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          <b>
            <c val="#008080">SECTION 2 - FAST FOURIER TRANSFORM</c>
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        <phrase>fast Fourier transform</phrase>
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            <b>
              <c val="#008080">Filtering vs. Exponential Smoothing</c>
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              <c val="#008080">Compares smoothing by detrending and lowpass filtering in the transform domain to a time-domain </c>
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            <c val="#008080">iterative </c>
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              <c val="#008080">exponential smoothing algorithm.</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">In this example, we illustrate the technique of smoothing a time series by windowing the discrete Fourier transform of the series. We also compare the results with another common time-domain smoothing technique: exponential smoothing. First generate some sample 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 first step in smoothing by filtering is to remove the linear trend; otherwise, the transform will mostly display frequency components of the linear signal, and not the higher frequency content. Make a vector representing the time of each sample and use the <b>slope</b> function: </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">Remove the trend:</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">Take the 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">Window the transform by zeroing the elements in the middle, saving only the beginning and the end, which represent the low frequencies:</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">Now invert and add the trend back in. (Note that since vectors add element wise, we can do this with a single plus.) The smoothed data is in the vector S:</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">Before we plot, we'll try another smoothing technique that uses Mathcad's seeded iteration.</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">To get seed values for the iteration, estimate the stationary and trend values, s and b, by looking at the beginning of the series: </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">Note that the exponential smoothing doesn't begin to track the data effectively until enough points have been averaged by the smoothing coefficients, <f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">a</c></inlineAttr></f> and <f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">b</c></inlineAttr></f>, at about time 1.</p>
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