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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">WAVELETS EXTENSION PACK</p>
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        <p style="Section Title" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Appendix D: Wavelet Filters</p>
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        <phrase>wavelet filters</phrase>
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                <c val="#000">daublet(n)</c>
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        <phrase>daublet function</phrase>
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                <c val="#000">symmlet(n)</c>
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                <c val="#000">coiflet(n)</c>
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        <phrase>coiflet function</phrase>
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                <c val="#000">Bspline(m,n)</c>
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        <phrase>Bspline function</phrase>
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                <c val="#000">bl(n)</c>
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        <phrase>bl function</phrase>
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            <c val="#000">These functions create the low pass filter coefficients for particular </c>
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          <b>
            <u>
              <link href="./wavefami.xmcd" title="Links to section" popup="false">wavelet families</link>
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            <c val="#000"> in particular wavelet classes. The phrase </c>
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            <i>
              <c val="#000">
                <inlineAttr font-weight="normal" line-through="false">wavelet class</inlineAttr>
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            <c val="#000"> refers to the collection of all wavelets that can be created by any one of the above functions. For example, the wavelet class created by the function </c>
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            <b>
              <inlineAttr line-through="false">
                <c val="#000">coiflet(n)</c>
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            <c val="#000"> is the class of coiflets (named after the mathematician Ronald Coifman). The phrase </c>
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            <inlineAttr font-weight="normal" line-through="false">
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                <f family="Times New Roman" charset="0">wavelet family</f>
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          <inlineAttr font-weight="normal">
            <c val="#000"> refers to all of the wavelet packets that are associated with a particular wavelet class and a particular value of </c>
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            <b>
              <inlineAttr line-through="false">
                <c val="#000">n</c>
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            <c val="#000">. </c>
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            <c val="#000">For example, </c>
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                <b>daublet(12)</b>
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            <c val="#000"> are the wavelets based upon the minimum phase Daubechies wavelets with 12 coefficients. All arguments to the functions are integers:</c>
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        </p>
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        <phrase>coiflets class</phrase>
        <phrase>Coifman, R.</phrase>
        <phrase>low pass filter coefficients</phrase>
        <phrase>wavelet class</phrase>
        <phrase>wavelet famly</phrase>
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                <c val="#000"> n</c>
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            <c val="#000">, an integer describing a particular wavelet family</c>
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          <c val="#000080">.</c>
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              <c val="#008080">•</c>
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                <c val="#000">m</c>
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            <c val="#000">, an integer describing a particular wavelet family.</c>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">daublet(n)</p>
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            <c val="#000">This function creates the </c>
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            <b>
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                <c val="#000">n</c>
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            <c val="#000">-coefficient minimum phase compactly supported wavelets popularized by Ingrid Daubechies. The values of </c>
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                <c val="#000">n</c>
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            <c val="#000"> this function accepts are even numbers in the range from 2 to 20:</c>
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        <phrase>Daubechies, I.</phrase>
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          <ml:id xml:space="preserve">n</ml:id>
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              <ml:real>2</ml:real>
              <ml:real>4</ml:real>
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            <ml:real>20</ml:real>
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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 function returns the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n </inlineAttr></b></f>coefficients of the low pass filter used in all wavelets transform algorithms:</p>
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            <ml:real>4</ml:real>
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              <ml:real>0.4829629131445341</ml:real>
              <ml:real>0.83651630373780772</ml:real>
              <ml:real>0.22414386804201339</ml:real>
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              <ml:real>0.3128715909143166</ml:real>
              <ml:real>0.67563073629731951</ml:real>
              <ml:real>0.585354683654216</ml:real>
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              <ml:real>-0.28401554296158238</ml:real>
              <ml:real>0.0004724845739124</ml:real>
              <ml:real>0.1287474266204893</ml:real>
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              <ml:real>0.0139810279174001</ml:real>
              <ml:real>0.0087460940474065</ml:real>
              <ml:real>-0.004870352993452</ml:real>
              <ml:real>-0.000391740373377</ml:real>
              <ml:real>0.0006754494064506</ml:real>
              <ml:real>-0.0001174767841248</ml:real>
