls20 ws0">have<span class="_ _0"> </span>the</div><div class="t m0 x1 h4 yb6 ff4 fs1 fc0 sc0 ls43 ws0">complex<span class="_ _d"> </span>m<span class="_"> </span>ultivariate<span class="_ _2"> </span>normal<span class="_ _d"> </span>distribut<span class="_"> </span>ion<span class="_ _2"> </span>w<span class="_ _7"></span>ith<span class="_ _2"> </span>t<span class="_ _7"></span>he<span class="_ _d"> </span>same</div><div class="t m0 x1 h4 yb7 ff4 fs1 fc0 sc0 ls13 ws0">covariance<span class="_ _a"> </span>matrix<span class="_ _2"> </span><span class="fff ls1f">R<span class="_ _12"></span><span class="ff4 ls44">,t<span class="_ _23"></span>h<span class="_ _23"></span>e<span class="_ _23"></span>n<span class="ff8 ls1f">Z<span class="_ _a"> </span></span><span class="ls8">follows<span class="_ _a"> </span>a<span class="_ _2"> </span>complex<span class="_ _a"> </span>Wishart</span></span></span></div><div class="t m0 x1 h6 yb8 ff4 fs1 fc0 sc0 lse ws0">distribution<span class="_ _d"> </span>denoted<span class="_ _b"> </span>by<span class="_ _d"> </span><span class="ff8 ls1f">Z<span class="_ _1"> </span><span class="ff5"><span class="_ _3"> </span><span class="ff6">W</span></span></span></div><div class="t m0 x49 h12 yb9 ff6 fs5 fc0 sc0 ls1f ws0">c</div><div class="t m0 x4a h12 yba ff6 fs5 fc0 sc0 ls1f ws0">t</div><div class="t m0 x4b h6 ybb ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff6">r<span class="ff9">;<span class="_ _14"> </span><span class="ff8">M<span class="_ _1e"></span></span>;<span class="_ _14"> </span><span class="fff">R</span></span></span>Þ<span class="_"> </span><span class="ff4 ls45">,w<span class="_ _24"></span>h<span class="_ _24"></span>e<span class="_ _25"></span>r<span class="_ _25"></span>e<span class="ff8 ls1f">M<span class="_ _d"> </span><span class="ff5">¼</span></span></span></div><div class="t m0 x1 h6 ybc ff5 fs1 fc0 sc0 ls1f ws0">½<span class="ffe">l</span></div><div class="t m0 x4c h10 ybd ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x4d h11 ybe ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _14"> </span><span class="ffe">l</span></div><div class="t m0 x37 h10 ybd ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 x4e h11 ybe ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _14"> </span><span class="ffb ls31">...</span>;<span class="_ _14"> </span><span class="ffe">l</span></div><div class="t m0 x1e h12 ybd ff6 fs5 fc0 sc0 ls1f ws0">r</div><div class="t m0 x4f h6 ybe ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 x14 h12 ybf ff6 fs5 fc0 sc0 ls1f ws0">T</div><div class="t m0 x39 h6 ybe ff4 fs1 fc0 sc0 ls1b ws0">.<span class="_ _6"> </span>Depending<span class="_ _6"> </span>upon<span class="_ _5"> </span><span class="ff8 ls1f">M<span class="_ _13"> </span><span class="ff5">¼<span class="_ _14"></span><span class="ff3">0<span class="_ _6"> </span></span></span></span><span class="ls6">or<span class="_ _6"> </span><span class="ff8 ls1f">M<span class="_ _4"> </span><span class="ff5">6<span class="_ _12"></span>¼<span class="_ _5"> </span><span class="ff3">0<span class="ff4">,<span class="_ _6"> </span></span>0<span class="_ _19"> </span><span class="ff4 lsc">being</span></span></span></span></span></div><div class="t m0 x1 h4 yc0 ff4 fs1 fc0 sc0 ls8 ws0">the<span class="_ _3"> </span>zero<span class="_ _3"> </span>matrix,<span class="_ _3"> </span>the<span class="_ _3"> </span>complex<span class="_ _3"> </span>Wi<span class="ls2">shart<span class="_ _3"> </span>distribution<span class="_ _1"> </span>is<span class="_ _5"> </span>c<span class="_ _7"></span>entral</span></div><div class="t m0 x1 h4 yc1 ff4 fs1 fc0 sc0 ls2 ws0">or<span class="_ _5"> </span>non-central,<span class="_ _3"> </span>respectively.<span class="_ _5"> </span>In<span class="_ _3"> </span>the<span class="_ _5"> </span>following,<span class="_ _5"> </span>we<span class="_ _3"> </span>will<span class="_ _5"> </span>simply</div><div class="t m0 x1 h4 yc2 ff4 fs1 fc0 sc0 ls1c ws0">use<span class="_ _1"> </span>complex<span class="_ _1"> </span>Wishart<span class="_ _1"> </span>distribut<span class="ls6">ion<span class="_ _1"> </span>when<span class="_ _1"> </span>referring<span class="_ _1"> </span>to<span class="_ _1"> </span>central</span></div><div class="t m0 x1 h4 yc3 ff4 fs1 fc0 sc0 ls3 ws0">complex<span class="_ _1"> </span>Wishart<span class="_ _1"> </span>distribution.</div><div class="t m0 x3 h4 yc4 ff4 fs1 fc0 sc0 ls1b ws0">The<span class="_ _4"> </span>non-central<span class="_ _8"> </span>complex<span class="_ _4"> </span>W<span class="_ _7"></span>ishart<span class="_ _4"> </span>distribution<span class="_ _8"> </span>has<span class="_ _8"> </span>been</div><div class="t m0 x1 h4 yc5 ff4 fs1 fc0 sc0 ls2 ws0">approximated<span class="_ _1d"> </span>by<span class="_ _1d"> </span>a<span class="_ _22"> </span>complex<span class="_ _1d"> </span><span class="ls27">Wishart<span class="_ _1d"> </span>distribution<span class="_ _0"> </span><span class="fc1 ls6">[14]</span></span></div><div class="t m0 x1 h4 yc6 ff4 fs1 fc0 sc0 lsc ws0">with<span class="_ _d"> </span>a<span class="_ _d"> </span>technique<span class="_ _d"> </span>that<span class="_ _d"> </span>represents<span class="_ _b"> </span>a<span class="_ _d"> </span>non-central<span class="_ _d"> </span>complex</div><div class="t m0 x1 h4 yc7 ff4 fs1 fc0 sc0 lse ws0">Wishart<span class="_ _2"> </span>distribution<span class="_ _a"> </span>by<span class="_ _a"> </span>normal<span class="_ _2"> </span>vectors.