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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/622ba8af3d2fbb00078bfb26/bg1.jpg"><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>
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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/622ba8af3d2fbb00078bfb26/bg2.jpg"><div class="t m0 x7 h2 y20 ff3 fs0 fc0 sc0 ls2d ws27">The Scientist and Engineer's Guide to Digital Signal Processing<span class="_ _0"></span><span class="ff1 ls0 ws0">170</span></div><div class="t m0 x8 h8 y21 ff1 fs6 fc0 sc0 ls2e ws28">The first thing is to simply <span class="ff3 ls2f ws0">look</span><span class="ls30 ws29"> at the data. Figure 9-1a shows 256 samples</span></div><div class="t m0 x8 h8 y22 ff1 fs6 fc0 sc0 ls31 ws2a">from our imaginary experiment. All that can be seen is a noisy waveform that</div><div class="t m0 x8 h8 y23 ff1 fs6 fc0 sc0 ls32 ws2b">conveys little information to the human eye. For reasons explained shortly, the</div><div class="t m0 x8 h9 y24 ff1 fs6 fc0 sc0 ls33 ws2c">next step is to multiply this signal by a smooth curve called a <span class="ff4 ls34 ws0">Hamming</span></div><div class="t m0 x8 h9 y25 ff4 fs6 fc0 sc0 ls35 ws0">window<span class="ff1 ls36 ws2d">, shown in (b). (Chapter 16 provides the equations for the Hamming</span></div><div class="t m0 x8 h8 y26 ff1 fs6 fc0 sc0 ls37 ws2e">and other windows; see Eqs. 16-1 and 16-2, and Fig. 16-2a). This results in</div><div class="t m0 x8 h8 y27 ff1 fs6 fc0 sc0 ls38 ws2f">a 256 point signal where the samples near the ends have been reduced in</div><div class="t m0 x8 h8 y28 ff1 fs6 fc0 sc0 ls39 ws30">amplitude, as shown in (c). </div><div class="t m0 x8 h8 y29 ff1 fs6 fc0 sc0 ls3a ws31">Taking the DFT, and converting to polar notation, results in the 129 point</div><div class="t m0 x8 h8 y2a ff1 fs6 fc0 sc0 ls3b ws32">frequency spectrum in (d). Unfortunately, this also looks like a noisy mess.</div><div class="t m0 x8 h8 y2b ff1 fs6 fc0 sc0 ls3c ws33">This is because there is not enough information in the original 256 points to</div><div class="t m0 x8 h8 y2c ff1 fs6 fc0 sc0 ls3d ws34">obtain a well behaved curve. Using a longer DFT does nothing to help this</div><div class="t m0 x8 h8 y2d ff1 fs6 fc0 sc0 ls3e ws35">problem. For example, if a 2048 point DFT is used, the frequency spectrum</div><div class="t m0 x8 h8 y2e ff1 fs6 fc0 sc0 ls3f ws36">becomes 1025 samples long. Even though the original 2048 points contain</div><div class="t m0 x8 h8 y2f ff1 fs6 fc0 sc0 ls40 ws37">more information, the greater number of samples in the spectrum dilutes the</div><div class="t m0 x8 h8 y30 ff1 fs6 fc0 sc0 ls41 ws38">information by the same factor. Longer DFTs provide better frequency</div><div class="t m0 x8 h8 y31 ff1 fs6 fc0 sc0 ls42 ws39">resolution, but the same noise level. </div><div class="t m0 x8 h8 y32 ff1 fs6 fc0 sc0 ls43 ws3a">The answer is to use more