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          <b>
            <c val="#008080">SECTION 2 - FAST FOURIER TRANSFORM</c>
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            <b>
              <c val="#008080">The Inverse Transformation ICFFT</c>
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        <phrase>Fast Fourier Transform</phrase>
        <phrase>Inverse Transformation ICFFT</phrase>
        <phrase>ICFFT, inverse transformation</phrase>
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              <c val="#000"> maps data in the frequency domain</c>
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          </f> to the time domain, and is the inverse of the <b><u><link href="./110202i.xmcd" title="Links to section" popup="false">CFFT</link></u></b> function.</p>
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          </f>, a real or complex-valued vector or matrix of any size, with constant frequency 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 Inverse Fourier transform. </p>
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            <c val="#800000">Example - 1D and 2D Data</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">Since the ICFFT is the inverse transform of the CFFT, we'd expect ICFFT to restore the original data input to the complex transform.</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">Swap in the variable <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">C</c></inlineAttr></b></f> in place of <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">d</c></inlineAttr></b></f> in the equations above to see this example for a complex matrix.</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">Just a<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">s with CFFT, the underlying sum behind ICFFT can be easily</c></inlineAttr></f> demonstrated in Mathcad.</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">In the<b> one-dimensional</b> case,</p>
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            <c val="#800000">Example - Filtering in Frequency</c>
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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">Generally, the ICFFT is used to obtain a time-domain result after modifying transformed data in the frequency domain by filtering or windowing. For example, the 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">We'll multiply the transform by a windowing function that rejects frequencies over 1. Since the frequency corresponding to element number i is </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">and then transform back to the time domain usin<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">g </c></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">ICFFT</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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