<span class="_ _a"> </span>Now<span class="_ _2"> </span>consider<span class="_ _a"> </span>a</div><div class="t m0 x1 h4 yc8 ff4 fs1 fc0 sc0 ls13 ws0">complex<span class="_ _a"> </span>Wishart<span class="_ _2"> </span>distribution</div><div class="t m0 x50 h4 yc9 ffb fs1 fc0 sc0 ls1f ws0">^</div><div class="t m0 x31 h6 yc8 ff8 fs1 fc0 sc0 ls1f ws0">Z<span class="_ _1"> </span><span class="ff5"><span class="_ _3"> </span><span class="ff6">W</span></span></div><div class="t m0 x51 h12 yca ff6 fs5 fc0 sc0 ls1f ws0">c</div><div class="t m0 x52 h12 ycb ff6 fs5 fc0 sc0 ls1f ws0">t</div><div class="t m0 x53 h6 ycc ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff6">r<span class="ff9">;</span></span></div><div class="t m0 x45 h4 ycd ffb fs1 fc0 sc0 ls1f ws0">^</div><div class="t m0 x3b h6 ycc fff fs1 fc0 sc0 ls1f ws0">R<span class="ff5">Þ<span class="ff4 ls46">,w<span class="_ _23"></span>h<span class="_ _23"></span>e<span class="_ _23"></span>r<span class="_ _26"></span>e</span></span></div><div class="t m0 x54 h4 ycd ffb fs1 fc0 sc0 ls1f ws0">^</div><div class="t m0 x55 h6 ycc fff fs1 fc0 sc0 ls1f ws0">R<span class="_ _a"> </span><span class="ff5">¼</span></div><div class="t m0 x1 h6 yce fff fs1 fc0 sc0 ls1f ws0">R<span class="_ _6"> </span><span class="ff5">þ</span></div><div class="t m0 x56 h10 ycf ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x56 h12 yd0 ff6 fs5 fc0 sc0 ls1f ws0">r</div><div class="t m0 x57 h15 yd1 ff8 fs1 fc0 sc0 ls1f ws0">M</div><div class="t m0 x58 h12 yd2 ff6 fs5 fc0 sc0 ls1f ws0">H</div><div class="t m0 x46 h4 yd1 ff8 fs1 fc0 sc0 ls1f ws0">M<span class="_ _1e"></span><span class="ff4 ls8">.<span class="_ _2"> </span>The<span class="_ _2"> </span>first<span class="_ _2"> </span>order<span class="_ _2"> </span>moments<span class="_ _2"> </span>of<span class="_ _2"> </span></span>Z<span class="_ _2"> </span><span class="ff4 ls8">and</span></div><div class="t m0 x59 h4 yd3 ffb fs1 fc0 sc0 ls1f ws0">^</div><div class="t m0 x5a h4 yd1 ff8 fs1 fc0 sc0 ls1f ws0">Z<span class="_ _b"> </span><span class="ff4 ls25">are</span></div><div class="t m0 x1 h6 yd4 ff4 fs1 fc0 sc0 ls6 ws0">identical,<span class="_ _d"> </span>whereas<span class="_ _d"> </span>the<span class="_ _d"> </span>second<span class="_ _2"> </span>order<span class="_ _d"> </span>moments<span class="_ _d"> </span>E<span class="_ _7"></span><span class="ff5 ls1f">ð<span class="ff8">Z</span></span></div><div class="t m0 x12 h12 yd5 ff6 fs5 fc0 sc0 ls47 ws0">ij</div><div class="t m0 x5b h15 yd6 ff8 fs1 fc0 sc0 ls1f ws0">Z</div><div class="t m0 x1a h12 yd5 ff6 fs5 fc0 sc0 ls40 ws0">kl</div><div class="t m0 x5c h6 yd6 ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x1 h6 yd7 ff4 fs1 fc0 sc0 ls8 ws0">and<span class="_ _1"> </span>E<span class="_ _7"></span><span class="ff5 ls1f">ð</span></div><div class="t m0 x5d h4 yd8 ffb fs1 fc0 sc0 ls1f ws0">^</div><div class="t m0 x38 h15 yd7 ff8 fs1 fc0 sc0 ls1f ws0">Z</div><div class="t m0 x5e h12 yd9 ff6 fs5 fc0 sc0 ls47 ws0">ij</div><div class="t m0 x5f h4 yda ffb fs1 fc0 sc0 ls1f ws0">^</div><div class="t m0 x46 h15 ydb ff8 fs1 fc0 sc0 ls1f ws0">Z</div><div class="t m0 x60 h12 yd9 ff6 fs5 fc0 sc0 ls40 ws0">kl</div><div class="t m0 x1f h6 ydc ff5 fs1 fc0 sc0 ls1f ws0">Þ<span class="_ _1"> </span><span class="ff4 lsf">differs<span class="_ _4"> </span>by</span></div><div class="t m0 x19 h10 ydd ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x19 h12 yde ff6 fs5 fc0 sc0 ls1f ws0">r</div><div class="t m0 x61 h6 ydc ff5 fs1 fc0 sc0 ls1f ws0">½<span class="ff8">M</span></div><div class="t m0 x62 h12 yd9 ff6 fs5 fc0 sc0 ls48 ws0">ik</div><div class="t m0 x63 h6 ydc ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff8">M</span></div><div class="t m0 x64 h12 yd9 ff6 fs5 fc0 sc0 ls49 ws0">jl</div><div class="t m0 x65 h6 ydc ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x66 h14 ydf ff5 fs5 fc0 sc0 ls1f ws0"></div><div class="t m0 x67 h6 ydc ff5 fs1 fc0 sc0 ls1f ws0">þ<span class="_ _6"> </span><span class="ff8">M</span></div><div class="t m0 x68 h12 yd9 ff6 fs5 fc0 sc0 ls47 ws0">il</div><div class="t m0 x69 h6 ydc ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff8">M</span></div><div class="t m0 x6a h12 yd9 ff6 fs5 fc0 sc0 ls49 ws0">jk</div><div class="t m0 x6b h6 ydc ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x6c h14 ydf ff5 fs5 fc0 sc0 ls1f ws0"></div><div class="t m0 x6d h6 ydc ff5 fs1 fc0 sc0 ls1f ws0"><span class="ff4 ls4a">.T<span class="_ _10"></span>h<span class="_ _f"></span>i<span class="_ _10"></span>s</span></div><div class="t m0 x1 h4 ye0 ff4 fs1 fc0 sc0 ls1b ws0">suggests<span class="_ _5"> </span>that<span class="_ _5"> </span>we<span class="_ _3"> </span>can<span class="_ _5"> </span>approx<span class="ls6">imate<span class="_ _5"> </span>the<span class="_ _5"> </span>distribution<span class="_ _3"> </span>of<span class="_ _5"> </span><span class="ff8 ls1f">Z<span class="_ _6"> </span></span>by<span class="_ _3"> </span>a</span></div><div class="t m0 x1 h8 ye1 ff4 fs1 fc0 sc0 ls42 ws0">complex<span class="_ _b"> </span>Wis<span class="_"> </span>hart<span class="_ _b"> </span>distributi<span class="_"> </span>on<span class="_ _b"> </span>with<span class="_ _d"> </span>parameters<span class="_ _b"> </span><span class="ff7 ls1f">r<span class="_ _12"></span><span