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										<sp count="6"/>PRECONDITIONING DATA</link>
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				<p style="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Smoothing by Windowing</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">Filtering, the process of convolving or multiplying a time series by another function, can be used for smoothing and other data preprocessing. For example, a lowpass window function multiplied in the Fourier transform domain can be used for smoothing. </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 first step in smoothing by filtering is to remove the linear trend; otherwise, the transform mostly displays frequency components of the linear signal, and not the higher frequency content. Make a vector representing the time of each sample and use the <f family="Verdana" charset="0">
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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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The complex DFT is used here, but we could have used the real-signal <f family="Verdana" charset="0">
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					</f>, as well, since our data is real and of length 2<f family="Verdana" charset="0">
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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 transformed spectrum will be samples long, and symmetric around the center, with DC on the ends of the spectrum, and higher frequencies in the center. Window the transform by zeroing the high frequency components 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This writes a window centered around 0, so the DC component is always included, and there are always an odd number of samples.<sp count="2"/>Then, the remaining window components are copied to the other end of the frequency cycle using the mod 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">Different window shapes change how much of the mid-frequency band is retained. Select a window:</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">Construct a window of width w using the selected window shape. Note that you have to select an appropriate width for each shape. </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">Apply the window by multiplying it with the transformed spectrum:</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. </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">Most of the noise has been removed, but also some of the periodicity of the original signal. You can select how much noise to remove vs. how much of the frequency content of the signal to retain by adjusting the window width. </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">This technique is appropriate for periodic signals with slight trends and noise at a frequency much higher than the original signal, but is less successful when the trend is strong, or the data has a shape which requires wide frequency content to describe it. You may wish to consider one of the other smoothing techniques described in this section for other types of data.</p>
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				<p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Other filtering techniques and windows</p>
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				<phrase>other filtering techniques and windows</phrase>
				<phrase>filtering techniques, other</phrase>
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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">This shows a single example of smoothing by windowing, where a window was used in the transform domain to low-pass filter the data. If you require more information on time-series filtering, filter design, and additional built-in functions for windowing and signal manipulation, many are available in the <b>
						<u>
							<link href="http://store.ptc.com/" title="Links to Web site" popup="false">Signal Processing Extension Pack</link>
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					</b> and the <b>
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							<link href="http://store.ptc.com/" title="Links to Web site" popup="false">Image Processing Extension Pack</link>
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