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list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./examples.xmcd" title="Links to Section Contents" popup="false"><sp count="4" />EXAMPLES</link></c></inlineAttr></u></f></p></text></region><region region-id="1559" left="36" top="102.75" width="111" height="13.5" align-x="36" align-y="114" 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="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"><f size="12"><c val="#800000">Cosine Smoothing</c></f></p></text><indexes><phrase>smoothing functions, user-written</phrase></indexes></region><region region-id="1561" left="36" top="116.25" width="134.25" height="36" align-x="40.5" align-y="126" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Verdana" charset="0" size="10"><inlineAttr line-through="false"><c val="#000">by James C. (Jim) Bach</c></inlineAttr></f></p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Verdana" charset="0" size="10"><inlineAttr line-through="false"><c val="#000">Delphi Electronics &amp; Safety</c></inlineAttr></f></p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Verdana" charset="0" size="10"><inlineAttr line-through="false"><c val="#000">Kokomo, IN, USA</c></inlineAttr></f></p></text></region><region region-id="1570" left="36" top="163.5" width="60.75" height="12.75" align-x="60.75" align-y="174" 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="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Introduction </p></text></region><region region-id="1666" left="36" top="188.25" width="310.5" height="96" align-x="52.5" align-y="198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad provides a number of built-in <b><u><link href="./smooth.xmcd" title="Links to section" popup="false">smoothing filters</link></u></b>, which are often sufficient for cleaning up your data/waveforms prior to the primary processing operation. Sometimes, however, the built-in filters oversmooth data, removing features that are not noise, and you cannot obtain the desired level or quality of filtering you need. For example, here is a noisy signal which we might want to filter before analyzing its key characteristics. </p></text><indexes><phrase>smoothing functions</phrase></indexes></region><region region-id="1679" left="66" top="303" width="139.5" height="12.75" align-x="96.75" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Signal</ml:id><ml:apply><ml:id xml:space="preserve">READPRN</ml:id><ml:str xml:space="preserve">signal.prn</ml:str></ml:apply></ml:define></math><rendering item-idref="4" /></region><region region-id="1678" left="222" top="303" width="74.25" height="12.75" align-x="230.25" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id 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push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Passing the signal through the three built-in smoothing filters we see:</p></text></region><region region-id="1691" left="66" top="405" width="47.25" height="16.5" align-x="96.75" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#80ffff" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Filt1">N</ml:id><ml:real>41</ml:real></ml:define></math><rendering item-idref="7" /></region><region region-id="1669" left="138" top="394.5" width="66" height="31.5" align-x="168.75" align-y="414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Filt2">N</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve" subscript="Filt1">N</ml:id><ml:real>300</ml:real></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="1692" left="66" top="435" width="170.25" height="16.5" align-x="120.75" align-y="444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Med">Filtered</ml:id><ml:apply><ml:id xml:space="preserve">medsmooth</ml:id><ml:sequence><ml:id xml:space="preserve">Signal</ml:id><ml:id xml:space="preserve" subscript="Filt1">N</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="1696" left="66" top="459" width="170.25" height="16.5" align-x="108.75" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="K">Filtered</ml:id><ml:apply><ml:id xml:space="preserve">ksmooth</ml:id><ml:sequence><ml:id xml:space="preserve">Time</ml:id><ml:id xml:space="preserve">Signal</ml:id><ml:id xml:space="preserve" subscript="Filt2">N</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="10" /></region><region region-id="1698" left="66" top="483" width="160.5" height="16.5" align-x="117.75" align-y="492" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Sup">Filtered</ml:id><ml:apply><ml:id xml:space="preserve">supsmooth</ml:id><ml:sequence><ml:id xml:space="preserve">Time</ml:id><ml:id xml:space="preserve">Signal</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="1699" left="0" top="516" width="6000" height="6" align-x="0" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1675" left="18" top="534" width="327.75" height="155.25" align-x="18" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="12" /><rendering item-idref="13" /></region><region region-id="1703" left="18" top="702" width="327.75" height="155.25" align-x="18" align-y="702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="14" /><rendering item-idref="15" /></region><region region-id="1701" left="18" top="858" width="333.75" height="155.25" align-x="18" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="16" /><rendering item-idref="17" /></region><region region-id="1714" left="36" top="1034.25" width="296.25" height="24" align-x="36" align-y="1044" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">With this dataset, there are problems with all of the above smoothing techniques.