CH8 - The Discrete Fourier Transform.rar

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DSP Guide to DSP Chapter 8
CH8 - The Discrete Fourier Transform.rar
  • CH8 - The Discrete Fourier Transform.pdf
<html xmlns=""> <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=""> <link rel="stylesheet" href=""> <link rel="stylesheet" href=""> <script src=""></script> <script src=""></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=""><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">141</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls1 ws0">CHAPTER</div><div class="t m0 x3 h4 y3 ff1 fs2 fc0 sc0 ls2 ws0">8</div><div class="t m0 x4 h5 y4 ff2 fs3 fc0 sc0 ls3 ws1">The Discrete Fourier Transform</div><div class="t m0 x5 h6 y5 ff1 fs4 fc0 sc0 ls4 ws2">Fourier analysis is a family of mathematical techniques, all based on decomposing signals into</div><div class="t m0 x5 h6 y6 ff1 fs4 fc0 sc0 ls5 ws3">sinusoids. The discrete Fourier transform (DFT) is the family member used with <span class="ff3 ls6 ws0">digitized</span></div><div class="t m0 x5 h7 y7 ff1 fs4 fc0 sc0 ls7 ws4">signals. This is the first of four chapters on the <span class="ff4 ls8 ws5">real DFT</span><span class="ls9 ws6">, a version of the discrete Fourier</span></div><div class="t m0 x5 h7 y8 ff1 fs4 fc0 sc0 lsa ws7">transform that uses real numbers to represent the input and output signals. The <span class="ff4 lsb ws8">complex DFT</span><span class="lsc ws0">,</span></div><div class="t m0 x5 h6 y9 ff1 fs4 fc0 sc0 lsd ws9">a more advanced technique that uses complex numbers, will be discussed in Chapter 31. In this</div><div class="t m0 x5 h6 ya ff1 fs4 fc0 sc0 lse wsa">chapter we look at the mathematics and algorithms of the Fourier decomposition, the heart of the</div><div class="t m0 x5 h6 yb ff1 fs4 fc0 sc0 lsf ws0">DFT.</div><div class="t m0 x5 h8 yc ff2 fs5 fc0 sc0 ls10 wsb">The Family of Fourier Transform</div><div class="t m0 x6 h9 yd ff1 fs6 fc0 sc0 ls11 wsc">Fourier analysis is named after <span class="ff4 ls12 wsd">Jean Baptiste Joseph Fourier</span><span class="ls13 wse"> (1768-1830),</span></div><div class="t m0 x6 ha ye ff1 fs6 fc0 sc0 ls14 wsf">a French mathematician and physicist. (Fourier is pronounced: <span class="_ _0"> </span><span class="ls15 ws10">, and is<span class="_ _1"></span><span class="fs0 ls16 ws0">f<span class="ls17">o<span class="ls18">r<span class="ff5 ls19">@<span class="_ _2"></span></span><span class="ls0">&#175;<span class="_ _3"></span><span class="ls1a">e<span class="ff5 ls19">@<span class="_ _2"></span></span><span class="ls0">&#175;<span class="_ _3"></span><span class="ls1b">a</span></span></span></span></span></span></span></span></div><div class="t m0 x6 ha yf ff1 fs6 fc0 sc0 ls1c ws11">always capitalized). While many contributed to the field, Fourier is honored</div><div class="t m0 x6 ha y10 ff1 fs6 fc0 sc0 ls1d ws12">for his mathematical discoveries and insight into the practical usefulness of the</div><div class="t m0 x6 ha y11 ff1 fs6 fc0 sc0 ls1e ws13">techniques. Fourier was interested in heat propagation, and presented a paper</div><div class="t m0 x6 ha y12 ff1 fs6 fc0 sc0 ls1f ws14">in 1807 to the Institut de France on the use of sinusoids to represent</div><div class="t m0 x6 ha y13 ff1 fs6 fc0 sc0 ls20 ws15">temperature distributions. The paper contained the controversial claim that any</div><div class="t m0 x6 ha y14 ff1 fs6 fc0 sc0 ls21 ws16">continuous periodic signal could be represented as the sum of properly chosen</div><div class="t m0 x6 ha y15 ff1 fs6 fc0 sc0 ls22 ws17">sinusoidal waves. Among the reviewers were two of history's most famous</div><div class="t m0 x6 ha y16 ff1 fs6 fc0 sc0 ls23 ws18">mathematicians, Joseph Louis Lagrange (1736-1813), and Pierre Simon de</div><div class="t m0 x6 ha y17 ff1 fs6 fc0 sc0 ls24 ws19">Laplace (1749-1827). </div><div class="t m0 x6 ha y18 ff1 fs6 fc0 sc0 ls25 ws1a">While Laplace and the other reviewers voted to publish the paper, Lagrange</div><div class="t m0 x6 ha y19 ff1 fs6 fc0 sc0 ls26 ws1b">adamantly protested. For nearly 50 years, Lagrange had insisted that such an</div><div class="t m0 x6 ha y1a ff1 fs6 fc0 sc0 ls27 ws1c">approach could not be used to represent signals with <span class="ff3 ls28 ws0">corners</span><span class="ls29 ws1d">, i.e.,</span></div><div class="t m0 x6 ha y1b ff1 fs6 fc0 sc0 ls2a ws1e">discontinuous slopes, such as in square waves. The Institut de France bowed</div><div class="t m0 x6 ha y1c ff1 fs6 fc0 sc0 ls2b ws1f">to the prestige of Lagrange, and rejected Fourier's work. It was only after</div><div class="t m0 x6 ha y1d ff1 fs6 fc0 sc0 ls2c ws20">Lagrange died that the paper was finally published, some 15 years later.</div><div class="t m0 x6 ha y1e ff1 fs6 fc0 sc0 ls2d ws21">Luckily, Fourier had other things to keep him busy, political activities,</div><div class="t m0 x6 ha y1f ff1 fs6 fc0 sc0 ls2e ws22">expeditions to Egypt with Napoleon, and trying to avoid the guillotine after the</div><div class="t m0 x6 ha y20 ff1 fs6 fc0 sc0 ls2f ws23">French Revolution (literally!).</div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div> </body> </html>