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      <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">
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">The result of </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr underline="false" line-through="false">
                <c val="#000">daublet</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000"> can be used in conjunction with other functions. For example with the </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr underline="false" line-through="false">
                <c val="#000">dwt</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000"> function:</c>
          </inlineAttr>
        </p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>15</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="10" />
    </region>
    <region region-id="4494" left="90" top="696.75" width="57" height="17.25" align-x="105.75" align-y="708" 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:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">v</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">rnd</ml:id>
            <ml:real>1</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
    <region region-id="4495" left="168" top="696.75" width="132" height="15.75" align-x="184.5" align-y="708" 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">w</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dwt</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">v</ml:id>
              <ml:real>2</ml:real>
              <ml:apply>
                <ml:id xml:space="preserve">daublet</ml:id>
                <ml:real>12</ml:real>
              </ml:apply>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="29" left="0" top="732" width="6000" height="6" align-x="0" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4496" left="6" top="744.75" width="55.5" height="14.25" align-x="6" align-y="756" 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="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">symmlet(n)</p>
      </text>
      <indexes>
        <phrase>symmlet function</phrase>
      </indexes>
    </region>
    <region region-id="4497" left="6" top="768.75" width="315.75" height="42.75" align-x="6" align-y="780" 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">
          <f family="Times New Roman" charset="0">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">This function creates the </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">n</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0">
            <c val="#000">
              <inlineAttr font-weight="normal" line-through="false">-coefficient least asymmetric compactly supported wavelet constructed by Ingrid Daubechies. The values of </inlineAttr>
            </c>
          </f>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">n </c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">this function accepts are even numbers in the range from 4 to 20:</c>
            </inlineAttr>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>Daubechies, I.</phrase>
      </indexes>
    </region>
    <region region-id="4506" left="24" top="822.75" width="63.75" height="15.75" align-x="37.5" align-y="834" 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="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">n</ml:id>
          <ml:range>
            <ml:sequence>
              <ml:real>4</ml:real>
              <ml:real>6</ml:real>
            </ml:sequence>
            <ml:real>20</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="4500" left="6" top="852.75" width="317.25" height="42.75" align-x="6" align-y="864" 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">
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">The function returns the </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr underline="false" line-through="false">
                <c val="#000">n </c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">coefficients of the low pass filter used in various wavelet transform algorithms. Note that the </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <inlineAttr underline="false" line-through="false">
              <b>
                <c val="#000">daublet(n)</c>
              </b>
            </inlineAttr>
          </f>
          <c val="#000">
            <inlineAttr font-weight="normal" underline="false"> and </inlineAttr>
          </c>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr underline="false" line-through="false">
                <c val="#000">symmlet(n)</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000"> filters are the same for </c>
          </inlineAttr>
          <b>
            <inlineAttr underline="false" line-through="false">
              <c val="#000">
                <f family="Times New Roman" charset="0">n</f>
              </c>
            </inlineAttr>
          </b>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">
              <sp />
            </c>
          </inlineAttr>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">= 4 and </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr underline="false" line-through="false">
                <c val="#000">n</c>
              </inlineAttr>
            </b>
          </f>
          <b>
            <inlineAttr underline="false" line-through="false">
              <c val="#000">
                <f family="Arial" charset="0">
                  <sp />
                </f>
              </c>
            </inlineAttr>
          </b>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">= 6.</c>
          </inlineAttr>
        </p>
      </text>
    </region>
    <region region-id="4571" left="12" top="911.25" width="126" height="78.75" align-x="72" align-y="954" 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:id xml:space="preserve">symmlet</ml:id>
            <ml:real>4</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="1">
              <ml:real>0.4829629131445341</ml:real>
              <ml:real>0.83651630373780772</ml:real>
              <ml:real>0.22414386804201339</ml:real>
              <ml:real>-0.12940952255126029</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="14" />
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    <region region-id="4572" left="12" top="1002.75" width="478.5" height="33.75" align-x="84.75" align-y="1020" 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:transpose />
            <ml:apply>
              <ml:id xml:space="preserve">symmlet</ml:id>
              <ml:real>16</ml:real>
            </ml:apply>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="1" cols="16">
              <ml:real>0.0018899503329009007</ml:real>
              <ml:real>-0.00030292051455166417</ml:real>
              <ml:real>-0.014952258336793822</ml:real>
              <ml:real>0.0038087520140603582</ml:real>
              <ml:real>0.049137179673480712</ml:real>
              <ml:real>-0.027219029916815851</ml:real>
              <ml:real>-0.051945838107879248</ml:real>
              <ml:real>0.36444189483598555</ml:real>
              <ml:real>0.77718575169981</ml:real>
              <ml:real>0.48135965125923963</ml:real>