</p></text></region><region region-id="1715" left="0" top="1068" width="6000" height="6" align-x="0" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1753" left="180" top="1080" width="123.75" height="126" align-x="180" align-y="1080" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="18" /><rendering item-idref="19" /></region><region region-id="1759" left="36" top="1094.25" width="147" height="36" align-x="36" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Median smoothing artifically 'squared-up' the high-frequency ripple.</p></text><indexes><phrase>median smoothing</phrase><phrase>smoothing, median</phrase></indexes></region><region region-id="1754" left="180" top="1200" width="123.75" height="126" align-x="180" align-y="1200" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="20" /><rendering item-idref="21" /></region><region region-id="1725" left="36" top="1208.25" width="147.75" height="36" align-x="36" align-y="1218" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Gaussian smoothing overattenuated the high-frequency ripple,</p></text><indexes><phrase>Gaussian smoothing</phrase><phrase>smoothing, Gaussian</phrase></indexes></region><region region-id="1755" left="180" top="1314" width="123.75" height="126" align-x="180" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="22" /><rendering item-idref="23" /></region><region region-id="1765" left="36" top="1328.25" width="117.75" height="36" align-x="36" align-y="1338" 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="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and artificially 'rounded-off' the square pulse.</p></text></region><region region-id="1756" left="180" top="1434" width="123.75" height="126" align-x="180" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="24" /><rendering item-idref="25" /></region><region region-id="1767" left="36" top="1448.25" width="163.5" height="36" align-x="36" align-y="1458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Super smoothing removed the</p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">high-frequency ripple (highly over-filtered),</p></text><indexes><phrase>super smoothing</phrase><phrase>smoothing, super</phrase></indexes></region><region region-id="1757" left="180" top="1536" width="123.75" height="126" align-x="180" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="26" /><rendering item-idref="27" /></region><region region-id="1773" left="36" top="1550.25" width="147.75" height="24" align-x="36" align-y="1560" 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="0" text-align="left" list-style-type="none" tabs="inherit">and artificially rounded the square pulse.</p></text></region><region region-id="1794" left="0" top="1656" width="6000" height="6" align-x="0" align-y="1656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1799" left="36" top="1663.5" width="76.5" height="12.75" align-x="47.25" align-y="1674" 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="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mean smoothing</p></text><indexes><phrase>mean smoothing</phrase><phrase>smoothing, mean</phrase></indexes></region><region region-id="1776" left="36" top="1682.25" width="317.25" height="72" align-x="36" align-y="1692" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">In cases such as this, you can write your own sliding window filters using a Mathcad program block (multiline function), as shown in the example below. This "Mean" smoother filters your data (input vector "Data") by taking the average (mean) of the data points within the sliding window (width specified by "Width"): </p></text><indexes><phrase>sliding window filter</phrase><phrase>filters, sliding window</phrase></indexes></region><region region-id="1573" left="36" top="1779" width="467.25" height="291.75" align-x="151.5" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:globalDefine><ml:function><ml:id xml:space="preserve">MeanSmooth</ml:id><ml:boundVars><ml:id xml:space="preserve">Data</ml:id><ml:id xml:space="preserve">Width</ml:id></ml:boundVars></ml:function><ml:program><ml:str xml:space="preserve">Calculate 1/2-width of filtering window</ml:str><ml:localDefine><ml:id xml:space="preserve" subscript="Half">Width</ml:id><ml:apply><ml:id xml:space="preserve">trunc</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">Width</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:localDefine><ml:str xml:space="preserve">Calculate where we need to start collapsing the window</ml:str><ml:localDefine><ml:id xml:space="preserve">NearEnd</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine><ml:str xml:space="preserve">Iterate through all of the data points</ml:str><ml:for><ml:id xml:space="preserve">Pt</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply></ml:range><ml:program><ml:str