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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 <i>Discrete Hartley Transform</i> and its inverse, respectively.</p>
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          <b>x</b>, a real-valued input array</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">Returns the transformed array, a real array having the same number of elements as the input. The Hartley transform was designed to contain the information inherent to the Fourier transform but to avoid entering the complex domain. In order to accomplish the latter goal, the Hartley transform restricts its attention to real input data. </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 Hartley transform for a sum of sine waves is compared with the magnitude of the complex Fourier 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">The discrete Fourier Transform produces much the same result, as we'd expect from the definitions. </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 close relationship between the Hartley transform and the Fourier transform can be demonstrated by showing how to compute phase and magnitude from the Hartley transform. This is done below with a small data set. </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">Compute and recenter the discrete Hartley transform</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">Then compute the odd and even parts of the discrete Hartley transform (these formulas assume an odd value of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">N</inlineAttr></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">respectively.</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 compare the results to that obtained via the Fourier transform by employing the <f family="Times New Roman" charset="0"><i><inlineAttr font-weight="normal" line-through="false">Signal Processing Extension Pack's</inlineAttr></i></f> functions <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">phase</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">, </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">mag</inlineAttr></b></f> and <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">phasecor</f></inlineAttr></b>: the results will agree or in some cases differ by <inlineAttr font-weight="normal" line-through="false">2</inlineAttr><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">p</inlineAttr></f><inlineAttr font-weight="normal" line-through="false">.</inlineAttr></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 thorough treatment of the Hartley transform and its applications, see Ronald Bracewell's book <i>The Hartley Transform</i> (Oxford University Press). </p>
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