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                <c val="#008080">SECTION 1 - SIGNALS AND SYSTEMS</c>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">These functions simulate, respectively, white noise, Gaussian noise, and 1/f noise.</p>
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          <b>N</b>, an integer argument between 2 and the maximum array size, inclusive</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The functions produce a vector of N elements, each element a random number selected according to the probability distribution specified by the function. White noise is useful as a flat-spectrum source for filter inputs. 1/f noise is typically found in hierarchically interacting or chaotic systems, and so can be used to simulate those systems.</p>
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          </f>The maximum array size for the noise generators will depend on your hardware's capabilities. </p>
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          </f>The sequence of numbers is determined by a seed value, l<f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">ike </inlineAttr></f>Mathcad's built-in function <b>rnd</b>. Information on the seed value can be found in the <i>Mathcad User's Guide</i>.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Consider a somewhat larger output vector</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The plot shows a traditional picture of white noise:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Note that the mean and standard deviation of the noise vector</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">do not deviate significantly from the theoretical values</p>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:sqrt />
            <ml:apply>
              <ml:div />
              <ml:real>1</ml:real>
              <ml:real>12</ml:real>
            </ml:apply>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.28867513459481287</ml:real>
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        </ml:eval>
      </math>
      <rendering item-idref="14" />
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      <pageBreak />
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#800000">Example - Gaussian Noise</c>
              </inlineAttr>
            </b>
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      </text>
      <indexes>
        <phrase>Gaussian noise, example</phrase>
        <phrase>noise, Gaussian, example</phrase>
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Similarly, <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">gaussn</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">(N) simulates</inlineAttr></f><f family="MS Serif" charset="0"><b><sp /></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">N</inlineAttr></f> independent sources, each following a Gaussian probability distribution of mean 0 and standard deviation 1. </p>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">gaussn</ml:id>
            <ml:real>6</ml:real>
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          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="6" cols="1">
              <ml:real>-0.4580637823325085</ml:real>
              <ml:real>1.4801818077953792</ml:real>
              <ml:real>0.27964007128785917</ml:real>
              <ml:real>0.47552098115698177</ml:real>
              <ml:real>0.37965556067517425</ml:real>
              <ml:real>-0.047524269118716891</ml:real>
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          </result>
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    <region region-id="258" left="6" top="1224.75" width="319.5" height="57" align-x="6" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Thus the elements in a <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">gaussn</inlineAttr></b></f> vector can <i>theoretically</i> be any real value but will most likely lie between -3 and +3. To see some more values, select the math region above and press <b>[F9]</b> to recalculate as many times as you'd like.</p>
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    <region region-id="259" left="6" top="1290.75" width="151.5" height="14.25" align-x="6" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Plot a sample Gaussian sequence:</p>
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        <ml:define>
          <ml:id xml:space="preserve">gn</ml:id>
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            <ml:id xml:space="preserve">gaussn</ml:id>
            <ml:real>1000</ml:real>
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    <region region-id="264" left="36" top="1338.75" width="78" height="15.75" align-x="52.5" align-y="1350" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">m</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
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              <ml:id xml:space="preserve">last</ml:id>
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Gaussian noise:</p>
      </text>
    </region>
    <region region-id="268" left="6" top="1392" width="325.5" height="145.5" align-x="6" align-y="1392" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="18" />
      <rendering item-idref="19" />
    </region>
    <region region-id="389" left="0" top="1560" width="6000" height="6" align-x="0" align-y="1560" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="269" left="6" top="1572.75" width="306" height="42.75" align-x="6" align-y="1584" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">To view the Gaussian data in its more familiar bell-shaped form, we use the following construction, computing a histogram by grouping the values in 20 bins of width .05:</p>
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    <region region-id="270" left="6" top="1620.75" width="72.75" height="14.25" align-x="6" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Number of bins:</p>
      </text>
    </region>
    <region region-id="287" left="36" top="1644.75" width="56.25" height="15.75" align-x="69.75" align-y="1656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">nslots</ml:id>
          <ml:real>20</ml:real>
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      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="288" left="114" top="1644.75" width="82.5" height="15.75" align-x="126" align-y="1656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">j</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:apply>
              <ml:minus />
              <ml:id xml:space="preserve">nslots</ml:id>
              <ml:real>1</ml:real>
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          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="277" left="6" top="1668.75" width="174.75" height="14.25" align-x="6" align-y="1680" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This vector holds the edges of the bins:</p>
      </text>
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    <region region-id="289" left="36" top="1690.5" width="113.25" height="33.75" align-x="67.5" align-y="1710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">slots</ml:id>
            <ml:id xml:space="preserve">j</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:plus />
            <ml:real>-3.5</ml:real>
            <ml:apply>
              <ml:mult style="default" />
              <ml:id xml:space="preserve">j</ml:id>
              <ml:apply>
                <ml:div />
                <ml:real>7</ml:real>
                <ml:id xml:space="preserve">nslots</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
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    <region region-id="281" left="6" top="1734.75" width="123.75" height="14.25" align-x="6" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Set the rightmost boundary:</p>
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    <region region-id="290" left="36" top="1758.75" width="78.75" height="17.25" align-x="89.25" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">slots</ml:id>
            <ml:id xml:space="preserve">nslots</ml:id>
          </ml:apply>
          <ml:real>3.5</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="23" />
    </region>
    <region region-id="52" left="0" top="1782" width="6000" height="6" align-x="0" align-y="1782" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="291" left="6" top="1794.75" width="81.75" height="14.25" align-x="6" align-y="1806" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Fill the histogram: </p>
