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                <c val="#008080">SECTION 1 - SIGNALS AND SYSTEMS</c>
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                <c val="#008080">Elementwise Magnitude and Phase</c>
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        <phrase>signals and systems</phrase>
        <phrase>element-wise magnitude and phase</phrase>
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        <phrase>mag function</phrase>
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        <phrase>phasecor function</phrase>
        <phrase>function, phasecor</phrase>
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        <phrase>makeri function</phrase>
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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 return, respectively, magnitudes, phase angles, phase angles corrected for jump discontinuities, and a restored data vector given vectors of magnitude and phase.</p>
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        <phrase>vector of magnitudes</phrase>
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        <phrase>phase angles, vector of</phrase>
        <phrase>vector of phase angles</phrase>
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            <b>
              <c val="#008080">•</c>
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            <b>
              <sp />
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          <b>x</b>, a real or complex-valued array of any size</p>
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              <c val="#008080">•</c>
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              <sp />
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          <b>phasevec</b>, a real-valued array of elementwise phases</p>
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          <b>magvec</b>, a real-valued array of element-wise magnitudes of the same length as <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">phasevec</inlineAttr></b></f></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">Each phase angle generated by <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">phase</inlineAttr></b></f> is selected in the range <f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">-p &lt; q £ +p</inlineAttr></f><inlineAttr font-weight="normal" line-through="false">. </inlineAttr>The function <b>makeri</b><b>(magvec, phasevec)</b> reverses the action of <b>mag</b> and <b>phase</b>. These functions are useful for manipulating the output of <b><u><link href="./110201i.xmcd" title="Links to section" popup="false">Fourier transform functions</link></u></b>.</p>
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            <b>
              <c val="#008080">•</c>
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          </f>biguity in angle selection can cause the phase of the output vector to differ from the phase of the input vector. The two vectors agree up to multiples of <f family="MS Serif" charset="0">2</f><f family="Symbol" charset="2">p</f>.</p>
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          </f>If an element in <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">magvec</inlineAttr></b></f> is negative, its absolute value is interpreted as the desired magnitude, and the corresponding element in <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">phasevec</inlineAttr></b></f> is advanced <f family="Symbol" charset="2">p</f> radians.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here's a simple complex vector that can be used to demonstrate the functions.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">mag</inlineAttr></b></f> function will find the element-wise magnitudes, and the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">phase</inlineAttr></b></f> function the element-wise phases.</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">Typically phase computations are done elementwise without regard to past phase history. This results in large jump discontinuities in the resulting phase data due to the change in angle from <f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">+p</inlineAttr></f> to <f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">-p</inlineAttr></f> when you cross the negative real axis. For example:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">Y</ml:id>
          <ml:matrix rows="4" cols="1">
            <ml:apply>
              <ml:plus />
              <ml:real>-1</ml:real>
              <ml:imag symbol="j">1</ml:imag>
            </ml:apply>
            <ml:apply>
              <ml:plus />
              <ml:real>-1</ml:real>
              <ml:imag symbol="j">0.01</ml:imag>
            </ml:apply>
            <ml:apply>
              <ml:minus />
              <ml:real>-1</ml:real>
              <ml:imag symbol="j">0.01</ml:imag>
            </ml:apply>
            <ml:apply>
              <ml:minus />
              <ml:real>-1</ml:real>
              <ml:imag symbol="j">1</ml:imag>
            </ml:apply>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">phase</ml:id>
            <ml:id xml:space="preserve">Y</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="1">
              <ml:real>2.3561944901923448</ml:real>
              <ml:real>3.131592986903128</ml:real>
              <ml:real>-3.131592986903128</ml:real>