class="ff4">,<span class="_ _d"> </span><span class="ff7">t<span class="_ _d"> </span></span><span class="ls8">and</span></span></span></div><div class="t m0 x1 h6 ye2 fff fs1 fc0 sc0 ls1f ws0">R<span class="_ _6"> </span><span class="ff5">þ</span></div><div class="t m0 x56 h10 ye3 ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x56 h12 ye4 ff6 fs5 fc0 sc0 ls1f ws0">r</div><div class="t m0 x57 h15 ye5 ff8 fs1 fc0 sc0 ls1f ws0">M</div><div class="t m0 x58 h12 ye6 ff6 fs5 fc0 sc0 ls1f ws0">H</div><div class="t m0 x46 h6 ye5 ff8 fs1 fc0 sc0 ls1f ws0">M<span class="_ _1e"></span><span class="ff4 ls24">,<span class="_ _1"> </span>i.e.,<span class="_ _1"> </span></span>Z<span class="_ _1"> </span><span class="ff5"><span class="_ _3"> </span><span class="ff6">W</span></span></div><div class="t m0 x19 h12 ye7 ff6 fs5 fc0 sc0 ls1f ws0">c</div><div class="t m0 x6e h12 ye8 ff6 fs5 fc0 sc0 ls1f ws0">t</div><div class="t m0 x6f h6 ye5 ff6 fs1 fc0 sc0 ls1f ws0">r<span class="ff9">;<span class="_ _14"> </span><span class="fff">R<span class="_ _6"> </span><span class="ff5">þ</span></span></span></div><div class="t m0 x70 h10 ye3 ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x70 h12 ye4 ff6 fs5 fc0 sc0 ls1f ws0">r</div><div class="t m0 x71 h15 ye5 ff8 fs1 fc0 sc0 ls1f ws0">M</div><div class="t m0 x72 h12 ye6 ff6 fs5 fc0 sc0 ls1f ws0">H</div><div class="t m0 x73 h15 ye5 ff8 fs1 fc0 sc0 ls1f ws0">M</div><div class="t m0 x2b h16 ye9 ff10 fs1 fc0 sc0 ls4b ws0"></div><div class="t m0 x74 h4 ye5 ff4 fs1 fc0 sc0 ls1f ws0">.</div><div class="t m0 x3 h4 yea ff4 fs1 fc0 sc0 lsc ws0">For<span class="_ _1d"> </span>the<span class="_ _1d"> </span>case<span class="_ _1d"> </span>where<span class="_ _1d"> </span><span class="ff8 ls1f">Z<span class="_ _0"> </span></span><span class="ls8">follows<span class="_ _1d"> </span>a<span class="_ _1d"> </span>complex<span class="_ _1d"> </span>Wishart</span></div><div class="t m0 x1 h4 yeb ff4 fs1 fc0 sc0 ls6 ws0">distribution,<span class="_ _4"> </span>the<span class="_ _8"> </span>probability<span class="_ _8"> </span><span class="ls4">density<span class="_ _8"> </span>function<span class="_ _8"> </span>(<span class="_ _14"> </span>p.d.f.)<span class="_ _4"> </span>of<span class="_ _8"> </span><span class="ffd ls1f">g</span></span></div><div class="t m1 x5c h9 yec ff7 fs2 fc0 sc0 ls1f ws0">k</div><div class="t m0 x1 h4 yed ff4 fs1 fc0 sc0 ls3e ws0">is<span class="_ _1"> </span>given<span class="_ _1"> </span>by<span class="_ _1"> </span><span class="fc1 ls12">[4]</span></div><div class="t m0 x4c h6 yee ff6 fs1 fc0 sc0 ls1f ws0">f<span class="_ _13"> </span><span class="ff5">ð<span class="_ _7"></span><span class="ffd">g</span></span></div><div class="t m0 x37 h12 yef ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x75 h6 yf0 ff5 fs1 fc0 sc0 ls4c ws0">Þ¼</div><div class="t m0 x76 h6 yf1 ff4 fs1 fc0 sc0 ls2b ws0">exp<span class="ff5 ls27">ð<span class="_ _27"></span><span class="ffd ls1f">g</span></span></div><div class="t m0 x77 h12 yf2 ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x2f h11 yf1 ff9 fs1 fc0 sc0 ls1f ws0">=<span class="ffd">g</span></div><div class="t m0 x18 hf yf2 ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 x78 h6 yf1 ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x42 h13 yf3 ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x76 hf yf4 ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 x43 h6 yf5 ff11 fs1 fc0 sc0 ls1f ws0">G<span class="ff5">ð<span class="ff6">r<span class="_ _6"> </span></span><span class="_ _6"> </span><span class="ff6">t<span class="_ _5"> </span></span>þ<span class="_ _6"> </span><span class="ff4">1</span>Þ</span></div><div class="t m0 x79 h13 yf6 ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x50 h12 yf7 ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x7a h13 yf5 ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x7b hf yf4 ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 x6f h16 yf8 ff10 fs1 fc0 sc0 ls4d ws0"> !</div><div class="t m0 x7c h14 yf9 ff6 fs5 fc0 sc0 ls1f ws0">r<span class="ff5"></span>t</div><div class="t m0 x7d h6 yf0 ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _1f"> </span><span class="ff6">k<span class="_ _2"> </span><span class="ff5">¼<span class="_ _a"> </span><span class="ff4">1</span></span></span>;<span class="_ _14"> </span><span class="ffb ls31">...</span>;<span class="_ _14"> </span><span class="ff6">t<span class="_ _18"> </span><span class="ff5">ð<span class="ff4">4</span>Þ</span></span></div><div class="t m0 x4 h4 yfa ff4 fs1 fc0 sc0 ls2d ws0">where</div><div class="t m0 x7e h13 yfb ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x7f hf yfc ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 x80 h6 yfd ff5 fs1 fc0 sc0 ls1f ws0">¼<span class="_ _a"> </span><span class="ffd">g</span></div><div class="t m0 x81 h12 yfc ff6 fs5 fc0 sc0 ls1f ws0">o</div><div class="t m0 x82 h4 yfe ffb fs1 fc0 sc0 ls1f ws0">^</div><div class="t m0 x83 h17 yff fff fs1 fc0 sc0 ls1f ws0">R</div><div class="t m0 xf h14 y100 ff5 fs5 fc0 sc0 ls1f ws0"><span class="ff4">1</span></div><div class="t m0 x84 h16 y101 ff10 fs1 fc0 sc0 ls4e ws0">hi</div><div class="t m0 x85 h12 y102 ff6 fs5 fc0 sc0 ls40 ws0">kk</div><div class="t m0 x86 h16 y101 ff10 fs1 fc0 sc0 ls1f ws0">.