of the original signal in a way that doesn't</div><div class="t m0 x8 h8 y33 ff1 fs6 fc0 sc0 ls44 ws3b">increase the number of points in the frequency spectrum. This can be done</div><div class="t m0 x8 h8 y34 ff1 fs6 fc0 sc0 ls45 ws3c">by breaking the input signal into many 256 point <span class="ff3 ls46 ws0">segments</span><span class="ls47 ws3d">. Each of these</span></div><div class="t m0 x8 h8 y35 ff1 fs6 fc0 sc0 ls48 ws3e">segments is multiplied by the Hamming window, run through a 256 point</div><div class="t m0 x8 h8 y36 ff1 fs6 fc0 sc0 ls49 ws3f">DFT, and converted to polar notation. The resulting frequency spectra are</div><div class="t m0 x8 h8 y37 ff1 fs6 fc0 sc0 ls4a ws40">then <span class="ff3 ls4b ws0">averaged</span><span class="ls4c ws41"> to form a single 129 point frequency spectrum. Figure (e)</span></div><div class="t m0 x8 h8 y38 ff1 fs6 fc0 sc0 ls4d ws42">shows an example of averaging 100 of the frequency spectra typified by (d).</div><div class="t m0 x8 h8 y39 ff1 fs6 fc0 sc0 ls4e ws43">The improvement is obvious; the noise has been reduced to a level that</div><div class="t m0 x8 h8 y3a ff1 fs6 fc0 sc0 ls4f ws44">allows interesting features of the signal to be observed. Only the</div><div class="t m0 x8 h8 y3b ff3 fs6 fc0 sc0 ls50 ws0">magnitude<span class="ff1 ls51 ws45"> of the frequency domain is averaged in this manner; the </span><span class="ls52">phase</span></div><div class="t m0 x8 h8 y3c ff1 fs6 fc0 sc0 ls53 ws46">is usually discarded because it doesn't contain useful information. The</div><div class="t m0 x8 h8 y3d ff1 fs6 fc0 sc0 ls54 ws47">random noise reduces in proportion to the <span class="ff3 ls55 ws0">square-root</span><span class="ls56 ws48"> of the number of</span></div><div class="t m0 x8 h8 y3e ff1 fs6 fc0 sc0 ls57 ws49">segments. While 100 segments is typical, some applications might average</div><div class="t m0 x8 h8 y3f ff3 fs6 fc0 sc0 ls58 ws0">millions<span class="ff1 ls59 ws4a"> of segments to bring out weak features. </span></div><div class="t m0 x8 h8 y40 ff1 fs6 fc0 sc0 ls5a ws4b">There is also a second method for reducing spectral noise. Start by taking a</div><div class="t m0 x8 h8 y41 ff1 fs6 fc0 sc0 ls5b ws4c">very long DFT, say 16,384 points. The resulting frequency spectrum is high</div><div class="t m0 x8 h8 y42 ff1 fs6 fc0 sc0 ls5c ws4d">resolution (8193 samples), but very noisy. A low-pass digital filter is then</div><div class="t m0 x8 h8 y43 ff1 fs6 fc0 sc0 ls5d ws4e">used to <span class="ff3 ls5e ws0">smooth</span><span class="ls5f ws4f"> the spectrum, reducing the noise at the expense of the</span></div><div class="t m0 x8 h8 y44 ff1 fs6 fc0 sc0 ls60 ws50">resolution. For example, the simplest digital filter might average 64 adjacent</div><div class="t m0 x8 h8 y45 ff1 fs6 fc0 sc0 ls61 ws51">samples in the original spectrum to produce each sample in the filtered</div><div class="t m0 x8 h8 y46 ff1 fs6 fc0 sc0 ls62 ws52">spectrum. Going through the calculations, this provides about the same noise</div><div class="t m0 x8 h8 y47 ff1 fs6 fc0 sc0 ls63 ws53">and resolution as the first method, where the 16,384 points would be broken</div><div class="t m0 x8 h8 y48 ff1 fs6 fc0 sc0 ls64 ws54">into 64 segments of 256 points each.