xml:space="preserve">Calculate beginning (Start) of sliding window</ml:str><ml:ifThen><ml:apply><ml:greaterOrEqual /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply><ml:localDefine><ml:id xml:space="preserve">Start</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:id xml:space="preserve">Start</ml:id><ml:real>0</ml:real></ml:localDefine></ml:otherwise><ml:str xml:space="preserve">Calculate ending (Stop) of sliding window</ml:str><ml:ifThen><ml:apply><ml:lessOrEqual /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve">NearEnd</ml:id></ml:apply><ml:localDefine><ml:id xml:space="preserve">Stop</ml:id><ml:apply><ml:plus /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:id xml:space="preserve">Stop</ml:id><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply></ml:localDefine></ml:otherwise><ml:str xml:space="preserve">Use 'submatrix' to extract the window of data, then take mean of that chunk</ml:str><ml:localDefine><ml:apply><ml:indexer /><ml:id xml:space="preserve">Out</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:apply><ml:id xml:space="preserve">submatrix</ml:id><ml:sequence><ml:id xml:space="preserve">Data</ml:id><ml:id xml:space="preserve">Start</ml:id><ml:id xml:space="preserve">Stop</ml:id><ml:real>0</ml:real><ml:real>0</ml:real></ml:sequence></ml:apply></ml:apply></ml:localDefine></ml:program></ml:for><ml:str xml:space="preserve">Return the filtered data</ml:str><ml:return><ml:id xml:space="preserve">Out</ml:id></ml:return></ml:program></ml:globalDefine></math><rendering item-idref="28" /></region><region region-id="1780" left="36" top="2078.25" width="333.75" height="12" align-x="36" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Testing this filter with the signal yields</p></text></region><region region-id="1783" left="66" top="2097" width="182.25" height="16.5" align-x="125.25" align-y="2106" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Mean">Filtered</ml:id><ml:apply><ml:id xml:space="preserve">MeanSmooth</ml:id><ml:sequence><ml:id xml:space="preserve">Signal</ml:id><ml:id xml:space="preserve" subscript="Filt1">N</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="29" /></region><region region-id="1787" left="18" top="2118" width="345.75" height="185.25" align-x="18" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="30" /><rendering item-idref="31" /></region><region region-id="1853" left="0" top="2322" width="6000" height="6" align-x="0" align-y="2322" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1854" left="36" top="2329.5" width="82.5" height="12.75" align-x="36" align-y="2340" 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="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cosine smoothing</p></text><indexes><phrase>cosine smoothing</phrase><phrase>cosine weighted mean smoother</phrase><phrase>smoothing, cosine</phrase></indexes></region><region region-id="1790" left="36" top="2348.25" width="348.75" height="48" align-x="39" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A variant of the mean smoother, shown above, is the "Cosine Weighted Mean" smoother, which weights the averaged data points so that those near the center of the window have more weight (importance) than those near the edges of the window:</p></text></region><region region-id="1791" left="36" top="2415" width="386.25" height="477.75" align-x="168" align-y="2424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">CosMeanSmooth</ml:id><ml:boundVars><ml:id xml:space="preserve">Data</ml:id><ml:id xml:space="preserve">Width</ml:id></ml:boundVars></ml:function><ml:program><ml:str xml:space="preserve">Calculate 1/2-width of filtering window</ml:str><ml:localDefine><ml:id xml:space="preserve" subscript="Half">Width</ml:id><ml:apply><ml:id xml:space="preserve">trunc</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">Width</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:localDefine><ml:str xml:space="preserve">Calculate where we need to start collapsing the window</ml:str><ml:localDefine><ml:id xml:space="preserve">NearEnd</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine><ml:str xml:space="preserve">Store-away pi divided by 2 because we'll need it often</ml:str><ml:localDefine><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">π</ml:id><ml:real>2</ml:real></ml:apply></ml:localDefine><ml:str xml:space="preserve">Iterate through all of the data points</ml:str><ml:for><ml:id xml:space="preserve">Pt</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply></ml:range><ml:program><ml:str xml:space="preserve">Calculate beginning (Start) of sliding window</ml:str><ml:ifThen><ml:apply><ml:greaterOrEqual /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply><ml:localDefine><ml:id xml:space="preserve">Start</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:id xml:space="preserve">Start</ml:id><ml:real>0</ml:real></ml:localDefine></ml:otherwise><ml:str xml:space="preserve">Calculate ending (Stop) of sliding window</ml:str><ml:ifThen><ml:apply><ml:lessOrEqual /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve">NearEnd</ml:id></ml:apply><ml:localDefine><ml:id xml:space="preserve">Stop</ml:id><ml:apply><ml:plus /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:id xml:space="preserve">Stop</ml:id><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply></ml:localDefine></ml:otherwise><ml:str xml:space="preserve">Calculate 'Area Correction Factor' . . .