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    </region>
    <region region-id="294" left="36" top="1818.75" width="92.25" height="15.75" align-x="56.25" align-y="1830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">pd</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">hist</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">slots</ml:id>
              <ml:id xml:space="preserve">gn</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
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      <rendering item-idref="24" />
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    <region region-id="344" left="6" top="1842.75" width="287.25" height="28.5" align-x="6" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Here is a plot of the distribution of our noise values, plotted as a bar graph: </p>
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    </region>
    <region region-id="391" left="24" top="1884" width="281.25" height="133.5" align-x="24" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="25" />
      <rendering item-idref="26" />
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    <region region-id="309" left="6" top="2028.75" width="285.75" height="28.5" align-x="6" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Compare the mean and standard deviation of the sample to the expected values of 0 and 1:</p>
      </text>
    </region>
    <region region-id="307" left="36" top="2064.75" width="94.5" height="15.75" align-x="87.75" align-y="2076" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">mean</ml:id>
            <ml:id xml:space="preserve">gn</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.054438110707130435</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="27" />
    </region>
    <region region-id="392" left="156" top="2064.75" width="94.5" height="15.75" align-x="207.75" align-y="2076" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">stdev</ml:id>
            <ml:id xml:space="preserve">gn</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.99518599098324911</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="28" />
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    <region region-id="393" left="0" top="2094" width="6000" height="6" align-x="0" align-y="2094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="395" left="6" top="2106.75" width="98.25" height="14.25" align-x="6" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#800000">Example - 1/f Noise</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>1/f noise, example</phrase>
        <phrase>noise, 1/f, example</phrase>
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    <region region-id="311" left="6" top="2124.75" width="292.5" height="57" align-x="6" align-y="2136" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">onefn</inlineAttr></b></f> computes an approximation of 1/f noise using a routine based on an algorithm developed by Richard Voss (see Martin Gardner's column in <i>Scientific American,</i> April 1978). Plot a sample below, adjusting to mean 0. </p>
      </text>
      <indexes>
        <phrase>onefn function</phrase>
        <phrase>function, onefn</phrase>
      </indexes>
    </region>
    <region region-id="317" left="36" top="2190.75" width="91.5" height="15.75" align-x="58.5" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">ofn</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">onefn</ml:id>
            <ml:real>1000</ml:real>
          </ml:apply>
        </ml:define>
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    <region region-id="318" left="162" top="2190.75" width="108" height="15.75" align-x="184.5" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">ofn</ml:id>
          <ml:apply>
            <ml:minus />
            <ml:id xml:space="preserve">ofn</ml:id>
            <ml:apply>
              <ml:id xml:space="preserve">mean</ml:id>
              <ml:id xml:space="preserve">ofn</ml:id>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="30" />
    </region>
    <region region-id="396" left="24" top="2214" width="262.5" height="145.5" align-x="24" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="31" />
      <rendering item-idref="32" />
    </region>
    <region region-id="322" left="6" top="2364.75" width="296.25" height="85.5" align-x="6" align-y="2376" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">1/f noise is found in the signals of many natural and man-made phenomena, including music, brain signals, weather, stock market movements, and other hierarchically interacting systems. For example, if one were to digitize samples of a music of a variety of styles (rock, jazz, classical), one would find that their power spectra are very close in distribution to 1/f noise.</p>
      </text>
    </region>
    <region region-id="326" left="6" top="2454.75" width="318" height="42.75" align-x="6" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">To compare the statistical structure of this noise with the white noise generated above, compute autocorrelations using <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">lcorr </inlineAttr></b></f>(see the section on <b><u><link href="./Corr.xmcd" title="Links to section" popup="false">Sample Correlation and Partial Autocorrelation</link></u></b>). </p>
      </text>
      <indexes>
        <phrase>lcorr function</phrase>
        <phrase>function, lcorr</phrase>
        <phrase>correlation, sample</phrase>
        <phrase>autocorrelation, partial</phrase>
      </indexes>
    </region>
    <region region-id="332" left="36" top="2508.75" width="90.75" height="15.75" align-x="52.5" align-y="2520" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">C</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">lcorr</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">wn</ml:id>
              <ml:id xml:space="preserve">wn</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="33" />
    </region>
    <region region-id="334" left="156" top="2508.75" width="92.25" height="15.75" align-x="172.5" align-y="2520" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">D</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">lcorr</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">ofn</ml:id>
              <ml:id xml:space="preserve">ofn</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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    </region>
    <region region-id="398" left="24" top="2532" width="272.25" height="145.5" align-x="24" align-y="2532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="35" />
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Unlike the white noise, the noise generated by <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">onefn</inlineAttr></b></f> is correlated, with the correlation decaying as the lag increases. </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Another way of looking at the differences between the white noise and 1/f noise is to compare their power spectra. Here we compute the magnitude squared of the FFT (frequency spectra) for each signal, and smooth the result by averaging in frequency groups of 20: </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The white noise spectrum is essentially flat, as we would expect:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The magnitude spectrum of our 1/f approximation falls off rapidly with increasing frequency; we plot the log of the magnitude, which is more informative here.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">As an experiment, we can try constructing a 1/f power spectrum, with random phases, and take the inverse transform to obtain the noise vector: </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Construct the first half of the transform. A 1/f power spectrum corresponds to a 1/sqrt(f) magnitude spectrum: </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">For a real signal, the second half is conjugate-symmetric around the midpoint:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The behavior of the autocorrelation is similar to our output from <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">onefn</inlineAttr></b></f>, with a correlation around .5 at lag 10. </p>
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