              <ml:real>-2.3561944901923448</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="449" left="6" top="942.75" width="300.75" height="85.5" align-x="6" align-y="954" 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">phaseco</inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">r(x)</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><sp /></inlineAttr></f>eliminates these jump discontinuities. <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">phasecor</inlineAttr></b></f> takes a real vector representing phase information as argument then walks through the vector adding multiples of <f family="MS Serif" charset="0"><inlineAttr font-weight="normal" line-through="false">2</inlineAttr></f><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">p</inlineAttr></f> (maintaining the integrity of the data) when necessary to produce a continuous data set. For example, with the small vector above we have</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">phasecor</ml:id>
            <ml:apply>
              <ml:id xml:space="preserve">phase</ml:id>
              <ml:id xml:space="preserve">Y</ml:id>
            </ml:apply>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="1">
              <ml:real>2.3561944901923448</ml:real>
              <ml:real>3.131592986903128</ml:real>
              <ml:real>3.1515923202764582</ml:real>
              <ml:real>3.9269908169872414</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="483" left="0" top="1128" width="6000" height="6" align-x="0" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="460" left="6" top="1146.75" width="309" height="57" align-x="6" align-y="1158" 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">makeri</inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">(magvec, phasevec)</inlineAttr></b></f> reverses the action of <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">mag</inlineAttr></b></f> and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">phase</inlineAttr></b></f>. Arguments <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">x</inlineAttr></b></f><b><f family="MS Serif" charset="0"><sp /></f></b>and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">y</inlineAttr></b></f>, real vectors of the same length, are interpreted as the magnitude and phase data respectively of a complex data vector. <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">makeri</inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">(x, y)</inlineAttr></b></f> returns this complex vector.</p>
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    <region region-id="461" left="6" top="1212.75" width="59.25" height="14.25" align-x="6" align-y="1224" 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">For example:</p>
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    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">makeri</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">Mag</ml:id>
              <ml:id xml:space="preserve">θ</ml:id>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="1">
              <ml:complex>
                <ml:real>-3.7</ml:real>
                <ml:imag>4.5310435091427964E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>3</ml:real>
                <ml:imag>4</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>1.5999999999999999</ml:real>
                <ml:imag>-2.9</ml:imag>
              </ml:complex>
              <ml:real>0.4</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="14" />
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">X</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="1">
              <ml:real>-3.7</ml:real>
              <ml:complex>
                <ml:real>3</ml:real>
                <ml:imag>4</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>1.6</ml:real>
                <ml:imag>-2.9</ml:imag>
              </ml:complex>
              <ml:real>0.4</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="487" left="0" top="1338" width="6000" height="6" align-x="0" align-y="1338" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#800000">Example - Noisy Signal</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>signal, noisy</phrase>
      </indexes>
    </region>
    <region region-id="417" left="6" top="1368.75" width="221.25" height="14.25" align-x="6" align-y="1380" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Construct an example signal and add white noise:</p>
      </text>
    </region>
    <region region-id="113" left="36" top="1392.75" width="30.75" height="15.75" align-x="50.25" align-y="1404" 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">k</ml:id>
          <ml:real>6</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="114" left="84" top="1392.75" width="45.75" height="15.75" align-x="101.25" align-y="1404" 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">N</ml:id>
          <ml:real>301</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="17" />
    </region>
    <region region-id="115" left="144" top="1392.75" width="67.5" height="15.75" align-x="157.5" align-y="1404" 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">n</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:apply>
              <ml:minus />
              <ml:id xml:space="preserve">N</ml:id>
              <ml:real>1</ml:real>
            </ml:apply>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="116" left="36" top="1413.75" width="146.25" height="34.5" align-x="54.75" align-y="1434" 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">x</ml:id>
            <ml:id xml:space="preserve">n</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:id xml:space="preserve">ceil</ml:id>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:id xml:space="preserve">n</ml:id>
                  <ml:id xml:space="preserve">k</ml:id>
                </ml:apply>
                <ml:id xml:space="preserve">N</ml:id>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">sin</ml:id>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:real>2</ml:real>
                      <ml:id xml:space="preserve">k</ml:id>
                    </ml:apply>
                    <ml:id xml:space="preserve">π</ml:id>
                  </ml:apply>
                  <ml:id xml:space="preserve">n</ml:id>
                </ml:apply>