</div><div class="t m0 x87 h6 y103 ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4">5</span>Þ</div><div class="t m0 x4 h6 y104 ff4 fs1 fc0 sc0 ls8 ws0">and<span class="_ _a"> </span><span class="ff11 ls1f">G<span class="ff4">(<span class="_ _12"></span><span class="ff7">r<span class="_ _12"></span><span class="ff5"><span class="ff7">t<span class="_ _12"></span><span class="ff4 ls6">+1)<span class="_ _a"> </span>represents<span class="_ _a"> </span>the<span class="_ _2"> </span>Gamma<span class="_ _a"> </span>function.<span class="_ _a"> </span>Also<span class="_ _a"> </span>we</span></span></span></span></span></span></div><div class="t m0 x4 h4 y105 ff4 fs1 fc0 sc0 ls6 ws0">can<span class="_ _1"> </span>obtain<span class="_ _1"> </span>the<span class="_ _1"> </span>cumulative<span class="_ _1"> </span>density<span class="_ _1"> </span>function<span class="_ _3"> </span>(c.d.f.)<span class="_ _1"> </span>of<span class="_ _4"> </span><span class="ffd ls1f">g</span></div><div class="t m1 x88 h9 y106 ff7 fs2 fc0 sc0 ls1f ws0">k</div><div class="t m0 x89 h4 y107 ff4 fs1 fc0 sc0 ls15 ws0">as</div><div class="t m0 x8a h6 y108 ff6 fs1 fc0 sc0 ls1f ws0">F<span class="_ _14"> </span><span class="ff5">ð<span class="ffd">g</span></span></div><div class="t m0 x8b h12 y109 ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x8c h6 y10a ff5 fs1 fc0 sc0 ls4c ws0">Þ¼</div><div class="t m0 x8d h6 y10b ffd fs1 fc0 sc0 ls1f ws0">g<span class="ff5">ð<span class="ff6">r<span class="_ _6"> </span></span><span class="_ _6"> </span><span class="ff6">t<span class="_ _5"> </span></span>þ<span class="_ _6"> </span><span class="ff4">1<span class="ff9">;<span class="_ _14"> </span></span></span></span>g</div><div class="t m0 x85 h12 y10c ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x8e h11 y10b ff9 fs1 fc0 sc0 ls1f ws0">=<span class="ffd">g</span></div><div class="t m0 x8f hf y10c ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 x90 h6 y10b ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x91 h6 y10d ff11 fs1 fc0 sc0 ls1f ws0">G<span class="ff5">ð<span class="ff6">r<span class="_ _6"> </span></span><span class="_ _6"> </span><span class="ff6">t<span class="_ _5"> </span></span>þ<span class="_ _6"> </span><span class="ff4">1</span>Þ</span></div><div class="t m0 x87 h6 y10e ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4">6</span>Þ</div><div class="t m0 x4 h8 y10f ff4 fs1 fc0 sc0 ls2d ws0">where<span class="_ _11"> </span><span class="ffd ls1f">g<span class="ff4">(<span class="_ _12"></span><span class="ff7">a<span class="_ _12"></span><span class="ff4">,<span class="_ _28"> </span><span class="ff7">z<span class="_ _12"></span><span class="ff4 ls3">)<span class="_ _0"> </span>denotes<span class="_ _11"> </span>the<span class="_ _0"> </span>incomplete<span class="_ _0"> </span>Gamma<span class="_ _0"> </span>function</span></span></span></span></span></span></div><div class="t m0 x4 h6 y110 ff4 fs1 fc0 sc0 ls14 ws0">defined<span class="_ _1"> </span>as<span class="_ _1"> </span><span class="ffd ls1f">g<span class="ff5">ð<span class="ff6">a<span class="ff9">;<span class="_ _28"> </span></span>z</span><span class="ls4c">Þ¼</span></span></span></div><div class="t m0 xb h16 y111 ff10 fs1 fc0 sc0 ls1f ws0">R</div><div class="t m0 x92 h12 y112 ff6 fs5 fc0 sc0 ls1f ws0">z</div><div class="t m0 x93 h10 y113 ff4 fs5 fc0 sc0 ls1f ws0">0</div><div class="t m0 x94 h18 y114 ff6 fs1 fc0 sc0 ls1f ws0">e</div><div class="t m0 x9 h14 y115 ff5 fs5 fc0 sc0 ls1f ws0"><span class="ff6">x</span></div><div class="t m0 x95 h18 y114 ff6 fs1 fc0 sc0 ls1f ws0">x</div><div class="t m0 x81 h14 y115 ff6 fs5 fc0 sc0 ls1f ws0">a<span class="ff5"><span class="ff4">1</span></span></div><div class="t m0 x96 h4 y114 ff4 fs1 fc0 sc0 ls1f ws0">d<span class="ff6">x</span>.</div><div class="t m0 x5 h4 y116 ff4 fs1 fc0 sc0 ls5 ws0">Supposing<span class="_ _11"> </span>quadrature<span class="_ _11"> </span>phase<span class="_ _11"> </span>shift<span class="_ _11"> </span>keying<span class="_ _11"> </span>(<span class="_ _7"></span>QPSK)<span class="_ _11"> </span>is</div><div class="t m0 x4 h4 y117 ff4 fs1 fc0 sc0 ls9 ws0">employed<span class="_ _3"> </span>as<span class="_ _3"> </span>the<span class="_ _3"> </span>modulation/demodulation<span class="_ _1"> </span>scheme,<span class="_ _3"> </span>we<span class="_ _3"> </span>can</div><div class="t m0 x4 h4 y118 ff4 fs1 fc0 sc0 ls1b ws0">then<span class="_ _3"> </span>obtain<span class="_ _3"> </span>the<span class="_ _5"> </span>average<span class="_ _3"> </span>BER<span class="_ _3"> </span>as<span class="_ _3"> </span>(see<span class="_ _3"> </span>Section<span class="_ _3"> </span>8.1<span class="_ _3"> </span>for<span class="_ _3"> </span>details)</div><div class="t m0 x97 h18 y119 ff6 fs1 fc0 sc0 ls1f ws0">P</div><div class="t m0 x98 h12 y11a ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x99 h6 y11b ff5 fs1 fc0 sc0 ls1f ws0">¼</div><div class="t m0 x9a h4 y11c ff4 fs1 fc0 sc0 ls1f ws0">1</div><div class="t m0 x9a h4 y11d ff4 fs1 fc0 sc0 ls1f ws0">2</div><div class="t m0 x9b h6 y11e ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 x94 h16 y11f ff10 fs1 fc0 sc0 ls1f ws0">ffiffiffiffiffiffi<span class="_ _29"></span>ffi</div><div class="t m0 x94 h13 y120 ffd fs1 fc0 sc0 ls1f ws0">g</div><div class="t m0 x9 hf y121 ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 x9 h13 y122 ffd fs1 fc0 sc0 ls1f ws0">p</div><div class="t m0 x9c h16 y123 ff10 fs1 fc0 sc0 ls1f ws0">r</div><div class="t m0 x9d h6 y124 ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 x9e h6 y125 ff11 fs1 fc0 sc0 ls1f ws0">G<span class="_ _22"> </span><span class="ff6">r<span class="_ _6"> </span><span class="ff5"><span class="_ _5"> </span></span>t<span class="_ _6"> </span><span class="ff5">þ</span></span></div><div class="t m0 x9f h4 y126 ff4 fs1 fc0 sc0 ls1f ws0">3</div><div class="t m0 x9f h4 y127 ff4 fs1 fc0 sc0 ls1f ws0">2</div><div class="t m0 xa0 h16 y128 ff10 fs1 fc0 sc0 ls4f ws0"></div><div class="t m0 xa1 h6 y129 ff11 fs1 fc0 sc0 ls1f ws0">G<span class="ff5">ð<span class="ff6">r<span class="_ _6"> </span></span><span class="_ _6"> </span><span class="ff6">t<span class="_ _5"> </span></span>þ<span class="_ _6"> </span><span class="ff4">1</span>Þ</span></div><div class="t m0 xa2 h6 y12a ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 xa3 h10 y12b ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 xa4 h18 y12c ff6 fs1 fc0 sc0 ls1f ws0">F</div><div class="t m0 x9c h10 y12b ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 x80 h4 y12d ff4 fs1 fc0 sc0 ls1f ws0">1</div><div class="t m0 x80 h4 y12e ff4 fs1 fc0 sc0 ls1f ws0">2</div><div class="t m0 xa5 h6 y12f ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _28"> </span><span class="ff6">r<span class="_ _5"> </span><span class="ff5"><span class="_ _6"> </span></span>t<span class="_ _5"> </span><span class="ff5">þ</span></span></div><div class="t m0 xa6 h4 y12d ff4 fs1 fc0 sc0 ls1f ws0">3</div><div class="t m0 xa6 h4 y12e ff4 fs1 fc0 sc0 ls1f ws0">2</div><div class="t m0 xa7 h4 y12f ffb fs1 fc0 sc0 ls1f ws0">;</div><div class="t m0 xa8 h4 y12d ff4 fs1 fc0 sc0 ls1f ws0">3</div><div class="t m0 xa8 h4 y12e ff4 fs1 fc0 sc0 ls1f ws0">2</div><div class="t m0 xa9 h6 y12f ffb fs1 fc0 sc0 ls1f ws0">;<span class="_ _1"> </span><span class="ff5"><span class="ffd">g</span></span></div><div class="t m0 xaa hf y130 ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 xab h16 y131 ff10 fs1 fc0 sc0 ls50 ws0"></div><div class="t m0 x87 h6 y132 ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4">7</span>Þ</div><div class="t m0 x4 h4 y133 ff4 fs1 fc0 sc0 ls2d ws0">where</div><div class="t m0 xac h10 y134 ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 xad h18 y135 ff6 fs1 fc0 sc0 ls1f ws0">F</div><div class="t m0 xae h10 y134 ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 xaf h6 y135 ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff6">a<span class="ff9">;<span class="_ _1"> </span></span>b<span class="ffb">;<span class="_ _1"> </span></span>c<span class="ffb">;<span class="_ _1"> </span></span>z</span>Þ<span class="_ _2"> </span><span class="ff4 ls1b">is<span class="_ _a"> </span>the<span class="_ _a"> </span>hypergeometric<span class="_ _2"> </span>function<span class="_ _a"> </span><span class="fc1 ls6">[15]</span><span class="ls1f">.</span></span></div><div class="t m0 x4 h4 y136 ff4 fs1 fc0 sc0 ls9 ws0">In<span class="_ _1"> </span>the<span class="_ _1"> </span>study<span class="_ _1"> </span>of<span class="_ _1"> </span>MIMO<span class="_ _1"> </span>system,<span class="_ _1"> </span>the<span class="_ _1"> </span>integral</div><div class="t m0 xad h6 y137 ff6 fs1 fc0 sc0 ls1f ws0">J<span class="_ _1e"></span><span class="ff5">ð</span>m<span class="ff9">;<span class="_ _14"> </span></span>a<span class="ff9">;<span class="_ _14"> </span></span>b<span class="ff5 ls4c">Þ¼</span></div><div class="t m0 xc h18 y138 ff6 fs1 fc0 sc0 ls1f ws0">a</div><div class="t m0 x80 h12 y139 ff6 fs5 fc0 sc0 ls1f ws0">m</div><div class="t m0 xb0 h6 y13a ff11 fs1 fc0 sc0 ls1f ws0">G<span class="ff5">ð<span class="ff6">m</span>Þ</span></div><div class="t m0 xa h16 y13b ff10 fs1 fc0 sc0 ls1f ws0">Z</div><div class="t m0 xb1 h14 y13c ff5 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 xb2 h10 y13d ff4 fs5 fc0 sc0 ls1f ws0">0</div><div class="t m0 xb3 h18 y13e ff6 fs1 fc0 sc0 ls1f ws0">e</div><div class="t m0 xb4 h14 y13f ff5 fs5 fc0 sc0 ls1f ws0"><span class="ff6 ls51">at</span></div><div class="t m0 xb5 h18 y13e ff6 fs1 fc0 sc0 ls1f ws0">t</div><div class="t m0 xb6 h14 y13f ff6 fs5 fc0 sc0 ls1f ws0">m<span class="ff5"><span class="ff4">1</span></span></div><div class="t m0 xb7 h18 y13e ff6 fs1 fc0 sc0 ls1f ws0">Q</div><div class="t m0 xb8 h16 y140 ff10 fs1 fc0 sc0 ls1f ws0">ffiffiffi<span class="_ _12"></span>ffi</div><div class="t m0 xb8 h18 y13e ff6 fs1 fc0 sc0 ls15 ws0">bt</div><div class="t m0 xb9 h6 y140 ff5 fs1 fc0 sc0 ls1f ws0">p</div><div class="t m0 xba h16 y141 ff10 fs1 fc0 sc0 ls52 ws0"></div><div class="t m0 xe h6 y142 ff4 fs1 fc0 sc0 ls1f ws0">d<span class="ff6">t<span class="_ _2a"> </span><span class="ff5">ð</span></span>8<span class="ff5">Þ</span></div><div class="t m0 x4 h4 y143 ff4 fs1 fc0 sc0 ls9 ws0">frequently<span class="_ _4"> </span>arises.