</ml:str><ml:str xml:space="preserve"> . . . compensate for size of sliding window</ml:str><ml:localDefine><ml:id xml:space="preserve">η</ml:id><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:id xml:space="preserve">Start</ml:id><ml:id xml:space="preserve">Stop</ml:id></ml:bounds></ml:apply></ml:apply></ml:localDefine><ml:str xml:space="preserve">Iterate through the window of data . . .</ml:str><ml:str xml:space="preserve">Take sum of of cosine-weighted data points</ml:str><ml:localDefine><ml:apply><ml:indexer /><ml:id xml:space="preserve">Out</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">η</ml:id><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">Data</ml:id><ml:id xml:space="preserve">j</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:id xml:space="preserve">Start</ml:id><ml:id xml:space="preserve">Stop</ml:id></ml:bounds></ml:apply></ml:apply></ml:localDefine></ml:program></ml:for><ml:str xml:space="preserve">Return the filtered data</ml:str><ml:return><ml:id xml:space="preserve">Out</ml:id></ml:return></ml:program></ml:define></math><rendering item-idref="32" /></region><region region-id="1803" left="0" top="2916" width="6000" height="6" align-x="0" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1858" left="36" top="2930.25" width="323.25" height="183" align-x="36" align-y="2940" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Note the truncated (narrowed) window widths at the ends of the data, as controlled by the variables "Start" (beginning of sliding window) and "Stop" (ending of sliding window). At the 1st data point, only <f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sup>1</sup></c></inlineAttr></f>/<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sub>2</sub></c></inlineAttr></f> of the specified window width is available for averaging. As the program block iterates through the data, the averaging window expands until it is the specified width, which occurs when it has reached the "Width<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sub>Half</sub></c></inlineAttr></f>" data point. It continues to slide across the data until it nears the end of the data, at which point (at "NearEnd" data point) it begins to collapse. When it finally reaches the last data point, the averaging window is once again only <sup>1</sup>/<sub>2</sub> of the specified width. It is because of this automatically expanding/collapsing window that the summation of weighted points needs to be multiplied by the scaling factor containing the "Start" and "Stop" indices:</p></text></region><region region-id="1860" left="66" top="3128.25" width="147.75" height="59.25" align-x="66" align-y="3144" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><region region-id="1463" left="66" top="3128.25" width="427.5" height="58.5" align-x="80.25" align-y="3144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:localDefine><ml:id xml:space="preserve">η</ml:id><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:id xml:space="preserve">Start</ml:id><ml:id xml:space="preserve">Stop</ml:id></ml:bounds></ml:apply></ml:apply></ml:localDefine></math><rendering item-idref="33" /></region></p></text></region><region region-id="1856" left="36" top="3194.25" width="132" height="12" align-x="36" align-y="3204" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Testing this filter, we see:</p></text></region><region region-id="1812" left="66" top="3213" width="190.5" height="16.5" align-x="117.75" align-y="3222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Cos">Filtered</ml:id><ml:apply><ml:id xml:space="preserve">CosMeanSmooth</ml:id><ml:sequence><ml:id xml:space="preserve">Signal</ml:id><ml:id xml:space="preserve" subscript="Filt1">N</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="1815" left="18" top="3234" width="339.75" height="197.25" align-x="18" align-y="3234" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="35" /><rendering item-idref="36" /></region><region region-id="1819" left="36" top="3433.5" width="329.25" height="120.75" align-x="36" align-y="3444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note, however, that the area compensation factor (<f family="Symbol" charset="2" size="11"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">h</c></inlineAttr></f><f family="Verdana" charset="0" size="11"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">) </c></inlineAttr></f><f family="Verdana" charset="0">needs to be adjusted for each shape by performing an iterated sum of the shape's values in order to find the area under the curve. Since the curve (window) expands and collapses dynamically at the data endpoints, this needs to be calculated on-the-fly, or, at least when near the ends. This process can be made more efficient (since the collapsed window occurs a small fraction of the time) by only repeatedly calculating the window area when near the ends of the data, as shown in