McGraw-Hill-DSP-Digital_Signal_Processin

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数字信号处理英文第二版课后习题答案 数字信号处理英文第二版课后习题答案 数字信号处理英文第二版课后习题答案
McGraw-Hill-DSP-Digital_Signal_Processing_Mitra_3rd_Solution(英文改编版).rar
  • McGraw-Hill-DSP-Digital_Signal_Processing_Mitra_3rd_Solution(英文改编版)
  • Chapter5_SM.pdf
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    503KB
  • Chapter4_SM.pdf
    331.4KB
  • Chapter3_SM.pdf
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  • Chapter8_SM.pdf
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内容介绍
<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/638c2231e53e5839a7221608/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/638c2231e53e5839a7221608/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Chapter 5 </div><div class="t m0 x2 h3 y2 ff2 fs1 fc1 sc0 ls1 ws1"> </div><div class="t m0 x2 h4 y3 ff3 fs1 fc2 sc0 ls1 ws1">5.1 <span class="_ _0"> </span><span class="ff2 fc1 ls2">Let </span></div><div class="t m0 x3 h3 y4 ff2 fs1 fc1 sc0 ls3 ws1">Then <span class="_ _1"> </span><span class="ls4 ws2"> Since <span class="_ _2"> </span><span class="ls5 ws3"> is </span></span></div><div class="t m0 x4 h3 y5 ff2 fs1 fc1 sc0 ls6 ws4">periodic in </div><div class="t m0 x5 h3 y6 ff2 fs1 fc1 sc0 ls1 ws1"> with a period <span class="_ _3"> </span><span class="ls7 ws5"> Therefore <span class="_ _4"> </span></span> </div><div class="t m0 x6 h3 y7 ff2 fs1 fc1 sc0 ls8 ws6"> hence <span class="_ _2"> </span><span class="ls9 ws7"> is also periodic in <span class="_ _5"> </span><span class="lsa ws8"> with a period <span class="_ _6"> </span><span class="ls1 ws1"> </span></span></span></div><div class="t m0 x7 h5 y8 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x8 h5 y3 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _7"></span>=</div><div class="t m0 x9 h6 y9 ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 xa h6 ya ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 xb h7 y9 ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 xc h7 ya ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x6 h8 y4 ff5 fs1 fc1 sc0 ls1 ws1">].<span class="_ _8"></span>[</div><div class="t m0 xd h8 yb ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xe h8 y4 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 xf h8 yc ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x10 h8 y4 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 x11 h8 yc ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x12 h9 y9 ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 x12 h9 ya ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x13 ha y4 ff6 fs1 fc1 sc0 ls1 ws1">r<span class="_ _b"></span>n<span class="_ _c"></span>h<span class="_ _d"></span>r<span class="_ _a"></span>x<span class="_ _e"></span>n<span class="_ _f"></span>y</div><div class="t m0 x14 h5 yd ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x15 h5 y4 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _10"></span>+<span class="_ _7"></span>=<span class="_ _11"></span>+</div><div class="t m0 x16 h6 y9 ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 x17 h6 ya ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x18 h7 y9 ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 x19 h7 ya ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x1a h8 y4 ff5 fs1 fc1 sc0 ls1 ws1">].<span class="_ _12"></span>[</div><div class="t m0 x1b h8 yb ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x1c h8 y4 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 x1d h8 yc ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x1e h8 y4 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _13"></span>[</div><div class="t m0 x1f h8 yc ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x20 h9 y9 ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 x20 h9 ya ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x21 h8 y4 ff6 fs1 fc1 sc0 ls1 ws1">r<span class="_ _14"></span>kN<span class="_ _15"></span>n<span class="_ _c"></span>h<span class="_ _d"></span>r<span class="_ _a"></span>x<span class="_ _16"></span>kN<span class="_ _15"></span>n<span class="_ _f"></span>y<span class="_ _17"> </span><span class="ff5">]<span class="_ _a"></span>[</span></div><div class="t m0 x22 h8 yb ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x23 ha y4 ff6 fs1 fc1 sc0 ls1 ws1">n<span class="_ _c"></span>h</div><div class="t m0 x24 hb ye ff6 fs3 fc1 sc0 ls1 ws1">n</div><div class="t m0 x25 hc yf ff5 fs4 fc1 sc0 ls1 ws1">].