              <ml:real>-0.061273359067913681</ml:real>
              <ml:real>-0.1432942383510657</ml:real>
              <ml:real>0.0076074873252853746</ml:real>
              <ml:real>0.031695087810347779</ml:real>
              <ml:real>-0.00054213233163559</ml:real>
              <ml:real>-0.0033824159513597117</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
        <resultFormat>
          <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" />
          <matrix display-style="auto" expand-nested-arrays="false" />
          <unit format-units="false" simplify-units="false" />
          <table item-idref="15" />
        </resultFormat>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="4501" left="6" top="1062.75" width="327.75" height="28.5" align-x="6" align-y="1074" 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">
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">The result of </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr font-style="normal" underline="false" line-through="false">
                <c val="#000">symmlet</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000"> can be used in </c>
          </inlineAttr>
          <c val="#000">conjunction with other functions.</c>
          <c val="#000">
            <inlineAttr font-weight="normal" underline="false"> For example with the </inlineAttr>
          </c>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr underline="false" line-through="false">
                <c val="#000">dwti</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000"> function:</c>
          </inlineAttr>
        </p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>63</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="17" />
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">v</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">rnd</ml:id>
            <ml:real>1</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="4342" left="186" top="1104.75" width="138" height="15.75" align-x="202.5" align-y="1116" 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="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">w</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dwti</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">v</ml:id>
              <ml:real>2</ml:real>
              <ml:apply>
                <ml:id xml:space="preserve">symmlet</ml:id>
                <ml:real>12</ml:real>
              </ml:apply>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="44" left="0" top="1140" width="6000" height="6" align-x="0" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4348" left="6" top="1152.75" width="44.25" height="14.25" align-x="6" align-y="1164" 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="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">coiflet(n)</p>
      </text>
      <indexes>
        <phrase>coiflet function</phrase>
      </indexes>
    </region>
    <region region-id="4510" left="6" top="1176.75" width="318.75" height="57" align-x="6" align-y="1188" 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">
          <inlineAttr font-weight="normal">
            <c val="#000">This function creates the </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">n</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000">-coefficient compactly supported coiflets which were created by Ingrid Daubechies following a suggestion of Ronald Coifman. The values of </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">n</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000"> this function accepts are the even integers 6, 12, 18, 24, and 30:</c>
          </inlineAttr>
        </p>
      </text>
      <indexes>
        <phrase>Daubechies, I.</phrase>
        <phrase>Coifman, R.</phrase>
      </indexes>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:sequence>
          <ml:apply>
            <ml:equal />
            <ml:id xml:space="preserve">n</ml:id>
            <ml:real>6</ml:real>
          </ml:apply>
          <ml:real>12</ml:real>
          <ml:real>18</ml:real>
          <ml:real>24</ml:real>
          <ml:real>30</ml:real>
        </ml:sequence>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="4511" left="6" top="1266.75" width="304.5" height="57" align-x="6" align-y="1278" 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">
          <inlineAttr font-weight="normal">
            <c val="#000">The function returns the </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">n </c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000">coefficients of the low pass filter used in various wavelet transform algorithms. They have simple sampling properties and are therefore particularly useful when processing continuous time data.</c>
          </inlineAttr>
        </p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">coiflet</ml:id>
            <ml:real>6</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="6" cols="1">
              <ml:real>-0.0727326195128539</ml:real>
              <ml:real>0.33789766245780922</ml:real>
              <ml:real>0.85257202021225542</ml:real>
              <ml:real>0.38486484686420286</ml:real>
              <ml:real>-0.0727326195128539</ml:real>
              <ml:real>-0.01565572813546454</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="21" />
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:transpose />
            <ml:apply>
              <ml:id xml:space="preserve">coiflet</ml:id>
              <ml:real>18</ml:real>
            </ml:apply>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="1" cols="18">
              <ml:real>-0.0037935128644910137</ml:real>
              <ml:real>0.0077825964273254174</ml:real>
              <ml:real>0.023452696141836264</ml:real>
              <ml:real>-0.065771911281855486</ml:real>
              <ml:real>-0.061123390002672855</ml:real>
              <ml:real>0.40517690240961679</ml:real>
              <ml:real>0.79377722262562045</ml:real>
              <ml:real>0.42848347637761869</ml:real>
              <ml:real>-0.071799821619312</ml:real>
              <ml:real>-0.082301927106885983</ml:real>
              <ml:real>0.034555027573061628</ml:real>
              <ml:real>0.0158805448636159</ml:real>
              <ml:real>-0.0090079761366615788</ml:real>
              <ml:real>-0.0025745176887502232</ml:real>
              <ml:real>0.0011175187708906014</ml:real>
              <ml:real>0.00046621696011288626</ml:real>
              <ml:real>-7.0983303138141238E-05</ml:real>
              <ml:real>-3.4599772836212559E-05</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
        <resultFormat>
          <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" />
          <matrix display-style="auto" expand-nested-arrays="false" />
          <unit format-units="false" simplify-units="false" />
          <table item-idref="22" />
        </resultFormat>
      </math>
      <rendering item-idref="23" />
    </region>
    <region region-id="4520" left="6" top="1524.75" width="372" height="28.5" align-x="6" align-y="1536" 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">
          <inlineAttr font-weight="normal" underline="false">