the following revision of the program block:</f></p></text></region><region region-id="1820" left="0" top="3564" width="6000" height="6" align-x="0" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1822" left="6" top="3585" width="412.5" height="581.25" align-x="138" align-y="3594" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">CosMeanSmooth</ml:id><ml:boundVars><ml:id xml:space="preserve">Data</ml:id><ml:id xml:space="preserve">Width</ml:id></ml:boundVars></ml:function><ml:program><ml:str xml:space="preserve">Calculate 1/2-width of filtering window</ml:str><ml:localDefine><ml:id xml:space="preserve" subscript="Half">Width</ml:id><ml:apply><ml:id xml:space="preserve">trunc</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">Width</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:localDefine><ml:str xml:space="preserve">Calculate where we need to start collapsing the window</ml:str><ml:localDefine><ml:id xml:space="preserve">NearEnd</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine><ml:str xml:space="preserve">Store-away pi divided by 2 because we'll need it often</ml:str><ml:localDefine><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">π</ml:id><ml:real>2</ml:real></ml:apply></ml:localDefine><ml:str xml:space="preserve">Calculate window area when free-and-clear of the ends of data</ml:str><ml:localDefine><ml:id xml:space="preserve" subscript="mid">η</ml:id><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:bounds></ml:apply></ml:apply></ml:localDefine><ml:str xml:space="preserve">Iterate through all of the data points</ml:str><ml:for><ml:id xml:space="preserve">Pt</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply></ml:range><ml:program><ml:str xml:space="preserve">Calculate beginning (Start) of sliding window</ml:str><ml:ifThen><ml:apply><ml:greaterOrEqual /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply><ml:localDefine><ml:id xml:space="preserve">Start</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:id xml:space="preserve">Start</ml:id><ml:real>0</ml:real></ml:localDefine></ml:otherwise><ml:str xml:space="preserve">Calculate ending (Stop) of sliding window</ml:str><ml:ifThen><ml:apply><ml:lessOrEqual /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve">NearEnd</ml:id></ml:apply><ml:localDefine><ml:id xml:space="preserve">Stop</ml:id><ml:apply><ml:plus /><ml:id xml:space="preserve">Pt</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:id xml:space="preserve">Stop</ml:id><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">Data</ml:id></ml:apply></ml:localDefine></ml:otherwise><ml:str xml:space="preserve">Calculate 'Area Correction Factor' . . .</ml:str><ml:str xml:space="preserve"> . . . compensate for size of sliding window</ml:str><ml:ifThen><ml:apply><ml:lessOrEqual /><ml:apply><ml:lessOrEqual /><ml:id xml:space="preserve" subscript="Half">Width</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:id xml:space="preserve">NearEnd</ml:id></ml:apply><ml:localDefine><ml:id xml:space="preserve">η</ml:id><ml:id xml:space="preserve" subscript="mid">η</ml:id></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:id xml:space="preserve">η</ml:id><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:id xml:space="preserve">Start</ml:id><ml:id xml:space="preserve">Stop</ml:id></ml:bounds></ml:apply></ml:apply></ml:localDefine></ml:otherwise><ml:str xml:space="preserve">Iterate through the window of data . . .</ml:str><ml:str xml:space="preserve">Take sum of of cosine-weighted data points</ml:str><ml:localDefine><ml:apply><ml:indexer /><ml:id xml:space="preserve">Out</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">η</ml:id><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">Data</ml:id><ml:id xml:space="preserve">j</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve">Pt</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:id xml:space="preserve">Start</ml:id><ml:id xml:space="preserve">Stop</ml:id></ml:bounds></ml:apply></ml:apply></ml:localDefine></ml:program></ml:for><ml:str xml:space="preserve">Return the filtered data</ml:str><ml:return><ml:id xml:space="preserve">Out</ml:id></ml:return></ml:program></ml:define></math><rendering item-idref="37" /></region><region region-id="1599" left="0" top="4188" width="6000" height="6" align-x="0" align-y="4188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1601" left="36" top="4202.25" width="394.5" height="12" align-x="36" align-y="4212" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Testing this more efficient version of the smoother:</p></text></region><region region-id="1603" left="36" top="4221" width="195" height="16.5" align-x="92.25" align-y="4230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Cos2">Filtered</ml:id><ml:apply><ml:id xml:space="preserve">CosMeanSmooth</ml:id><ml:sequence><ml:id xml:space="preserve">Signal</ml:id><ml:id xml:space="preserve" subscript="Filt1">N</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="1511" left="6" top="4242" width="357.75" height="239.25" align-x="6" align-y="4242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="39" /><rendering item-idref="40" /></region><region