<span class="_ _1"> </span>In<span class="_ _4"> </span>particu<span class="lsc">lar,<span class="_ _1"> </span>it<span class="_ _4"> </span>has<span class="_ _4"> </span>been<span class="_ _1"> </span>shown<span class="_ _4"> </span>that<span class="_ _1"> </span>the</span></div><div class="t m0 x4 h4 y144 ff4 fs1 fc0 sc0 ls5 ws0">integral<span class="_ _1"> </span>has<span class="_ _1"> </span>the<span class="_ _1"> </span>closed-form<span class="_ _1"> </span>result<span class="_ _1"> </span><span class="fc1">[16]</span></div><div class="t m0 xbb h6 y145 ff6 fs1 fc0 sc0 ls1f ws0">J<span class="_ _1e"></span><span class="ff5">ð</span>m<span class="_ _7"></span><span class="ff9">;<span class="_ _14"> </span></span>a<span class="ff9">;<span class="_ _14"> </span></span>b<span class="ff5 ls4c">Þ¼</span></div><div class="t m0 xbc h16 y146 ff10 fs1 fc0 sc0 ls1f ws0">ffiffiffiffiffiffiffi<span class="_ _2b"></span>ffi</div><div class="t m0 xbc h11 y147 ff6 fs1 fc0 sc0 ls1f ws0">c<span class="ff9">=<span class="ffd">p</span></span></div><div class="t m0 xbd h16 y146 ff10 fs1 fc0 sc0 ls1f ws0">p</div><div class="t m0 xbe h6 y148 ff4 fs1 fc0 sc0 ls1f ws0">2<span class="ff5">ð</span>1<span class="_ _6"> </span><span class="ff5">þ<span class="_ _6"> </span><span class="ff6">c</span>Þ</span></div><div class="t m0 xbf h14 y149 ff6 fs5 fc0 sc0 ls1f ws0">m<span class="ff5">þ<span class="ff4">1<span class="ff9">=</span>2</span></span></div><div class="t m0 xb2 h6 y14a ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 xb1 h6 y14b ff11 fs1 fc0 sc0 ls1f ws0">G<span class="ff5">ð<span class="ff6">m<span class="_ _6"> </span></span>þ<span class="_ _6"> </span><span class="ff4">1<span class="ff9">=</span>2</span>Þ</span></div><div class="t m0 xa1 h6 y14c ff11 fs1 fc0 sc0 ls1f ws0">G<span class="ff5">ð<span class="ff6">m<span class="_ _6"> </span></span>þ<span class="_ _6"> </span><span class="ff4">1</span>Þ</span></div><div class="t m0 xbe h6 y14d ff5 fs1 fc0 sc0 ls1f ws0"></div><div class="t m0 xc0 h10 y14e ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 xc1 h18 y14f ff6 fs1 fc0 sc0 ls1f ws0">F</div><div class="t m0 xa3 h10 y14e ff4 fs5 fc0 sc0 ls1f ws0">1</div><div class="t m0 xc2 h6 y14f ff4 fs1 fc0 sc0 ls1f ws0">1<span class="ff9">;<span class="_ _14"> </span><span class="ff6">m<span class="_ _6"> </span><span class="ff5">þ</span></span></span></div><div class="t m0 xc3 h4 y150 ff4 fs1 fc0 sc0 ls1f ws0">1</div><div class="t m0 xc3 h4 y151 ff4 fs1 fc0 sc0 ls1f ws0">2</div><div class="t m0 xc4 h6 y152 ffb fs1 fc0 sc0 ls1f ws0">;<span class="_ _1"> </span><span class="ff6">m<span class="_ _6"> </span><span class="ff5">þ<span class="_ _6"> </span><span class="ff4">1</span></span></span>;</div><div class="t m0 xc5 h4 y150 ff4 fs1 fc0 sc0 ls1f ws0">1</div><div class="t m0 xc6 h6 y151 ff4 fs1 fc0 sc0 ls1f ws0">1<span class="_ _6"> </span><span class="ff5">þ<span class="_ _6"> </span><span class="ff6">c</span></span></div><div class="t m0 xa4 h16 y153 ff10 fs1 fc0 sc0 ls53 ws0"></div><div class="t m0 xc7 h6 y154 ff9 fs1 fc0 sc0 ls1f ws0">;<span class="_ _1f"> </span><span class="ff6">c<span class="_ _a"> </span><span class="ff5">¼</span></span></div><div class="t m0 xc8 h18 y150 ff6 fs1 fc0 sc0 ls1f ws0">b</div><div class="t m0 xc9 h4 y151 ff4 fs1 fc0 sc0 ls1f ws0">2<span class="ff6">a</span></div><div class="t m0 x87 h6 y155 ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4">9</span>Þ</div><div class="t m0 x4 h8 y156 ff4 fs1 fc0 sc0 ls2c ws0">When<span class="_ _4"> </span><span class="ff7 ls1f">m<span class="_ _4"> </span></span><span class="ls1b">is<span class="_ _4"> </span>restricted<span class="_ _4"> </span>to<span class="_ _4"> </span>positive<span class="_ _4"> </span>integer<span class="_ _4"> </span>values,<span class="_ _4"> </span>(9)<span class="_ _8"> </span>can<span class="_ _4"> </span>be</span></div><div class="t m0 x4 h4 y157 ff4 fs1 fc0 sc0 ls15 ws0">further<span class="_ _1"> </span>simplified<span class="_ _1"> </span>to</div><div class="t m0 xca h6 y158 ff6 fs1 fc0 sc0 ls1f ws0">J<span class="_ _1e"></span><span class="ff5">ð</span>m<span class="ff9">;<span class="_ _14"> </span></span>a<span class="ff9">;<span class="_ _14"> </span></span>b<span class="ff5 ls4c">Þ¼</span></div><div class="t m0 xc1 h4 y159 ff4 fs1 fc0 sc0 ls1f ws0">1</div><div class="t m0 xc1 h4 y15a ff4 fs1 fc0 sc0 ls1f ws0">2</div><div class="t m0 xcb h6 y15b ff4 fs1 fc0 sc0 ls1f ws0">1<span class="_ _6"> </span><span class="ff5"><span class="_ _6"> </span><span class="ffd">m</span></span></div><div class="t m0 x95 h16 y15c ff10 fs1 fc0 sc0 ls1f ws0">X</div><div class="t m0 x95 h14 y15d ff6 fs5 fc0 sc0 ls1f ws0">m<span class="ff5"><span class="ff4">1</span></span></div><div class="t m0 x95 h14 y15e ff6 fs5 fc0 sc0 ls1f ws0">k<span class="_ _28"> </span><span class="ff5">¼<span class="_ _13"></span><span class="ff4">0</span></span></div><div class="t m0 xb4 h4 y15f ff4 fs1 fc0 sc0 ls1f ws0">2<span class="ff6">k</span></div><div class="t m0 xcc h18 y160 ff6 fs1 fc0 sc0 ls1f ws0">k</div><div class="t m0 xa1 h16 y161 ff10 fs1 fc0 sc0 ls54 ws0"></div><div class="t m0 xc6 h6 y159 ff4 fs1 fc0 sc0 ls1f ws0">1<span class="_ _6"> </span><span class="ff5"><span class="_ _6"> </span><span class="ffd">m</span></span></div><div class="t m0 xcd h10 y162 ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 x7 h4 y163 ff4 fs1 fc0 sc0 ls1f ws0">4</div><div class="t m0 xa8 h16 y164 ff10 fs1 fc0 sc0 ls55 ws0"></div><div class="t m0 xce h12 y165 ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 xcf h16 y166 ff10 fs1 fc0 sc0 ls56 ws0">"#</div><div class="t m0 x88 h6 y167 ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4 ls15">10<span class="_ _7"></span></span>Þ</div><div class="t m0 x4 h4 y168 ff4 fs1 fc0 sc0 ls2d ws0">where<span class="_ _1"> </span><span class="ffd ls1f">m</span></div><div class="t m0 xad h19 y169 ff12 fs1 fc0 sc0 ls1f ws0">9</div><div class="t m0 xd0 h16 y16a ff10 fs1 fc0 sc0 ls1f ws0">ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi<span class="_ _29"></span>ffi</div><div class="t m0 xd0 h6 y16b ff6 fs1 fc0 sc0 ls1f ws0">c<span class="ff9">=<span class="ff5">ð<span class="ff4">1<span class="_ _6"> </span></span>þ<span class="_ _6"> </span></span></span>c<span class="ff5">Þ</span></div><div class="t m0 xd1 h16 y16a ff10 fs1 fc0 sc0 ls1f ws0">p</div><div class="t m0 xd2 h4 y16b ff4 fs1 fc0 sc0 ls2a ws0">and</div><div class="t m0 xd3 h4 y16c ff4 fs1 fc0 sc0 ls1f ws0">2<span class="ff6">k</span></div><div class="t m0 xc4 h18 y16d ff6 fs1 fc0 sc0 ls1f ws0">k</div><div class="t m0 x9d h16 y16e ff10 fs1 fc0 sc0 ls54 ws0"></div><div class="t m0 xd4 h4 y16b ff4 fs1 fc0 sc0 ls6 ws0">represents<span class="_ _1"> </span>the<span class="_ _1"> </span>binomial</div><div class="t m0 x4 h4 y16f ff4 fs1 fc0 sc0 ls6 ws0">coefficient.