</div><div class="t m0 x8 h8 y49 ff1 fs6 fc0 sc0 ls65 ws55">Which method should you use? The first method is easier, because the</div><div class="t m0 x8 h8 y4a ff1 fs6 fc0 sc0 ls66 ws56">digital filter isn't needed. The second method has the <span class="ff3 ls67 ws0">potential</span><span class="ls68 ws57"> of better</span></div><div class="t m0 x8 h8 y4b ff1 fs6 fc0 sc0 ls69 ws58">performance, because the digital filter can be tailored to optimize the trade-</div><div class="t m0 x8 h8 y4c ff1 fs6 fc0 sc0 ls6a ws59">off between noise and resolution. However, this improved performance is</div><div class="t m0 x8 h8 y4d ff1 fs6 fc0 sc0 ls6b ws5a">seldom worth the trouble. This is because both noise and resolution can</div><div class="t m0 x8 h8 y4e ff1 fs6 fc0 sc0 ls6c ws5b">be improved by using <span class="ff3 ls6d ws5c">more data</span><span class="ls6e ws5d"> from the input signal. For example,</span></div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div>
<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622ba8af3d2fbb00078bfb26/bg3.jpg"><div class="t m0 x9 h2 y20 ff3 fs0 fc0 sc0 ls6f ws5e">Chapter 9- Applications of the DFT<span class="_ _1"> </span><span class="ff1 ls0 ws0">171</span></div><div class="t m0 xa ha y4f ff5 fs7 fc0 sc0 ls70 ws5f">Sample number</div><div class="t m0 xb hb y50 ff5 fs8 fc0 sc0 ls71 ws0">0<span class="_ _2"> </span>32<span class="_ _3"> </span>64<span class="_ _3"> </span>96<span class="_ _4"> </span><span class="ls72">128<span class="_ _5"> </span>160<span class="_ _5"> </span>192<span class="_ _5"> </span>224<span class="_ _5"> </span><span class="fc1 sc0">256</span></span></div><div class="t m0 xc hb y51 ff5 fs8 fc0 sc0 ls73 ws0">-0.5</div><div class="t m0 xd hb y52 ff5 fs8 fc0 sc0 ls74 ws0">0.0</div><div class="t m0 xd hb y53 ff5 fs8 fc0 sc0 ls74 ws0">0.5</div><div class="t m0 xd hb y54 ff5 fs8 fc0 sc0 ls74 ws0">1.0</div><div class="t m0 xd hb y55 ff5 fs8 fc0 sc0 ls74 ws0">1.5</div><div class="t m0 xe hc y50 ff1 fs8 fc0 sc0 ls75 ws0">255</div><div class="t m0 xf hd y56 ff1 fs9 fc0 sc0 ls76 ws60">b. Hamming window</div><div class="t m0 xa ha y57 ff5 fs7 fc0 sc0 ls70 ws5f">Sample number</div><div class="t m0 xb hb y58 ff5 fs8 fc0 sc0 ls71 ws0">0<span class="_ _2"> </span>32<span class="_ _3"> </span>64<span class="_ _3"> </span>96<span class="_ _4"> </span><span class="ls72">128<span class="_ _5"> </span>160<span class="_ _5"> </span>192<span class="_ _5"> </span>224<span class="_ _5"> </span><span class="fc1 sc0">256</span></span></div><div class="t m0 xc hb y59 ff5 fs8 fc0 sc0 ls73 ws0">-1.0</div><div class="t