<span class="_ _8"></span>[</div><div class="t m0 x20 hc y10 ff5 fs4 fc1 sc0 ls1 ws1">~</div><div class="t m0 x1e hc yf ff5 fs4 fc1 sc0 ls1 ws1">]<span class="_ _16"></span>[</div><div class="t m0 x26 hc y10 ff5 fs4 fc1 sc0 ls1 ws1">~</div><div class="t m0 x27 hd yf ff5 fs4 fc1 sc0 ls1 ws1">,<span class="_ _18"> </span><span class="ff6">r<span class="_ _b"></span>n<span class="_ _c"></span>h<span class="_ _19"></span>r<span class="_ _14"></span>kN<span class="_ _1a"></span>n<span class="_ _c"></span>h<span class="_ _1b"></span>N<span class="_ _1c"> </span><span class="ff4">&#8722;<span class="_ _1d"></span>=<span class="_ _19"></span>&#8722;<span class="_ _1e"></span>+<span class="_ _1f"> </span><span class="ff5 fs5">]<span class="_ _13"></span>[</span></span></span></div><div class="t m0 x28 he y11 ff5 fs5 fc1 sc0 ls1 ws1">~</div><div class="t m0 x29 hf yf ff6 fs5 fc1 sc0 ls1 ws1">kN<span class="_ _15"></span>n<span class="_ _f"></span>y<span class="_ _20"> </span><span class="ff4">+</span></div><div class="t m0 x2a h5 y12 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x2b h5 y13 ff4 fs1 fc1 sc0 ls1 ws1">=<span class="_ _19"></span>&#8722;<span class="_ _21"></span>=</div><div class="t m0 x2c h6 y14 ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 x2d h6 y15 ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x2e h7 y14 ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 x2f h7 y15 ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x30 h8 y7 ff5 fs1 fc1 sc0 ls1 ws1">],<span class="_ _22"></span>[</div><div class="t m0 x31 h8 y16 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x32 h8 y7 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _15"></span>[</div><div class="t m0 x33 h8 y17 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x34 h8 y7 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 x35 h8 y16 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x36 h9 y14 ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 x36 h9 y15 ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x37 ha y7 ff6 fs1 fc1 sc0 ls1 ws1">n<span class="_ _f"></span>y<span class="_ _19"></span>r<span class="_ _b"></span>n<span class="_ _c"></span>h<span class="_ _d"></span>r<span class="_ _a"></span>x</div><div class="t m0 x38 he y18 ff5 fs5 fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 x1f he y19 ff5 fs5 fc1 sc0 ls1 ws1">~</div><div class="t m0 x39 h10 y18 ff6 fs5 fc1 sc0 ls1 ws1">n<span class="_ _f"></span>y</div><div class="t m0 x3a hb y1a ff6 fs3 fc1 sc0 ls1 ws1">n</div><div class="t m0 x3b h11 y18 ff5 fs6 fc1 sc0 ls1 ws1">.<span class="_ _23"></span><span class="ff6">N</span></div><div class="t m0 x2 h4 y1b ff3 fs1 fc2 sc0 ls1 ws1"> </div><div class="t m0 x2 h4 y1c ff3 fs1 fc2 sc0 ls1 ws1">5.2 <span class="_ _0"> </span><span class="fc3 lsb">(a) </span></div><div class="t m0 x3c h4 y1d ff3 fs1 fc3 sc0 ls1 ws1"> </div><div class="t m0 x3d h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">,<span class="_ _a"></span>13<span class="_ _24"></span>]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x3e h8 y1e ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x3f h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x40 h8 y1f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x41 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x42 h8 y1e ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x43 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x44 h8 y1f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x45 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x46 h8 y1e ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x47 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x48 h8 y1f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x49 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x4a h8 y1e ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x4b h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x4c h8 y1f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x4d h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x4e h8 y1e ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x4f h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x50 h8 y1f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x51 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _29"></span>[</div><div class="t m0 x52 h8 y1e ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x53 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 xc h8 y1f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x54 h8 y1c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x55 h8 y1f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x56 h7 y20 ff5 fs2 fc1 sc0 ls1 ws1">4</div><div class="t m0 x9 h7 y21 ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x57 h5 y1d ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _2a"></span>=<span class="_ _2b"></span>+<span class="_ _2c"></span>+<span class="_ _2c"></span>+<span class="_ _2d"></span>+<span class="_ _2e"></span>=</div><div class="t m0 x58 h5 y22 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x59 h5 y1d ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _2f"></span>=</div><div class="t m0 x24 h6 y23 ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x5a ha y1d ff6 fs1 fc1 sc0 ls1 ws1">h<span class="_ _30"></span>x<span class="_ _31"></span>h<span class="_ _32"></span>x<span class="_ _33"></span>h<span class="_ _30"></span>x<span class="_ _31"></span>h<span class="_ _34"></span>x<span class="_ _31"></span>h<span class="_ _30"></span>x<span class="_ _19"></span>r<span class="_ _35"></span>h<span class="_ _d"></span>r<span class="_ _36"></span>x<span class="_ _37"></span>y</div><div class="t m0 x33 h9 y23 ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x2 h4 y24 ff3 fs1 fc2 sc0 ls1 ws1"> <span class="_ _38"> </span><span class="fc3"> <span class="_ _39"></span><span class="ff5 fc1">,<span class="_ _a"></span>13<span class="_ _24"></span>]<span class="_ _26"></span>2<span class="_ _27"></span>[</span></span></div><div class="t m0 x5b h8 y25 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x5c h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x5d h8 y26 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x5e h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x5f h8 y25 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x60 h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x61 h8 y26 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x62 h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x63 h8 y25 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x64 h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x65 h8 y26 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x66 h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x67 h8 y25 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x68 h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x69 h8 y26 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x26 h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x6a h8 y25 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x6b h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x6c h8 y26 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x6d h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _3a"></span>1<span class="_ _25"></span>[</div><div class="t m0 x6e h8 y25 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x9 h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 x34 h8 y26 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x2a h8 y24 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x6f h8 y26 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x70 h7 y27 ff5 fs2 fc1 sc0 ls1 ws1">4</div><div class="t m0 x71 h7 y28 ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x72 h5 y24 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _2a"></span>=<span class="_ _12"></span>+<span class="_ _2b"></span>+<span class="_ _3b"></span>+<span class="_ _2d"></span>+<span class="_ _3c"></span>=</div><div class="t m0 x73 h5 y29 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x74 h5 y24 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _3d"></span>=</div><div class="t m0 x10 h6 y2a ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x75 ha y24 ff6 fs1 fc1 sc0 ls1 ws1">h<span class="_ _30"></span>x<span class="_ _33"></span>h<span class="_ _32"></span>x<span class="_ _3e"></span>h<span class="_ _30"></span>x<span class="_ _31"></span>h<span class="_ _34"></span>x<span class="_ _3f"></span>h<span class="_ _30"></span>x<span class="_ _19"></span>r<span class="_ _15"></span>h<span class="_ _d"></span>r<span class="_ _36"></span>x<span class="_ _40"></span>y</div><div class="t m0 x35 h9 y2a ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x2 h4 y2b ff3 fs1 fc3 sc0 ls1 ws1"> <span class="_ _41"> </span> <span class="_ _39"></span><span class="ff5 fc1">,<span class="_ _a"></span>13<span class="_ _24"></span>]<span class="_ _26"></span>3<span class="_ _26"></span>[</span></div><div class="t m0 x76 h8 y2c ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x75 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x1a h8 y2d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x77 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x78 h8 y2c ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x79 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x7a h8 y2d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x7b h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x7c h8 y2c ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x7d h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x7e h8 y2d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x7f h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x80 h8 y2c ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x81 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x82 h8 y2d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x83 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x84 h8 y2c ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x85 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x86 h8 y2d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x31 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _42"></span>2<span class="_ _27"></span>[</div><div class="t m0 x87 h8 y2c ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x5 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 x88 h8 y2d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x89 h8 y2b ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x6f h8 y2d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x8a h7 y2e ff5 fs2 fc1 sc0 ls1 ws1">4</div><div class="t m0 x8b h7 y2f ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x8c h5 y2b ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _2a"></span>=<span class="_ _2e"></span>+<span class="_ _2c"></span>+<span class="_ _3b"></span>+<span class="_ _43"></span>+<span class="_ _2e"></span>=</div><div class="t