<span class="_ _5"> </span>Substituting<span class="_ _5"> </span>(4)<span class="_ _3"> </span>into<span class="_ _5"> </span>(35)<span class="_ _3"> </span>and<span class="_ _5"> </span>using<span class="_ _3"> </span>the<span class="_ _5"> </span>results<span class="_ _5"> </span>in</div><div class="t m0 x4 h4 y170 ff4 fs1 fc0 sc0 ls3 ws0">(8),<span class="_ _1"> </span>we<span class="_ _1"> </span>have<span class="_ _1"> </span>another<span class="_ _1"> </span>expression<span class="_ _1"> </span>for<span class="_ _1"> </span>the<span class="_ _3"> </span>average<span class="_ _1"> </span>BER,<span class="_ _1"> </span>i.e.</div><div class="t m0 xd5 h18 y171 ff6 fs1 fc0 sc0 ls1f ws0">P</div><div class="t m0 xd6 h12 y172 ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 x8c h6 y173 ff5 fs1 fc0 sc0 ls1f ws0">¼<span class="_ _a"> </span><span class="ff6">J<span class="_ _1e"></span></span>ð<span class="ff6">r<span class="_ _5"> </span></span><span class="_ _6"> </span><span class="ff6">t<span class="_ _5"> </span></span>þ<span class="_ _19"> </span><span class="ff4">1<span class="_"> </span><span class="ff9">;<span class="_ _14"> </span></span>1<span class="ff9">;<span class="_ _14"> </span></span>2<span class="ffd">g</span></span></div><div class="t m0 xa9 hf y174 ff6 fs5 fc0 sc0 ls1f ws0">o<span class="ff9">;</span>k</div><div class="t m0 xc5 h6 y173 ff5 fs1 fc0 sc0 ls57 ws0">Þð<span class="_ _2c"></span><span class="ff4 ls15">11<span class="_ _7"></span><span class="ff5 ls1f">Þ</span></span></div><div class="t m0 x4 h4 y175 ff4 fs1 fc0 sc0 ls3 ws0">We<span class="_ _a"> </span>note<span class="_ _2"> </span>that<span class="_ _a"> </span>(7)<span class="_ _2"> </span>and<span class="_ _a"> </span>(11)<span class="_ _2"> </span>are<span class="_ _a"> </span>equivalent<span class="_ _2"> </span>with<span class="_ _a"> </span>the<span class="_ _2"> </span>same</div><div class="t m0 x4 h4 y176 ff4 fs1 fc0 sc0 ls58 ws0">p<span class="_ _2d"></span>a<span class="_ _2d"></span>r<span class="_ _2d"></span>a<span class="_ _2d"></span>m<span class="_ _2d"></span>e<span class="_ _2d"></span>t<span class="_ _2d"></span>e<span class="_ _2d"></span>r<span class="_ _2d"></span>s<span class="_ _2d"></span>.O<span class="_ _2d"></span>n<span class="_ _2d"></span>c<span class="_ _2d"></span>ew<span class="_ _2d"></span>eh<span class="_ _2d"></span>a<span class="_ _2d"></span>v<span class="_ _2d"></span>et<span class="_ _2d"></span>h<span class="_ _2d"></span>ep<span class="_ _2d"></span>.<span class="_ _2d"></span>d<span class="_ _2d"></span><span class="ls4">.f.<span class="_ _3"> </span>of<span class="_ _5"> </span>post-detection<span class="_ _5"> </span>SNR,</span></div><div class="t m0 x4 h4 y177 ff4 fs1 fc0 sc0 ls4 ws0">the<span class="_ _8"> </span>average<span class="_ _8"> </span>bit<span class="_ _8"> </span>error<span class="_ _8"> </span>rate<span class="_ _8"> </span>or<span class="_ _8"> </span>symbol<span class="_ _8"> </span>error<span class="_ _8"> </span>rate<span class="_ _8"> </span>(SER)<span class="_ _8"> </span>for</div><div class="t m0 x4 h4 y178 ff4 fs1 fc0 sc0 ls3 ws0">other<span class="_ _1"> </span>modulation<span class="_ _3"> </span>schemes<span class="_ _1"> </span>(e.g.,<span class="_ _1"> </span>M-PSK,<span class="_ _1"> </span>M-QAM)<span class="_ _3"> </span>can<span class="_ _1"> </span>be</div><div class="t m0 x4 h4 y179 ff4 fs1 fc0 sc0 ls3 ws0">obtained<span class="_ _b"> </span>by<span class="_ _d"> </span>replacing<span class="_ _b"> </span>the<span class="_ _b"> </span>cor<span class="ls2">responding<span class="_ _b"> </span>expression<span class="_ _d"> </span>for</span></div><div class="t m0 x4 h4 y17a ff4 fs1 fc0 sc0 ls2 ws0">conditional<span class="_ _1"> </span>BER<span class="_ _1"> </span>or<span class="_ _1"> </span>SER<span class="_ _1"> </span>in<span class="_ _1"> </span>(30)<span class="_ _1"> </span><span class="fc1 ls27">[17]</span><span class="ls1f">.</span></div><div class="t m0 x5 h4 y17b ff4 fs1 fc0 sc0 ls2 ws0">In<span class="_ _1"> </span>the<span class="_ _4"> </span>following,<span class="_ _4"> </span>we<span class="_ _1"> </span>also<span class="_ _4"> </span>attempt<span class="_ _1"> </span>to<span class="_ _4"> </span>derive<span class="_ _1"> </span>the<span class="_ _4"> </span>diversity</div><div class="t m0 x4 h4 y17c ff4 fs1 fc0 sc0 ls2 ws0">order<span class="_ _6"> </span>of<span class="_ _6"> </span>the<span class="_ _5"> </span>MIMO<span class="_ _6"> </span>system<span class="_ _6"> </span>under<span class="_ _5"> </span>study.