m0 xc hb y5a ff5 fs8 fc0 sc0 ls73 ws0">-0.5</div><div class="t m0 xd hb y5b ff5 fs8 fc0 sc0 ls74 ws0">0.0</div><div class="t m0 xd hb y5c ff5 fs8 fc0 sc0 ls74 ws0">0.5</div><div class="t m0 xd hb y5d ff5 fs8 fc0 sc0 ls74 ws0">1.0</div><div class="t m0 xe hc y5e ff1 fs8 fc0 sc0 ls75 ws0">255</div><div class="t m0 xf hd y3a ff1 fs9 fc0 sc0 ls77 ws61">c. Windowed signal</div><div class="t m0 x10 ha y57 ff5 fs7 fc0 sc0 ls78 ws0">Frequency</div><div class="t m0 x11 hb y5e ff5 fs8 fc0 sc0 ls71 ws0">0<span class="_ _6"> </span><span class="ls74">0.1<span class="_ _7"> </span>0.2<span class="_ _7"> </span>0.3<span class="_ _7"> </span>0.4<span class="_ _7"> </span>0.5</span></div><div class="t m0 x12 hb y59 ff5 fs8 fc0 sc0 ls71 ws0">0</div><div class="t m0 x12 hb y5f ff5 fs8 fc0 sc0 ls71 ws0">1</div><div class="t m0 x12 hb y60 ff5 fs8 fc0 sc0 ls71 ws0">2</div><div class="t m0 x12 hb y61 ff5 fs8 fc0 sc0 ls71 ws0">3</div><div class="t m0 x12 hb y62 ff5 fs8 fc0 sc0 ls71 ws0">4</div><div class="t m0 x12 hb y5b ff5 fs8 fc0 sc0 ls71 ws0">5</div><div class="t m0 x12 hb y63 ff5 fs8 fc0 sc0 ls71 ws0">6</div><div class="t m0 x12 hb y64 ff5 fs8 fc0 sc0 ls71 ws0">7</div><div class="t m0 x12 hb y65 ff5 fs8 fc0 sc0 ls71 ws0">8</div><div class="t m0 x12 hb y3a ff5 fs8 fc0 sc0 ls71 ws0">9</div><div class="t m0 x13 hb y5d ff5 fs8 fc0 sc0 ls71 ws0">10</div><div class="t m0 x14 hd y3a ff1 fs9 fc0 sc0 ls79 ws62">d. Single spectrum</div><div class="t m0 x10 ha y66 ff5 fs7 fc0 sc0 ls78 ws0">Frequency</div><div class="t m0 x11 hb y67 ff5 fs8 fc0 sc0 ls71 ws0">0<span class="_ _6"> </span><span class="ls74">0.1<span class="_ _7"> </span>0.2<span class="_ _7"> </span>0.3<span class="_ _7"> </span>0.4<span class="_ _7"> </span>0.5</span></div><div class="t m0 x12 hb y68 ff5 fs8 fc0 sc0 ls71 ws0">0</div><div class="t m0 x12 hb y69 ff5 fs8 fc0 sc0 ls71 ws0">1</div><div class="t m0 x12 hb y6a ff5 fs8 fc0 sc0 ls71 ws0">2</div><div class="t m0 x12 hb y6b ff5 fs8 fc0 sc0 ls71 ws0">3</div><div class="t m0 x12 hb y6c ff5 fs8 fc0 sc0 ls71 ws0">4</div><div class="t m0 x12 hb y6d ff5 fs8 fc0 sc0 ls71 ws0">5</div><div class="t m0 x12 hb y6e ff5 fs8 fc0 sc0 ls71 ws0">6</div><div class="t m0 x12 hb y6f ff5 fs8 fc0 sc0 ls71 ws0">7</div><div class="t m0 x12 hb y70 ff5 fs8 fc0 sc0 ls71 ws0">8</div><div class="t m0 x12 hb y71 ff5 fs8 fc0 sc0 ls71 ws0">9</div><div class="t m0 x13 hb y72 ff5 fs8 fc0 sc0 ls71 ws0">10</div><div class="t m0 x14 hd y71 ff1 fs9 fc0 sc0 ls7a ws63">e. Averaged spectrum</div><div class="t m0 x5 he y73 ff1 fsa fc0 sc0 ls7b ws64">FIGURE 9-1</div><div class="t m0 x5 he y74 ff1 fsa fc0 sc0 ls7c ws65">An example of spectral analysis. Figure (a) shows</div><div class="t m0 x5 he y47 ff1 fsa fc0 sc0 ls7d ws66">256 samples taken from a (simulated) undersea</div><div class="t m0 x5 he y75 ff1 fsa fc0 sc0 ls7e ws67">microphone at a rate of 160 samples per second.