m0 x70 h5 y30 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x8d h5 y2b ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _44"></span>=</div><div class="t m0 x8e h6 y31 ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x8f ha y2b ff6 fs1 fc1 sc0 ls1 ws1">h<span class="_ _30"></span>x<span class="_ _3e"></span>h<span class="_ _32"></span>x<span class="_ _31"></span>h<span class="_ _3a"></span>x<span class="_ _3f"></span>h<span class="_ _34"></span>x<span class="_ _31"></span>h<span class="_ _30"></span>x<span class="_ _45"></span>r<span class="_ _10"></span>h<span class="_ _d"></span>r<span class="_ _a"></span>x<span class="_ _37"></span>y</div><div class="t m0 x90 h9 y31 ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x2 h4 y32 ff3 fs1 fc3 sc0 ls1 ws1"> <span class="_ _46"> </span> <span class="_ _47"></span><span class="ff5 fc1">,<span class="_ _a"></span>13<span class="_ _24"></span>]<span class="_ _26"></span>4<span class="_ _27"></span>[</span></div><div class="t m0 x91 h8 y33 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x92 h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x93 h8 y34 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x94 h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x95 h8 y33 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x96 h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x97 h8 y34 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x98 h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x7c h8 y33 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x7d h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x7e h8 y34 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x7f h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 x99 h8 y33 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x9a h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x9b h8 y34 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x9c h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x9d h8 y33 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x9e h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x30 h8 y34 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x9f h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _48"></span>3<span class="_ _26"></span>[</div><div class="t m0 xb h8 y33 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x5 h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 x88 h8 y34 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x2d h8 y32 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x6f h8 y34 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x8e h7 y35 ff5 fs2 fc1 sc0 ls1 ws1">4</div><div class="t m0 x8b h7 y36 ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 xa0 h5 y32 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _2a"></span>=<span class="_ _12"></span>+<span class="_ _2c"></span>+<span class="_ _2d"></span>+<span class="_ _16"></span>+<span class="_ _49"></span>=</div><div class="t m0 x10 h5 y37 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x52 h5 y32 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _4a"></span>=</div><div class="t m0 x70 h6 y38 ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 xa1 ha y32 ff6 fs1 fc1 sc0 ls1 ws1">h<span class="_ _30"></span>x<span class="_ _31"></span>h<span class="_ _32"></span>x<span class="_ _3f"></span>h<span class="_ _30"></span>x<span class="_ _31"></span>h<span class="_ _34"></span>x<span class="_ _4b"></span>h<span class="_ _30"></span>x<span class="_ _45"></span>r<span class="_ _4c"></span>h<span class="_ _d"></span>r<span class="_ _a"></span>x<span class="_ _4d"></span>y</div><div class="t m0 x90 h9 y38 ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x2 h4 y39 ff3 fs1 fc3 sc0 ls1 ws1"> <span class="_ _4e"> </span> <span class="_ _4f"></span><span class="ff5 fc1">.<span class="_ _d"></span>13<span class="_ _24"></span>]<span class="_ _27"></span>0<span class="_ _26"></span>[</span></div><div class="t m0 x8f h8 y3a ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x75 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 xa2 h8 y3b ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x77 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 xa3 h8 y3a ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x5f h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 xa4 h8 y3b ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x25 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 xa5 h8 y3a ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x19 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>2<span class="_ _27"></span>[</div><div class="t m0 xa6 h8 y3b ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xa7 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _28"></span>3<span class="_ _26"></span>[</div><div class="t m0 x80 h8 y3a ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x81 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _25"></span>1<span class="_ _25"></span>[</div><div class="t m0 x82 h8 y3b ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x83 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x84 h8 y3a ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x85 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>0<span class="_ _27"></span>[</div><div class="t m0 x86 h8 y3b ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xa8 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _42"></span>4<span class="_ _27"></span>[</div><div class="t m0 x87 h8 y3a ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x5 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _9"></span>[</div><div class="t m0 x88 h8 y3b ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x89 h8 y39 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _26"></span>4<span class="_ _27"></span>[</div><div class="t m0 x6f h8 y3b ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x8a h7 y3c ff5 fs2 fc1 sc0 ls1 ws1">4</div><div class="t m0 x8b h7 y3d ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x8c h5 y39 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _2a"></span>=<span class="_ _12"></span>+<span class="_ _50"></span>+<span class="_ _3b"></span>+<span class="_ _50"></span>+<span class="_ _2e"></span>=</div><div class="t m0 x70 h5 y3e ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x8d h5 y39 ff4 fs1 fc1 sc0 ls1 ws1">&#8722;<span class="_ _44"></span>=</div><div class="t m0 x8e h6 y3f ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x8f ha y39 ff6 fs1 fc1 sc0 ls1 ws1">h<span class="_ _30"></span>x<span class="_ _3f"></span>h<span class="_ _32"></span>x<span class="_ _31"></span>h<span class="_ _30"></span>x<span class="_ _4b"></span>h<span class="_ _34"></span>x<span class="_ _3e"></span>h<span class="_ _3a"></span>x<span class="_ _45"></span>r<span class="_ _51"></span>h<span class="_ _d"></span>r<span class="_ _36"></span>x<span class="_ _37"></span>y</div><div class="t m0 x90 h9 y3f ff6 fs2 fc1 sc0 ls1 ws1">r</div><div class="t m0 x2 h4 y40 ff3 fs1 fc3 sc0 ls1 ws1"> <span class="_ _2"> </span><span class="ff2 fc1 lsc">Therefore, </span></div><div class="t m0 xa9 h12 y41 ff5 fs7 fc1 sc0 ls1 ws1">.<span class="_ _25"></span>4<span class="_ _52"></span>0<span class="_ _53"></span>},<span class="_ _27"></span>,<span class="_ _a"></span>13<span class="_ _8"></span>,<span class="_ _a"></span>13<span class="_ _8"></span>,<span class="_ _a"></span>13<span class="_ _8"></span>,<span class="_ _a"></span>13<span class="_ _8"></span>,<span class="_ _a"></span>13<span class="_ _32"></span>{<span class="_ _54"></span>]<span class="_ _a"></span>[</div><div class="t m0 x58 h12 y42 ff5 fs7 fc1 sc0 ls1 ws1">~</div><div class="c x79 y43 w2 h13"><div class="t m0 x0 h14 y44 ff4 fs7 fc1 sc0 ls1 ws1">&#8804;</div></div><div class="c x61 y43 w2 h13"><div class="t m0 x0 h14 y44 ff4 fs7 fc1 sc0 ls1 ws1">&#8804;</div></div><div class="c xaa y43 w2 h13"><div class="t m0 x0 h14 y44 ff4 fs7 fc1 sc0 ls1 ws1">&#8722;</div></div><div class="c xab y43 w2 h13"><div class="t m0 x0 h14 y44 ff4 fs7 fc1 sc0 ls1 ws1">&#8722;</div></div><div class="c xac y43 w2 h13"><div class="t m0 x0 h14 y44 ff4 fs7 fc1 sc0 ls1 ws1">&#8722;</div></div><div class="t m0 xad h14 y41 ff4 fs7 fc1 sc0 ls1 ws1">&#8722;<span class="_ _55"></span>&#8722;<span class="_ _29"></span>=<span class="_ _56"> </span><span class="ff6">n<span class="_ _57"></span>n<span class="_ _2a"></span>y</span></div><div class="t m0 x77 h3 y45 ff2 fs1 fc1 sc0 ls1 ws1"> </div><div class="t m0 x2 h3 y46 ff2 fs1 fc1 sc0 ls1 ws1"> </div><div class="t m0 x4 h4 y47 ff3 fs1 fc3 sc0 ls2 ws9">(b) </div><div class="t m0 xab h12 y48 ff5 fs7 fc1 sc0 ls1 ws1">.<span class="_ _25"></span>4<span class="_ _52"></span>0<span class="_ _53"></span>},<span class="_ _27"></span>,<span class="_ _4f"></span>1<span class="_ _1b"></span>,<span class="_ _4f"></span>1<span class="_ _1b"></span>,<span class="_ _4f"></span>1<span class="_ _1b"></span>,<span class="_ _4f"></span>1<span class="_ _1b"></span>,<span class="_ _4f"></span>1<span class="_ _28"></span>{<span class="_ _54"></span>]<span class="_ _d"></span>[</div><div class="t m0 xae h12 y49 ff5 fs7 fc1 sc0 ls1 ws1">~</div><div class="c x39 y4a w2 h13"><div class="t m0 x0 h14 y4b ff4 fs7 fc1 sc0 ls1 ws1">&#8804;</div></div><div class="c xaf y4a w2 h13"><div class="t m0 x0 h14 y4b ff4 fs7 fc1 sc0 ls1 ws1">&#8804;</div></div><div class="t m0 x8b h14 y48 ff4 fs7 fc1 sc0 ls1 ws1">=<span class="_ _58"> </span><span class="ff6">n<span class="_ _59"></span>n<span class="_ _2a"></span>y</span></div><div class="t m0 xb0 h4 y4c ff3 fs1 fc3 sc0 ls1 ws1"> </div><div class="t m0 x2 h4 y4d ff3 fs1 fc2 sc0 ls1 ws1"> </div><div class="t m0 x2 h4 y4e ff3 fs1 fc2 sc0 ls1 ws1">5.3 <span class="_"> </span><span class="ff2 fc1 lsd">Since </span></div><div class="t m0 xb1 h3 y4f ff2 fs1 fc1 sc0 ls5 wsa"> hence all the terms whic<span class="lsc wsb">h are not in the range </span></div><div class="t m0 xb2 h8 y4e ff5 fs1 fc1 sc0 ls1 ws1">],<span class="_ _22"></span>[</div><div class="t m0 xb3 h8 y50 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x53 h8 y4e ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _5a"></span>[</div><div class="t m0 x35 h8 y50 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x30 ha