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          <b>
            <c val="#008080">SECTION 2 - FAST FOURIER TRANSFORM</c>
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              <c val="#008080">Forward Transform CFFT</c>
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        <phrase>Forward Transform CFFT</phrase>
        <phrase>Fast Fourier Transform</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">Returns the complex Fourier transform of <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">A</c></inlineAttr></b></f>, where </p>
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          </f>, a real or complex-valued vector or matrix of any size, with constant time or space between samples</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 result has the same number of rows and columns as <b>A</b>. For matrix arguments, <inlineAttr font-style="normal">CFFT</inlineAttr> returns the two-dimensional Fourier transform. The algorithm used is a prime divisor algorithm.</p>
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            <c val="#800000">Notes</c>
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              <sp />
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          </f>For real-valued vectors with length 2<sup>m</sup>, use <b><u><link href="./110204i.xmcd" title="Links to section" popup="false">FFT</link></u></b> instead for speed. </p>
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            <c val="#800000">Example - 1-D complex sinusoidal data</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Create some simulated complex data with N data points at sample spacing T:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">returning a complex vector in frequency space. The sampling frequency and the frequency corresponding to the kth entry in the transformed vector are then given by </p>
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          <ml:apply>
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            <ml:apply>
              <ml:div />
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              <ml:id xml:space="preserve">N</ml:id>
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      <plot disable-calc="false" item-idref="19" />
      <indexes>
        <phrase>transform magnitude</phrase>
      </indexes>
      <rendering item-idref="20" />
    </region>
    <region region-id="558" left="6" top="1020.75" width="278.25" height="42.75" align-x="6" align-y="1032" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The upper half of the frequency-domain samples represent the negative frequencies. Note that this complex signal has no negative frequency content.</p>
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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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The calculations underlying the transformation can be computed directly. The vector <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">DD</c></inlineAttr></b></f> below contains the result of a direct Mathcad calculation of the Fourier transform of the vector <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">d</c></inlineAttr></b></f>.</p>
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                              <ml:apply>
                                <ml:mult style="default" />
                                <ml:real>-2</ml:real>
                                <ml:id xml:space="preserve">π</ml:id>
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                            </ml:apply>
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                              <ml:div />
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                              <ml:id xml:space="preserve">N</ml:id>
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                          <ml:id xml:space="preserve">m</ml:id>
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                    <ml:real>1</ml:real>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Comparing a particular sample in the vectors, </p>
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              <ml:real>-0.0018818455384237865</ml:real>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">As you may have noticed, the computation using sums takes longer than <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">the </c></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">CFFT</c></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"> function since the function computes the sums indirectly, in an efficient manner</c></inlineAttr></f>. </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <c val="#800000">Example - 2D real data</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>2-D real data</phrase>
        <phrase>data, 2-D real</phrase>
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    </region>
    <region region-id="185" left="6" top="1350.75" width="279.75" height="42.75" align-x="6" align-y="1362" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"><sp /></c></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">CFFT</c></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"> fun</c></inlineAttr></f>ction also works for complex data and for two- dimensional arrays of data. For example, we can transform a two-dimensional rectangular window.</p>
      </text>
      <indexes>
        <phrase>CFFT function and 2-D arrays of data</phrase>
        <phrase>CFFT function and complex data</phrase>
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          <ml:id xml:space="preserve">N</ml:id>
          <ml:real>19</ml:real>
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                <ml:mult style="default" />
                <ml:real>2</ml:real>
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        <outputs />
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    <region region-id="193" left="6" top="1692.75" width="100.5" height="14.25" align-x="6" align-y="1704" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Computing the<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"><sp /></c></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">CFFT</c></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">:</c></inlineAttr></f></p>
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      <indexes>
        <phrase>CFFT, computing the</phrase>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">CB</ml:id>
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            <ml:id xml:space="preserve">CFFT</ml:id>
            <ml:id xml:space="preserve">B</ml:id>
          </ml:apply>
        </ml:define>
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      <rendering item-idref="34" />
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">and plotting magnitude as a surface (note the use of the vectorize operator):</p>
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            <ml:vectorize />
            <ml:apply>
              <ml:absval />
              <ml:id xml:space="preserve">CB</ml:id>
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    <region region-id="491" left="6" top="1992.75" width="324" height="57" align-x="6" align-y="2004" 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="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The transform of the 2D boxcar above should be a 2D sinc function. You may notice that the image above doesn't match that expectation. The frequency origin for the 2D transform is at (0,0). To shift the origin to the center, use the recenter function.</p>
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      <indexes>
        <phrase>recenter function</phrase>
        <phrase>function, recenter</phrase>
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        <ml:define>
          <ml:id xml:space="preserve">CB_ctr</ml:id>
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              <ml:id xml:space="preserve">CB_ctr</ml:id>
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          </ml:apply>
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            <c val="#800000">Properties of the CFFT</c>
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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">1.</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 real data vector <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">x</c></inlineAttr></b></f>, the ve<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">ctor </c></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">CFFT</c></inlineAttr></b></f><b><inlineAttr line-through="false"><c val="#000"><f family="Times New Roman" charset="0">(x)</f></c></inlineAttr></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"> is, i</c></inlineAttr></f>n general, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">complex. It is conjugate symmetric </c></inlineAttr></f>about the middle value. In the case of a two-dimensional real array, each column of the result is conjugate symmetric. </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">For any square array input <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">A</c></inlineAttr></b></f>, <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">CFFT</c></inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">(A)</c></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"> is symm</c></inlineAttr></f>etric. That is,</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">According to the Nyquist sampling theorem, the sampling frequency must be at least twice the highest frequency you wish resolved by the Fourier transform.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Last, we point out a problem that you may or may not come across. The nature of the prime divisor algorithm used in the fast Fourier transform is such that it will slow down when the number of data points is a large prime number. Bear this in mind if you need to work with very large prime numbers of data points.</p>
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