            <c val="#000">The result of </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr underline="false" line-through="false">
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            <b>
              <inlineAttr underline="false" line-through="false">
                <c val="#000">dwt</c>
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            <c val="#000"> function:</c>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Bspline(m,n)</p>
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      <indexes>
        <phrase>Bspline function</phrase>
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            <c val="#000">This function creates both the </c>
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          <i>
            <inlineAttr font-weight="normal">
              <c val="#000">analysis</c>
            </inlineAttr>
          </i>
          <inlineAttr font-weight="normal">
            <c val="#000"> and </c>
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          <i>
            <inlineAttr font-weight="normal">
              <c val="#000">synthesis</c>
            </inlineAttr>
          </i>
          <inlineAttr font-weight="normal">
            <c val="#000"> low pass filters of the time limited spline-based biorthogonal wavelets constructed by Ingrid Daubechies. </c>
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          <inlineAttr font-weight="normal">
            <c val="#000">Available pairs of </c>
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          <b>
            <inlineAttr line-through="false">
              <c val="#000">
                <f family="Times New Roman" charset="0">m</f>
              </c>
            </inlineAttr>
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            <c val="#000"> and </c>
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            <b>
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                <c val="#000">n</c>
              </inlineAttr>
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          <inlineAttr font-weight="normal">
            <c val="#000"> are:</c>
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        </p>
      </text>
      <indexes>
        <phrase>analysis low pass filters</phrase>
        <phrase>synthesis low pass filters</phrase>
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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 first number indicates the degree of the polynomial for the wavelet (the wavelet for <b>Bspline(2,6)</b> has degree 1). The second number indicates the length of the support of the wavelet (the wavelet for <b>Bspline(2,8)</b> has wider support than the wavelet for <b>Bspline(2,6)</b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">)</inlineAttr></f>.</p>
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            <c val="#000">The function returns a 2-column matrix, whose first column specifies the coefficients of the </c>
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              <c val="#000">analysis</c>
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            <c val="#000"> low pass filter, and whose second column specifies the coefficients of the </c>
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              <inlineAttr font-weight="normal">synthesis</inlineAttr>
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            <inlineAttr font-weight="normal"> low pass filter used in various wavelet transform algorithms. The major benefit of using the </inlineAttr>
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                <c val="#000">Bspline</c>
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            <c val="#000"> can be used in in conjunction with other functions. In particular, the symmetry of </c>
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            <inlineAttr underline="false" line-through="false">
              <b>
                <c val="#000">Bspline</c>
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            <inlineAttr font-weight="normal" underline="false"> wavelets allows them to be used in the symmetric wavelet transform function </inlineAttr>
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                <c val="#000">dwts</c>
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            <c val="#000">:</c>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">bl(n)</p>
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        <phrase>bl function</phrase>
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            <c val="#000">This function creates an approximation of the low pass filter of the Battle-Lemarie wavelet based upon the degree </c>
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                <c val="#000">n</c>
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            <c val="#000"> cardinal B-spline. The Battle-Lemarie wavelets are not time limited and therefore the associated low pass filter is an IIR filter. This means that a finite approximation to the low pass filter must be made in order to implement Battle-Lemarie wavelets on a computer. Because of these truncation errors, </c>
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            <c val="#000"> only creates Battle-Lemarie wavelets for degrees 0 through 6:</c>
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        <phrase>Battle-Lemarie wavelet</phrase>
        <phrase>IIR filter</phrase>
        <phrase>wavelet, Battle-Lemarie</phrase>
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                <c val="#000">Note:</c>
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            <f family="Times New Roman" charset="0">Unlike the previously discussed wavelet filters, the Battle-Lemarie wavelets will not lead to perfect reconstruction (because of the finite approximation). Moreover, use of the </f>
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            <inlineAttr font-weight="normal" line-through="false"> wavelet should be limited to the discrete wavelet transform functions, and not with general wavelet packet transform functions.</inlineAttr>
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            <c val="#000">The function returns an approximation of the low pass filter used in various wavelet transform algorithms. The number of coefficients in the approximation is 40 when </c>
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            <c val="#000"> is odd. One of the benefits of using a Battle-Lemarie wavelet is that it is a symmetric, orthonormal wavelet. You can therefore use the symmetric wavelet transform </c>
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                <f family="Times New Roman" charset="0">dwts</f>
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            <c val="#000"> without encountering inconveniences associated with biorthogonal wavelets.</c>
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        <phrase>orthonormal wavelet, symmetric</phrase>
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