y4e ff6 fs1 fc1 sc0 ls1 ws1">n<span class="_ _5b"></span>rN<span class="_ _14"></span>n</div><div class="t m0 xb4 h9 y51 ff6 fs2 fc1 sc0 ls1 ws1">k<span class="_ _5c"></span>k</div><div class="t m0 x6d h5 y4f ff4 fs1 fc1 sc0 ls1 ws1">&#968;<span class="_ _22"></span>=<span class="_ _3f"></span>+<span class="_ _24"></span>&#968;</div><div class="t m0 xb5 h11 y52 ff5 fs6 fc1 sc0 ls1 ws1">,<span class="_ _39"></span>1<span class="_ _5d"></span>,<span class="_ _53"></span>,<span class="_ _39"></span>1<span class="_ _5e"></span>,<span class="_ _5e"></span>0</div><div class="c xb6 y53 w3 h15"><div class="t m0 x0 hf y44 ff4 fs5 fc1 sc0 ls1 ws1">&#8722;</div></div><div class="t m0 xb7 h16 y52 ff6 fs6 fc1 sc0 ls1 ws1">N<span class="_ _5f"></span><span class="ff7 fs5">K</span></div><div class="t m0 xb8 h3 y4f ff2 fs1 fc1 sc0 ls1 ws1"> </div><div class="t m0 x4 h3 y54 ff2 fs1 fc1 sc0 lsd wsc">can be accumulated to<span class="_ _60"> </span><span class="ls6 ws4"> where <span class="_ _61"></span><span class="ff5 ls1 ws1">],<span class="_ _22"></span>[</span></span></div><div class="t m0 xb9 h8 y55 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xba ha y54 ff6 fs1 fc1 sc0 ls1 ws1">n</div><div class="t m0 x30 h9 y56 ff6 fs2 fc1 sc0 ls1 ws1">k</div><div class="t m0 x51 h5 y54 ff4 fs1 fc1 sc0 ls1 ws1">&#968;</div><div class="t m0 xbb h17 y57 ff5 fs8 fc1 sc0 ls1 ws1">.<span class="_ _25"></span>1<span class="_ _62"></span>0</div><div class="c x14 y58 w4 h18"><div class="t m0 x0 h19 y4b ff4 fs9 fc1 sc0 ls1 ws1">&#8722;</div></div><div class="c x1e y58 w4 h18"><div class="t m0 x0 h19 y4b ff4 fs9 fc1 sc0 ls1 ws1">&#8804;</div></div><div class="c x80 y58 w4 h18"><div class="t m0 x0 h19 y4b ff4 fs9 fc1 sc0 ls1 ws1">&#8804;</div></div><div class="t m0 xbc h1a y57 ff6 fs8 fc1 sc0 ls1 ws1">N<span class="_ _19"></span>k</div><div class="t m0 xbd h3 y54 ff2 fs1 fc1 sc0 lsc wsd"> Hence, in this case the Fourier </div><div class="t m0 x4 h3 y59 ff2 fs1 fc1 sc0 ls9 ws7">series representation involves only </div><div class="c xbe y5a w5 h1b"><div class="t m0 x0 h1c y5b ff6 fsa fc1 sc0 ls1 ws1">N</div></div><div class="t m0 x38 h3 y59 ff2 fs1 fc1 sc0 ls6 ws4"> complex exponential sequences. Let </div><div class="t m0 xbf h8 y5c ff5 fs1 fc1 sc0 ls1 ws1">,<span class="_ _43"></span>]<span class="_ _a"></span>[</div><div class="t m0 xc0 h8 y5d ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x10 h8 y5e ff5 fs1 fc1 sc0 ls1 ws1">1</div><div class="t m0 x89 h8 y5c ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 x6f h8 y5f ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x9 h7 y60 ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 xc1 h7 y61 ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x50 h7 y62 ff5 fs2 fc1 sc0 ls1 ws1">/<span class="_ _32"></span>2</div><div class="t m0 xc2 h5 y63 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 xc3 h5 y5c ff4 fs1 fc1 sc0 ls1 ws1">=</div><div class="t m0 x24 h6 y60 ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 x12 h6 y61 ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x51 h6 y62 ff4 fs2 fc1 sc0 ls1 ws1">&#960;</div><div class="t m0 xc4 h9 y60 ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 xc4 h9 y61 ff6 fs2 fc1 sc0 ls1 ws1">k</div><div class="t m0 x6b h9 y62 ff6 fs2 fc1 sc0 ls1 ws1">N<span class="_ _63"></span>kn<span class="_ _34"></span>j</div><div class="t m0 xc5 ha y5c ff6 fs1 fc1 sc0 ls1 ws1">e<span class="_ _2a"></span>k<span class="_ _22"></span>X</div><div class="t m0 x73 ha y64 ff6 fs1 fc1 sc0 ls1 ws1">N</div><div class="t m0 x36 h3 y5c ff6 fs1 fc1 sc0 ls1 ws1">n<span class="_ _2a"></span>x<span class="_ _64"> </span><span class="ff2 ls8 ws6"> then </span></div><div class="t m0 xc6 h1d y65 ff5 fsb fc1 sc0 ls1 ws1">.<span class="_ _65"></span>]<span class="_ _a"></span>[</div><div class="t m0 x77 h1d y66 ff5 fsb fc1 sc0 ls1 ws1">~</div><div class="t m0 xc7 h1d y67 ff5 fsb fc1 sc0 ls1 ws1">1</div><div class="t m0 x99 h1d y68 ff5 fsb fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 x82 h1d y69 ff5 fsb fc1 sc0 ls1 ws1">~</div><div class="t m0 x8 h1d y6a ff5 fsb fc1 sc0 ls1 ws1">1</div><div class="t m0 x70 h1d y6b ff5 fsb fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 xae h1d y6c ff5 fsb fc1 sc0 ls1 ws1">~</div><div class="t m0 x29 h1e y6d ff5 fsc fc1 sc0 ls1 ws1">1</div><div class="t m0 xc8 h1e y6e ff5 fsc fc1 sc0 ls1 ws1">0</div><div class="t m0 xc9 h1e y6f ff5 fsc fc1 sc0 ls1 ws1">/<span class="_ _9"></span>)<span class="_ _34"></span>(<span class="_ _36"></span>2</div><div class="t m0 x42 h1e y6d ff5 fsc fc1 sc0 ls1 ws1">1</div><div class="t m0 xca h1e y6e ff5 fsc fc1 sc0 ls1 ws1">0</div><div class="t m0 x9d h1e y6d ff5 fsc fc1 sc0 ls1 ws1">1</div><div class="t m0 x3 h1e y6e ff5 fsc fc1 sc0 ls1 ws1">0</div><div class="t m0 x4d h1e y6d ff5 fsc fc1 sc0 ls1 ws1">1</div><div class="t m0 xcb h1e y6e ff5 fsc fc1 sc0 ls1 ws1">0</div><div class="t m0 x1d h1e y6f ff5 fsc fc1 sc0 ls1 ws1">/<span class="_ _9"></span>)<span class="_ _34"></span>(<span class="_ _36"></span>2</div><div class="t m0 x2a h1e y6d ff5 fsc fc1 sc0 ls1 ws1">1</div><div class="t m0 xcc h1e y6e ff5 fsc