<span class="_ _6"> </span>Diversity<span class="_ _6"> </span>order<span class="_ _6"> </span>has</div><div class="t m0 x4 h4 y17d ff4 fs1 fc0 sc0 ls9 ws0">been<span class="_ _6"> </span>defined<span class="_ _5"> </span>as<span class="_ _5"> </span>the<span class="_ _5"> </span>absolute<span class="_ _5"> </span>values<span class="_ _6"> </span>of<span class="_ _5"> </span>the<span class="_ _5"> </span>slopes<span class="_ _5"> </span>of<span class="_ _5"> </span>the<span class="_ _5"> </span>error</div><div class="t m0 x4 h4 y17e ff4 fs1 fc0 sc0 ls4 ws0">probability<span class="_ _4"> </span>curves<span class="_ _4"> </span>plotted<span class="_ _4"> </span>on<span class="_ _4"> </span>a<span class="_ _8"> </span>log-log<span class="_ _4"> </span>scale<span class="_ _4"> </span>at<span class="_ _4"> </span>high<span class="_ _4"> </span>SNR</div><div class="t m0 x4 h4 y17f ff4 fs1 fc0 sc0 lse ws0">values<span class="_ _1"> </span><span class="fc1 ls27">[18]</span><span class="ls59">,<span class="_ _1"> </span>i.e.</span></div><div class="t m0 xa4 h18 y180 ff6 fs1 fc0 sc0 ls1f ws0">D</div><div class="t m0 xc2 h12 y181 ff6 fs5 fc0 sc0 ls1f ws0">e</div><div class="t m0 xbf h19 y182 ff12 fs1 fc0 sc0 ls1f ws0">9</div><div class="t m0 xd7 h4 y183 ff4 fs1 fc0 sc0 ls2a ws0">lim</div><div class="t m0 xd8 h1a y184 ffd fs5 fc0 sc0 ls1f ws0">s</div><div class="t m0 xd7 h1b y185 ff4 fs6 fc0 sc0 ls1f ws0">2</div><div class="t m0 xa h14 y184 ff5 fs5 fc0 sc0 ls1f ws0">!<span class="ff4">0</span></div><div class="t m0 xb1 h1c y185 ff5 fs6 fc0 sc0 ls1f ws0">þ</div><div class="t m0 x96 h6 y186 ff4 fs1 fc0 sc0 ls59 ws0">log<span class="_ _7"></span><span class="ff5 ls1f">ð<span class="ff6">P</span></span></div><div class="t m0 xb6 h12 y187 ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 xa8 h6 y186 ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x96 h6 y188 ff4 fs1 fc0 sc0 ls3c ws0">log<span class="_ _7"></span><span class="ff5 ls1f">ð<span class="ffd">s</span></span></div><div class="t m0 xb6 h10 y189 ff4 fs5 fc0 sc0 ls1f ws0">2</div><div class="t m0 xa8 h6 y18a ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x88 h6 y18b ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4 ls15">12<span class="_ _7"></span></span>Þ</div><div class="t m0 x4 h4 y18c ff4 fs1 fc0 sc0 ls2d ws0">where<span class="_ _1"> </span><span class="ffd ls1f">s</span></div><div class="t m1 xad h7 y18d ff4 fs2 fc0 sc0 ls1f ws0">2</div><div class="t m0 xd9 h4 y18e ff4 fs1 fc0 sc0 ls6 ws0">denotes<span class="_ _4"> </span>the<span class="_ _1"> </span>noise<span class="_ _4"> </span>power.<span class="_ _1"> </span>Alternatively,<span class="_ _4"> </span>based<span class="_ _1"> </span>on</div><div class="t m0 x4 h4 y18f ff4 fs1 fc0 sc0 ls6 ws0">(3),<span class="_ _1"> </span>(12)<span class="_ _1"> </span>can<span class="_ _1"> </span>be<span class="_ _1"> </span>expressed<span class="_ _1"> </span>as</div><div class="t m0 xc1 h18 y190 ff6 fs1 fc0 sc0 ls1f ws0">D</div><div class="t m0 x7e h12 y191 ff6 fs5 fc0 sc0 ls1f ws0">e</div><div class="t m0 xda h6 y192 ff5 fs1 fc0 sc0 ls3a ws0">¼<span class="_ _3"> </span><span class="ff4 ls2a">li<span class="_ _7"></span>m</span></div><div class="t m0 x95 h1a y193 ffd fs5 fc0 sc0 ls1f ws0">g</div><div class="t m0 x9d h1d y194 ff6 fs6 fc0 sc0 ls1f ws0">o</div><div class="t m0 xb2 h14 y193 ff5 fs5 fc0 sc0 ls5a ws0">!þ1</div><div class="t m0 xdb h6 y195 ff4 fs1 fc0 sc0 ls59 ws0">log<span class="_ _7"></span><span class="ff5 ls1f">ð<span class="ff6">P</span></span></div><div class="t m0 xdc h12 y196 ff6 fs5 fc0 sc0 ls1f ws0">k</div><div class="t m0 xdd h6 y195 ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 xdb h6 y197 ff4 fs1 fc0 sc0 ls59 ws0">log<span class="_ _7"></span><span class="ff5 ls1f">ð<span class="ffd">g</span></span></div><div class="t m0 xdc h12 y198 ff6 fs5 fc0 sc0 ls1f ws0">o</div><div class="t m0 xdd h6 y199 ff5 fs1 fc0 sc0 ls1f ws0">Þ</div><div class="t m0 x88 h6 y19a ff5 fs1 fc0 sc0 ls1f ws0">ð<span class="ff4 ls15">13<span class="_ _7"></span></span>Þ</div><div class="t m0 x4 h4 y19b ff4 fs1 fc0 sc0 ls6 ws0">Considering<span class="_ _22"> </span>the<span class="_ _17"> </span>limit<span class="_ _22"> </span>of<span class="_ _22"> </span>the<span class="_ _17"> </span>hypergeometr<span class="ls14">ic<span class="_ _22"> </span>function</span></div><div class="t m1 x4 h7 y19c ff4 fs2 fc0 sc0 ls1f ws0">2</div><div class="t m0 xbb h8 y60 ff7 fs1 fc0 sc0 ls1f ws0">F</div><div class="t m1 xde h7 y19c ff4 fs2 fc0 sc0 ls1f ws0">1</div><div class="t m0 xdf h8 y60 ff4 fs1 fc0 sc0 lsc ws0">in<span class="_ _0"> </span>(9),<span class="_ _1d"> </span>it<span class="_ _0"> </span>can<span class="_ _1d"> </span>be<span class="_ _0"> </span>readily<span class="_ _1d"> </span>shown<span class="_ _0"> </span>that<span class="_ _0"> </span>as<span class="_ _1d"> </span><span class="ff7 ls1f">c<span class="_ _0"> </span></span>tends</div><div class="t m0 x1 he y19d ff7 fs3 fc0 sc0 ls5b ws0">IEE<span class="_ _5"> </span>Proc.-Commun.,<span class="_ _6"> </span>Vol.<span class="_ _5"> </span>153,<span class="_ _5"> </span>No.<span class="_ _5"> </span>1,<span class="_ _5"> </span>February<span class="_ _6"> </span>2006<span class="_ _1a"> </span><span class="ff4 fs4 ls38">75</span></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div><div class="d m2"></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div>