</div><div class="t m0 x5 he y76 ff1 fsa fc0 sc0 ls7f ws68">This signal is multiplied by the Hamming window</div><div class="t m0 x5 he y77 ff1 fsa fc0 sc0 ls80 ws69">shown in (b), resulting in the windowed signal in</div><div class="t m0 x5 he y78 ff1 fsa fc0 sc0 ls81 ws6a">(c). The frequency spectrum of the windowed</div><div class="t m0 x5 he y79 ff1 fsa fc0 sc0 ls82 ws6b">signal is found using the DFT, and is displayed in</div><div class="t m0 x5 he y7a ff1 fsa fc0 sc0 ls83 ws6c">(d) (magnitude only). Averaging 100 of these</div><div class="t m0 x5 he y7b ff1 fsa fc0 sc0 ls84 ws6d">spectra reduces the random noise, resulting in the</div><div class="t m0 x5 he y7c ff1 fsa fc0 sc0 ls85 ws6e">averaged frequency spectrum shown in (e).</div><div class="t m0 xa ha y7d ff5 fs7 fc0 sc0 ls70 ws5f">Sample number</div><div class="t m0 xb hb y7e ff5 fs8 fc0 sc0 ls71 ws0">0<span class="_ _2"> </span>32<span class="_ _3"> </span>64<span class="_ _3"> </span>96<span class="_ _4"> </span><span class="ls72">128<span class="_ _5"> </span>160<span class="_ _5"> </span>192<span class="_ _5"> </span>224<span class="_ _5"> </span><span class="fc1 sc0">256</span></span></div><div class="t m0 xc hb y7f ff5 fs8 fc0 sc0 ls73 ws0">-1.0</div><div class="t m0 xc hb y80 ff5 fs8 fc0 sc0 ls73 ws0">-0.5</div><div class="t m0 xd hb y81 ff5 fs8 fc0 sc0 ls74 ws0">0.0</div><div class="t m0 xd hb y82 ff5 fs8 fc0 sc0 ls74 ws0">0.5</div><div class="t m0 xd hb y83 ff5 fs8 fc0 sc0 ls74 ws0">1.0</div><div class="t m0 xe hc y7e ff1 fs8 fc0 sc0 ls75 ws0">255</div><div class="t m0 xf hd y84 ff1 fs9 fc0 sc0 ls86 ws6f">a. Measured signal</div><div class="t m0 x15 hf y85 ff1 fsb fc0 sc0 ls87 ws0">DFT</div><div class="t m1 x16 h10 y86 ff1 fs7 fc0 sc0 ls75 ws0">Amplitude</div><div class="t m1 x17 h10 y87 ff1 fs7 fc0 sc0 ls75 ws0">Amplitude</div><div class="t m1 x16 h10 y88 ff1 fs7 fc0 sc0 ls75 ws0">Amplitude<span class="_ _8"></span>Amplitude</div><div class="t m1 x17 h10 y89 ff1 fs7 fc0 sc0 ls75 ws0">Amplitude</div><div class="t m0 x18 h11 y8a ff1 fs5 fc0 sc0 ls88 ws70">Time Domain<span class="_ _9"> </span><span class="ls89 ws71">Frequency Domain</span></div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div>
<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622ba8af3d2fbb00078bfb26/bg4.jpg"><div class="t m0 x7 h2 y20 ff3 fs0 fc0 sc0 ls2d ws27">The Scientist and Engineer's Guide to Digital Signal Processing<span class="_ _0"></span><span class="ff1 ls0 ws0">172</span></div><div class="t m0 x8 h8 y21 ff1 fs6 fc0 sc0 ls8a ws72">imagine breaking the acquired data into 10,000 segments of 16,384 samples</div><div class="t m0 x8 h8 y22 ff1 fs6 fc0 sc0 ls8b ws73">each. This resulting frequency spectrum is high resolution (8193 points) <span class="ff3 ls8c ws0">and</span></div><div class="t m0 x8 h8 y23 ff1 fs6 fc0 sc0 ls8d ws74">low noise (10,000 averages). Problem solved! For this reason, we will only</div><div class="t m0 x8 h8 y24 ff1 fs6 fc0 sc0 ls8e ws75">look at