fc1 sc0 ls1 ws1">0</div><div class="t m0 xcd h1e y6f ff5 fsc fc1 sc0 ls1 ws1">/<span class="_ _5f"></span>2</div><div class="t m0 xce h1f y70 ff4 fsb fc1 sc0 ls1 ws1">&#9119;</div><div class="t m0 xce h1f y71 ff4 fsb fc1 sc0 ls1 ws1">&#9119;</div><div class="t m0 xce h1f y72 ff4 fsb fc1 sc0 ls1 ws1">&#9120;</div><div class="t m0 xce h1f y73 ff4 fsb fc1 sc0 ls1 ws1">&#9118;</div><div class="t m0 xcf h1f y70 ff4 fsb fc1 sc0 ls1 ws1">&#9116;</div><div class="t m0 xcf h1f y71 ff4 fsb fc1 sc0 ls1 ws1">&#9116;</div><div class="t m0 xcf h1f y72 ff4 fsb fc1 sc0 ls1 ws1">&#9117;</div><div class="t m0 xcf h1f y73 ff4 fsb fc1 sc0 ls1 ws1">&#9115;</div><div class="t m0 x5b h1f y74 ff4 fsb fc1 sc0 ls1 ws1">&#8721;<span class="_ _49"></span>&#8721;</div><div class="t m0 xd0 h1f y75 ff4 fsb fc1 sc0 ls1 ws1">=</div><div class="t m0 xba h1f y74 ff4 fsb fc1 sc0 lse ws1">&#8721;&#8721;</div><div class="t m0 x9f h1f y75 ff4 fsb fc1 sc0 ls1 ws1">=</div><div class="t m0 xd1 h1f y74 ff4 fsb fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x76 h20 y6d ff4 fsc fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 x91 h20 y6e ff4 fsc fc1 sc0 ls1 ws1">=</div><div class="t m0 xd2 h20 y6f ff4 fsc fc1 sc0 ls1 ws1">&#8722;<span class="_ _29"></span>&#960;</div><div class="t m0 xd3 h20 y6d ff4 fsc fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 x78 h20 y6e ff4 fsc fc1 sc0 ls1 ws1">=</div><div class="t m0 xb1 h20 y6d ff4 fsc fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 xd4 h20 y6e ff4 fsc fc1 sc0 ls1 ws1">=</div><div class="t m0 xd5 h20 y6d ff4 fsc fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 xd6 h20 y6e ff4 fsc fc1 sc0 ls1 ws1">=</div><div class="t m0 xd7 h20 y6f ff4 fsc fc1 sc0 ls1 ws1">&#8722;<span class="_ _29"></span>&#960;</div><div class="t m0 xd8 h20 y6d ff4 fsc fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 xd9 h20 y6e ff4 fsc fc1 sc0 ls1 ws1">=</div><div class="t m0 xc h20 y6f ff4 fsc fc1 sc0 ls1 ws1">&#960;<span class="_ _35"></span>&#8722;</div><div class="t m0 x5c h21 y6d ff6 fsc fc1 sc0 ls1 ws1">N</div><div class="t m0 x5c h21 y6e ff6 fsc fc1 sc0 ls1 ws1">n</div><div class="t m0 xda h21 y6f ff6 fsc fc1 sc0 ls1 ws1">N<span class="_ _66"></span>n<span class="_ _9"></span>r<span class="_ _67"></span>k<span class="_ _35"></span>j</div><div class="t m0 x96 h21 y6d ff6 fsc fc1 sc0 ls1 ws1">N</div><div class="t m0 xdb h21 y6e ff6 fsc fc1 sc0 ls1 ws1">k</div><div class="t m0 xb2 h21 y6d ff6 fsc fc1 sc0 ls1 ws1">N</div><div class="t m0 xb2 h21 y6e ff6 fsc fc1 sc0 ls1 ws1">n</div><div class="t m0 xdc h21 y6d ff6 fsc fc1 sc0 ls1 ws1">N</div><div class="t m0 xdc h21 y6e ff6 fsc fc1 sc0 ls1 ws1">k</div><div class="t m0 xdd h21 y6f ff6 fsc fc1 sc0 ls1 ws1">N<span class="_ _66"></span>n<span class="_ _68"></span>r<span class="_ _f"></span>k<span class="_ _35"></span>j</div><div class="t m0 x6f h21 y6d ff6 fsc fc1 sc0 ls1 ws1">N</div><div class="t m0 x6f h21 y6e ff6 fsc fc1 sc0 ls1 ws1">n</div><div class="t m0 xde h21 y6f ff6 fsc fc1 sc0 ls1 ws1">N<span class="_ _69"></span>rn<span class="_ _63"></span>j</div><div class="t m0 xdf h22 y65 ff6 fsb fc1 sc0 ls1 ws1">e<span class="_ _6a"></span>k<span class="_ _22"></span>X</div><div class="t m0 xa4 h22 y76 ff6 fsb fc1 sc0 ls1 ws1">N</div><div class="t m0 x4b h22 y65 ff6 fsb fc1 sc0 ls1 ws1">e<span class="_ _2a"></span>k<span class="_ _22"></span>X</div><div class="t m0 x37 h22 y76 ff6 fsb fc1 sc0 ls1 ws1">N</div><div class="t m0 x71 h22 y65 ff6 fsb fc1 sc0 ls1 ws1">e<span class="_ _2a"></span>n<span class="_ _2a"></span>x</div><div class="t m0 x4 h3 y77 ff2 fs1 fc1 sc0 ls1 ws1"> </div><div class="t m0 x4 h3 y78 ff2 fs1 fc1 sc0 lsa ws8">Now, from Eq. (5.11), the inner summation is equal to </div><div class="c xe0 y79 w5 h1b"><div class="t m0 x0 h1c y7a ff6 fsa fc1 sc0 ls1 ws1">N</div></div><div class="t m0 xe1 h3 y7b ff2 fs1 fc1 sc0 lsb wse"> if <span class="_ _6b"> </span><span class="ff5 ls1 ws1">,<span class="_ _4f"></span><span class="ff6">r<span class="_ _69"></span>k</span></span></div><div class="c xe2 y79 w6 h23"><div class="t m0 x0 h5 y7c ff4 fs1 fc1 sc0 ls1 ws1">=</div></div><div class="t m0 xe3 h3 y7b ff2 fs1 fc1 sc0 lsf wsf"> otherwise it is equal </div><div class="t m0 x4 h3 y7d ff2 fs1 fc1 sc0 ls10 ws1">to </div><div class="t m0 x2f h3 y7e ff2 fs1 fc1 sc0 ls1 ws1"> Thus, <span class="_ _6c"> </span> Next, we observe <span class="_ _6d"> </span> </div><div class="t m0 x4 h3 y7f ff2 fs1 fc1 sc0 ls1 ws1"> </div><div class="t m0 x2c h17 y80 ff5 fs8 fc1 sc0 ls1 ws1">.<span class="_ _25"></span>0</div><div class="t m0 xe4 h8 y7e ff5 fs1 fc1 sc0 ls1 ws1">].