CH9 - Applications of the DFT.rar

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DSP Guide to DSP Chapter 9
CH9 - Applications of the DFT.rar
  • CH9 - Applications of the DFT.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">169</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">9</div><div class="t m0 x4 h5 y4 ff2 fs3 fc0 sc0 ls3 ws1">Applications of the DFT</div><div class="t m0 x5 h6 y5 ff1 fs4 fc0 sc0 ls4 ws2">The Discrete Fourier Transform (DFT) is one of the most important tools in Digital Signal</div><div class="t m0 x5 h6 y6 ff1 fs4 fc0 sc0 ls5 ws3">Processing. This chapter discusses three common ways it is used. First, the DFT can calculate</div><div class="t m0 x5 h6 y7 ff1 fs4 fc0 sc0 ls6 ws4">a signal's <span class="ff3 ls7 ws5">frequency spectrum</span><span class="ls8 ws6">. This is a direct examination of information encoded in the</span></div><div class="t m0 x5 h6 y8 ff1 fs4 fc0 sc0 ls9 ws7">frequency, phase, and amplitude of the component sinusoids. For example, human speech and</div><div class="t m0 x5 h6 y9 ff1 fs4 fc0 sc0 lsa ws8">hearing use signals with this type of encoding. Second, the DFT can find a system's frequency</div><div class="t m0 x5 h6 ya ff1 fs4 fc0 sc0 lsb ws9">response from the system's impulse response, and vice versa. This allows systems to be analyzed</div><div class="t m0 x5 h6 yb ff1 fs4 fc0 sc0 lsc wsa">in the <span class="ff3 lsd wsb">frequency domain</span><span class="lse wsc">, just as convolution allows systems to be analyzed in the <span class="ff3 lsf wsd">time domain</span><span class="ls10 ws0">.</span></span></div><div class="t m0 x5 h6 yc ff1 fs4 fc0 sc0 ls11 wse">Third, the DFT can be used as an intermediate step in more elaborate signal processing</div><div class="t m0 x5 h6 yd ff1 fs4 fc0 sc0 ls12 wsf">techniques. The classic example of this is <span class="ff3 ls13 ws10">FFT convolution</span><span class="ls14 ws11">, an algorithm for convolving signals</span></div><div class="t m0 x5 h6 ye ff1 fs4 fc0 sc0 ls15 ws12">that is hundreds of times faster than conventional methods. </div><div class="t m0 x5 h7 yf ff2 fs5 fc0 sc0 ls16 ws13">Spectral Analysis of Signals</div><div class="t m0 x6 h8 y10 ff1 fs6 fc0 sc0 ls17 ws14">It is very common for information to be encoded in the sinusoids that form</div><div class="t m0 x6 h8 y11 ff1 fs6 fc0 sc0 ls18 ws15">a signal. This is true of naturally occurring signals, as well as those that</div><div class="t m0 x6 h8 y12 ff1 fs6 fc0 sc0 ls19 ws16">have been created by humans. Many things oscillate in our universe. For</div><div class="t m0 x6 h8 y13 ff1 fs6 fc0 sc0 ls1a ws17">example, speech is a result of vibration of the human vocal cords; stars</div><div class="t m0 x6 h8 y14 ff1 fs6 fc0 sc0 ls1b ws18">and planets change their brightness as they rotate on their axes and revolve</div><div class="t m0 x6 h8 y15 ff1 fs6 fc0 sc0 ls1c ws19">around each other; ship's propellers generate periodic displacement of the</div><div class="t m0 x6 h8 y16 ff1 fs6 fc0 sc0 ls1d ws1a">water, and so on. The <span class="ff3 ls1e ws0">shape</span><span class="ls1f ws1b"> of the time domain waveform is not important</span></div><div class="t m0 x6 h8 y17 ff1 fs6 fc0 sc0 ls20 ws1c">in these signals; the key information is in the <span class="ff3 ls21 ws0">frequency</span><span class="ls2 ws1d">, <span class="ff3 ls22 ws0">phase</span><span class="ls23 ws1e"> and</span></span></div><div class="t m0 x6 h8 y18 ff3 fs6 fc0 sc0 ls24 ws0">amplitude<span class="ff1 ls25 ws1f"> of the component sinusoids. The DFT is used to extract this</span></div><div class="t m0 x6 h8 y19 ff1 fs6 fc0 sc0 ls26 ws20">information. </div><div class="t m0 x6 h8 y1a ff1 fs6 fc0 sc0 ls27 ws21">An example will show how this works. Suppose we want to investigate the</div><div class="t m0 x6 h8 y1b ff1 fs6 fc0 sc0 ls28 ws22">sounds that travel through the ocean. To begin, a microphone is placed in the</div><div class="t m0 x6 h8 y1c ff1 fs6 fc0 sc0 ls29 ws23">water and the resulting electronic signal amplified to a reasonable level, say a</div><div class="t m0 x6 h8 y1d ff1 fs6 fc0 sc0 ls2a ws24">few volts. An analog low-pass filter is then used to remove all frequencies</div><div class="t m0 x6 h8 y1e ff1 fs6 fc0 sc0 ls2b ws25">above 80 hertz, so that the signal can be digitized at 160 samples per second.</div><div class="t m0 x6 h8 y1f ff1 fs6 fc0 sc0 ls2c ws26">After acquiring and storing several thousand samples, what next?</div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div> </body> </html>
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