the averaged segment method in this discussion. </div><div class="t m0 x8 h8 y26 ff1 fs6 fc0 sc0 ls8f ws76">Figure 9-2 shows an example spectrum from our undersea microphone,</div><div class="t m0 x8 h8 y27 ff1 fs6 fc0 sc0 ls90 ws77">illustrating the features that commonly appear in the frequency spectra of</div><div class="t m0 x8 h8 y28 ff1 fs6 fc0 sc0 ls91 ws78">acquired signals. Ignore the sharp peaks for a moment. Between 10 and 70</div><div class="t m0 x8 h9 y8b ff1 fs6 fc0 sc0 ls92 ws79">hertz, the signal consists of a relatively flat region. This is called <span class="ff4 ls93 ws7a">white noise</span></div><div class="t m0 x8 h8 y29 ff1 fs6 fc0 sc0 ls94 ws7b">because it contains an equal amount of all frequencies, the same as white light.</div><div class="t m0 x8 h8 y2a ff1 fs6 fc0 sc0 ls75 ws7c">It results from the noise on the time domain waveform being <span class="ff3 ls95 ws0">uncorrelated</span><span class="ls96 ws7d"> from</span></div><div class="t m0 x8 h8 y2b ff1 fs6 fc0 sc0 ls97 ws7e">sample-to-sample. That is, knowing the noise value present on any one sample</div><div class="t m0 x8 h8 y2c ff1 fs6 fc0 sc0 ls98 ws7f">provides no information on the noise value present on any other sample. For</div><div class="t m0 x8 h8 y2d ff1 fs6 fc0 sc0 ls99 ws80">example, the random motion of electrons in electronic circuits produces white</div><div class="t m0 x8 h8 y2e ff1 fs6 fc0 sc0 ls9a ws81">noise. As a more familiar example, the sound of the water spray hitting the</div><div class="t m0 x8 h8 y2f ff1 fs6 fc0 sc0 ls9b ws82">shower floor is white noise. The white noise shown in Fig. 9-2 could be</div><div class="t m0 x8 h8 y30 ff1 fs6 fc0 sc0 ls9c ws83">originating from any of several sources, including the analog electronics, or the</div><div class="t m0 x8 h8 y31 ff1 fs6 fc0 sc0 ls9d ws84">ocean itself.</div><div class="t m0 x8 h8 y32 ff1 fs6 fc0 sc0 ls9e ws85">Above 70 hertz, the white noise rapidly decreases in amplitude. This is a result</div><div class="t m0 x8 h8 y33 ff1 fs6 fc0 sc0 ls9f ws86">of the roll-off of the antialias filter. An ideal filter would pass all frequencies</div><div class="t m0 x8 h8 y34 ff1 fs6 fc0 sc0 lsa0 ws87">below 80 hertz, and block all frequencies above. In practice, a perfectly sharp</div><div class="t m0 x8 h8 y35 ff1 fs6 fc0 sc0 lsa1 ws88">cutoff isn't possible, and you should expect to see this gradual drop. If you</div><div class="t m0 x8 h8 y36 ff1 fs6 fc0 sc0 lsa2 ws89">don't, suspect that an aliasing problem is present. </div><div class="t m0 x8 h9 y38 ff1 fs6 fc0 sc0 lsa3 ws8a">Below about 10 hertz, the noise rapidly increases due to a curiosity called <span class="ff4 lsa4 ws0">1/f</span></div><div class="t m0 x8 h9 y39 ff4 fs6 fc0 sc0 lsa5 ws0">noise<span class="ff1 lsa6 ws8b"> (one-over-f noise). 