region-id="1827" left="36" top="4490.25" width="330.75" height="48" align-x="36" align-y="4500" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Notice that this filter doesn't suffer (as much) from the same problems of the built-in smoothing filters. The high-frequency ripple isn't as severely attenuated and the square pulse is more accurately reproduced:</p></text></region><region region-id="1831" left="36" top="4548" width="123.75" height="126" align-x="36" align-y="4548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="41" /><rendering item-idref="42" /></region><region region-id="1832" left="204" top="4548" width="123.75" height="126" align-x="204" align-y="4548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="43" /><rendering item-idref="44" /></region><region region-id="1839" left="0" top="4680" width="6000" height="6" align-x="0" align-y="4680" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1614" left="36" top="4688.25" width="342" height="36" align-x="36" align-y="4698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The graph below illustrates three varients of "Weighted Mean" smoothing windows that are eaily implemented with the programming block above:<sp count="2" /></p></text><indexes><phrase>weighted mean smoothing windows</phrase></indexes></region><region region-id="1838" left="60" top="4734" width="243.75" height="300" align-x="60" align-y="4734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><metafile mapping-mode="8" x-extent="174.8693" y-extent="215.4614" item-idref="45" /><rendering item-idref="46" /></region><region region-id="1620" left="36" top="5054.25" width="345" height="24" align-x="36" align-y="5064" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We can find the area under the curves, presuming untruncated (full-width) windows, as:</p></text></region><region region-id="1653" left="66" top="5083.5" width="30" height="28.5" align-x="80.25" align-y="5100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">π</ml:id><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="47" /></region><region region-id="1654" left="144" top="5091" width="61.5" height="12.75" align-x="175.5" align-y="5100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Width</ml:id><ml:real>10000</ml:real></ml:define></math><rendering item-idref="48" /></region><region region-id="1658" left="228" top="5091" width="102" height="16.5" align-x="276.75" align-y="5100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="Half">Width</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="auto-select" /><ml:real>50</ml:real><ml:id xml:space="preserve">%</ml:id></ml:apply><ml:id xml:space="preserve">Width</ml:id></ml:apply></ml:define></math><rendering item-idref="49" /></region><region region-id="1841" left="36" top="5119.5" width="83.25" height="12.75" align-x="36" align-y="5130" 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="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">"Cosine" Window</p></text></region><region region-id="1842" left="66" top="5143.5" width="186" height="58.5" align-x="197.25" align-y="5190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:div /><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:id xml:space="preserve">Width</ml:id></ml:bounds></ml:apply><ml:id xml:space="preserve">Width</ml:id></ml:apply><ml:unitOverride><ml:id xml:space="preserve">%</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>63.661976713159405</ml:real></result></ml:eval></math><rendering item-idref="50" /></region><region region-id="1843" left="36" top="5209.5" width="123.75" height="12.75" align-x="36" align-y="5220" 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="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">"Cosine Squared" Window</p></text></region><region region-id="1844" left="66" top="5236.5" width="161.25" height="61.5" align-x="201.75" align-y="5286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:div /><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">j</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:apply><ml:id xml:space="preserve">cos</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Π</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id xml:space="preserve">j</ml:id><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply><ml:id xml:space="preserve" subscript="Half">Width</ml:id></ml:apply></ml:apply></ml:apply><ml:real>2</ml:real></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:id xml:space="preserve">Width</ml:id></ml:bounds></ml:apply><ml:id xml:space="preserve">Width</ml:id></ml:apply><ml:unitOverride><ml:id xml:space="preserve">%</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>50.000000000000242</ml:real></result></ml:eval></math><rendering item-idref="51" /></region><region region-id="1845" left="0" top="5310" width="6000" height="6" align-x="0" align-y="5310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1847" left="36" top="5323.5" width="112.5" height="12.75" align-x="36" align-y="5334" 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="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">"Linear Ramp" Window</p></text></region><region region-id="1849" 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