<span class="_ _2a"></span>[</div><div class="t m0 x82 h8 y81 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x59 h8 y7e ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 xe5 h8 y82 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xe6 h7 y83 ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 xe7 h7 y84 ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x4f h7 y85 ff5 fs2 fc1 sc0 ls1 ws1">/<span class="_ _5f"></span>2</div><div class="t m0 xe8 ha y7e ff6 fs1 fc1 sc0 ls1 ws1">r<span class="_ _6e"></span>X<span class="_ _6f"></span>e<span class="_ _6e"></span>n<span class="_ _f"></span>x</div><div class="t m0 x7 h9 y83 ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 xa h9 y84 ff6 fs2 fc1 sc0 ls1 ws1">n</div><div class="t m0 xe9 h9 y85 ff6 fs2 fc1 sc0 ls1 ws1">N<span class="_ _69"></span>rn<span class="_ _63"></span>j</div><div class="t m0 xea h5 y7e ff4 fs1 fc1 sc0 ls1 ws1">=</div><div class="t m0 x5 h5 y86 ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 xb h6 y83 ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 xc h6 y84 ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 xb2 h6 y85 ff4 fs2 fc1 sc0 ls1 ws1">&#960;<span class="_ _35"></span>&#8722;</div><div class="t m0 x40 h8 y7e ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _55"></span>[</div><div class="t m0 x5f h8 y81 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x93 h5 y7e ff6 fs1 fc1 sc0 ls1 ws1">N<span class="_ _70"></span>k<span class="_ _22"></span>X<span class="_ _71"> </span><span class="ff7">l<span class="_ _72"></span><span class="ff4">+</span></span></div><div class="t m0 xce h8 y87 ff5 fs1 fc1 sc0 ls1 ws1">].<span class="_ _22"></span>[</div><div class="t m0 xeb h8 y88 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xec h8 y87 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 xed h8 y89 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 x81 h8 y87 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 x9b h8 y89 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xc4 h8 y87 ff5 fs1 fc1 sc0 ls1 ws1">]<span class="_ _a"></span>[</div><div class="t m0 x35 h8 y89 ff5 fs1 fc1 sc0 ls1 ws1">~</div><div class="t m0 xee h7 y8a ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 x15 h7 y8b ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 xef h7 y8c ff5 fs2 fc1 sc0 ls1 ws1">/<span class="_ _32"></span>2<span class="_ _73"></span>2</div><div class="t m0 xf0 h7 y8a ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 x4d h7 y8b ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x17 h7 y8c ff5 fs2 fc1 sc0 ls1 ws1">/<span class="_ _32"></span>2</div><div class="t m0 x2e h7 y8a ff5 fs2 fc1 sc0 ls1 ws1">1</div><div class="t m0 x2f h7 y8b ff5 fs2 fc1 sc0 ls1 ws1">0</div><div class="t m0 x13 h7 y8c ff5 fs2 fc1 sc0 ls1 ws1">/<span class="_ _9"></span>)<span class="_ _74"></span>(<span class="_ _36"></span>2</div><div class="t m0 xf1 ha y87 ff6 fs1 fc1 sc0 ls1 ws1">k<span class="_ _22"></span>X<span class="_ _6f"></span>e<span class="_ _6e"></span>n<span class="_ _2a"></span>x<span class="_ _75"></span>e<span class="_ _49"></span>e<span class="_ _6e"></span>n<span class="_ _f"></span>x<span class="_ _76"></span>e<span class="_ _6e"></span>n<span class="_ _f"></span>x</div><div class="t m0 xf2 h9 y8a ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 xf2 h9 y8b ff6 fs2 fc1 sc0 ls1 ws1">n</div><div class="t m0 xf3 h9 y8c ff6 fs2 fc1 sc0 ls1 ws1">N<span class="_ _63"></span>kn<span class="_ _34"></span>j<span class="_ _75"></span>n<span class="_ _34"></span>j</div><div class="t m0 xdc h9 y8a ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 x26 h9 y8b ff6 fs2 fc1 sc0 ls1 ws1">n</div><div class="t m0 x7d h9 y8c ff6 fs2 fc1 sc0 ls1 ws1">N<span class="_ _63"></span>kn<span class="_ _34"></span>j</div><div class="t m0 x36 h9 y8a ff6 fs2 fc1 sc0 ls1 ws1">N</div><div class="t m0 x36 h9 y8b ff6 fs2 fc1 sc0 ls1 ws1">n</div><div class="t m0 x6 h9 y8c ff6 fs2 fc1 sc0 ls1 ws1">N<span class="_ _66"></span>n<span class="_ _6e"></span>N<span class="_ _63"></span>k<span class="_ _35"></span>j</div><div class="t m0 xf4 h5 y87 ff4 fs1 fc1 sc0 ls1 ws1">=</div><div class="t m0 xe2 h5 y8d ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 xdb h5 y87 ff4 fs1 fc1 sc0 ls1 ws1">=</div><div class="t m0 x9c h5 y8d ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x84 h5 y87 ff4 fs1 fc1 sc0 ls1 ws1">=</div><div class="t m0 x2a h5 y8d ff4 fs1 fc1 sc0 ls1 ws1">&#8721;</div><div class="t m0 x6f h5 y87 ff4 fs1 fc1 sc0 ls1 ws1">=</div><div class="t m0 xf5 h6 y8a ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 xa9 h6 y8b ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 xf6 h6 y8c ff4 fs2 fc1 sc0 ls1 ws1">&#960;<span class="_ _35"></span>&#8722;<span class="_ _77"></span>&#960;<span class="_ _35"></span>&#8722;</div><div class="t m0 xd5 h6 y8a ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 x83 h6 y8b ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 xf7 h6 y8c ff4 fs2 fc1 sc0 ls1 ws1">&#960;<span class="_ _35"></span>&#8722;</div><div class="t m0 x2c h6 y8a ff4 fs2 fc1 sc0 ls1 ws1">&#8722;</div><div class="t m0 x2d h6 y8b ff4 fs2 fc1 sc0 ls1 ws1">=</div><div class="t m0 x2b h6 y8c ff4 fs2 fc1 sc0 ls1 ws1">+<span class="_ _29"></span>&#960;<span class="_ _35"></span>&#8722;<span class="_ _78"> </span><span class="ff7">l<span class="_ _79"></span>l</span></div><div class="t m0 x2 h4 y8e ff3 fs1 fc2 sc0 ls1 ws1"> </div><div class="t m0 xaf h4 y8f ff3 fs1 fc0 sc0 lsd wsc">Not for sale. </div><div class="t m0 x3c h3 y90 ff2 fs1 fc1 sc0 ls1 ws1">109</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>
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