1/f noise is a mystery. It has been measured in very</span></div><div class="t m0 x8 h8 y3a ff1 fs6 fc0 sc0 lsa7 ws8c">diverse systems, such as traffic density on freeways and electronic noise in</div><div class="t m0 x8 h8 y3b ff1 fs6 fc0 sc0 lsa8 ws8d">transistors. It probably could be measured in all systems, if you look low</div><div class="t m0 x8 h8 y3c ff1 fs6 fc0 sc0 lsa9 ws77">enough in frequency. In spite of its wide occurrence, a general theory and</div><div class="t m0 x8 h8 y3d ff1 fs6 fc0 sc0 lsaa ws8e">understanding of 1/f noise has eluded researchers. The cause of this noise can</div><div class="t m0 x8 h8 y3e ff1 fs6 fc0 sc0 lsab ws8f">be identified in some specific systems; however, this doesn't answer the</div><div class="t m0 x8 h8 y3f ff1 fs6 fc0 sc0 lsac ws90">question of why 1/f noise is everywhere. For common analog electronics and</div><div class="t m0 x8 h8 y8c ff1 fs6 fc0 sc0 lsad ws91">most physical systems, the transition between white noise and 1/f noise occurs</div><div class="t m0 x8 h8 y40 ff1 fs6 fc0 sc0 ls3a ws92">between about 1 and 100 hertz. </div><div class="t m0 x8 h8 y42 ff1 fs6 fc0 sc0 lsae ws93">Now we come to the sharp peaks in Fig. 9-2. The easiest to explain is at 60</div><div class="t m0 x8 h8 y43 ff1 fs6 fc0 sc0 lsaf ws94">hertz, a result of electromagnetic interference from commercial electrical</div><div class="t m0 x8 h8 y44 ff1 fs6 fc0 sc0 lsb0 ws95">power. Also expect to see smaller peaks at multiples of this frequency (120,</div><div class="t m0 x8 h8 y45 ff1 fs6 fc0 sc0 lsb1 ws96">180, 240 hertz, etc.) since the power line waveform is not a <span class="ff3 lsb2 ws0">perfect</span><span class="lsb3 ws97"> sinusoid.</span></div><div class="t m0 x8 h8 y46 ff1 fs6 fc0 sc0 lsb4 ws98">It is also common to find interfering peaks between 25-40 kHz, a favorite for</div><div class="t m0 x8 h8 y47 ff1 fs6 fc0 sc0 lsb5 ws99">designers of switching power supplies. Nearby radio and television stations</div><div class="t m0 x8 h8 y48 ff1 fs6 fc0 sc0 lsb6 ws9a">produce interfering peaks in the megahertz range. Low frequency peaks can be</div><div class="t m0 x8 h8 y8d ff1 fs6 fc0 sc0 lsb7 ws9b">caused by components in the system vibrating when shaken. This is called</div><div class="t m0 x8 h8 y49 ff3 fs6 fc0 sc0 lsb8 ws0">microphonics<span class="ff1 lsb9 ws9c">, and typically creates peaks at 10 to 100 hertz.</span></div><div class="t m0 x8 h8 y4b ff1 fs6 fc0 sc0 lsba ws9d">Now we come to the actual signals. There is a strong peak at 13 hertz, with</div><div class="t m0 x8 h8 y4c ff1 fs6 fc0 sc0 lsbb ws9e">weaker peaks at 26 and 39 hertz. As discussed in the next chapter, this is the</div><div class="t m0 x8 h8 y4d ff1 fs6 fc0 sc0 lsbc ws9f">frequency spectrum of a nonsinusoidal periodic waveform. The peak at 13</div><div class="t m0 x8 h8 y4e ff1 fs6 fc0 sc0 lsbd wsa0">hertz is called the fundamental frequency, while the peaks at 26 and 39</div></div><div class="pi" data-data='{"ctm":[1.839080,0.000000,0.000000,1.839080,0.000000,0.000000]}'></div></div>