                <ml:id xml:space="preserve">N</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="419" left="192" top="1422.75" width="100.5" height="15.75" align-x="205.5" align-y="1434" 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">x</ml:id>
          <ml:apply>
            <ml:plus />
            <ml:id xml:space="preserve">x</ml:id>
            <ml:apply>
              <ml:mult style="default" />
              <ml:real>4</ml:real>
              <ml:apply>
                <ml:id xml:space="preserve">whiten</ml:id>
                <ml:id xml:space="preserve">N</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="421" left="6" top="1458.75" width="303.75" height="42.75" align-x="6" align-y="1470" 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">Function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">whiten</inlineAttr></b></f> is one of the <b><u><link href="./Noise.xmcd" title="Links to section" popup="false">pseudo-random signal generators</link></u></b> included in this Extension Pack. Then take the Fourier Transform using <b><u><link href="./110202i.xmcd" title="Links to section" popup="false">CFFT</link></u></b>:</p>
      </text>
      <indexes>
        <phrase>whiten function</phrase>
        <phrase>function, whiten</phrase>
        <phrase>pseudo-random signal generators</phrase>
        <phrase>signal generators, pseudo-random</phrase>
        <phrase>CFFT</phrase>
      </indexes>
    </region>
    <region region-id="500" left="36" top="1512.75" width="65.25" height="15.75" align-x="47.25" align-y="1524" 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">f</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">CFFT</ml:id>
            <ml:id xml:space="preserve">x</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="407" left="18" top="1536" width="296.25" height="121.5" align-x="18" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="22" />
      <rendering item-idref="23" />
    </region>
    <region region-id="130" left="6" top="1668.75" width="258.75" 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">The magnitude of our signal's Fourier transform looks like</p>
      </text>
    </region>
    <region region-id="132" left="36" top="1692.75" width="63.75" height="15.75" align-x="57.75" align-y="1704" 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">mg</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">mag</ml:id>
            <ml:id xml:space="preserve">f</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="24" />
    </region>
    <region region-id="135" left="6" top="1716" width="342.75" height="133.5" align-x="6" align-y="1716" 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" />
    </region>
    <region region-id="488" left="0" top="1872" width="6000" height="6" align-x="0" align-y="1872" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="140" left="6" top="1884.75" width="306.75" height="42.75" align-x="6" align-y="1896" 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 peaks are at <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">k and N - k,</inlineAttr></f> as we would expect. It may be more convenient to display the spectrum recentered so that the <b>dc</b> term is in the middle. For this, we use the <b><u><link href="./Recenter.xmcd" title="Links to section" popup="false">recenter function</link></u></b>.</p>
      </text>
      <indexes>
        <phrase>dc term</phrase>
        <phrase>recenter function</phrase>
        <phrase>function, recenter</phrase>
      </indexes>
    </region>
    <region region-id="336" left="36" top="1932.75" width="98.25" height="15.75" align-x="63.75" align-y="1944" 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">mg1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">recenter</ml:id>
            <ml:id xml:space="preserve">mg</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="27" />
    </region>
    <region region-id="337" left="6" top="1956" width="345.75" height="149.25" align-x="6" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="28" />
      <rendering item-idref="29" />
    </region>
    <region region-id="174" left="6" top="2118.75" width="201.75" height="14.25" align-x="6" align-y="2130" 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 phase data for the FFT is plotted below:</p>
      </text>
    </region>
    <region region-id="176" left="36" top="2142.75" width="69" height="15.75" align-x="55.5" align-y="2154" 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">ph</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">phase</ml:id>
            <ml:id xml:space="preserve">f</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="30" />
    </region>
    <region region-id="411" left="24" top="2172" width="290.25" height="121.5" align-x="24" align-y="2172" 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="189" left="6" top="2304.75" width="203.25" height="14.25" align-x="6" align-y="2316" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To correct the phase into a continuous signal,</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">phc</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">phasecor</ml:id>
            <ml:id xml:space="preserve">ph</ml:id>
          </ml:apply>
        </ml:define>
      </math>
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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">Given the magnitude and phase vectors of our sample signal, we can use <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">makeri</inlineAttr></b></f> to reconstruct the FFT of the original signal. </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">A segment of both the inverse FFT construction and the original forward one is displayed below.</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 inverse Fourier transform can then be used to recover the original signal.</p>
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