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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">Wavelet Packet Bases</p>
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        <phrase>wavelet packet bases</phrase>
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          </f>The function <f family="Times New Roman" charset="0"><c val="#008080"><b><inlineAttr line-through="false">pgridplot</inlineAttr></b></c></f> is defined in the file <b><u><link href="./Wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b> and in this worksheet. It is not part of the <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">Wavelets Extension Pack</inlineAttr></i></f> DLL.</p>
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            <inlineAttr line-through="false">creates the wavelet packet basis matrix corresponding to all wavelet packets at a given level. </inlineAttr>
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          <b>wavebs</b> creates the wavelet packet basis matrix corresponding to the wavelet transform at a given level<f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">. </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false"><b><c val="#008080">pgridplot</c></b></inlineAttr></f><inlineAttr line-through="false"><f family="Times New Roman" charset="0" size="12"> produces a plot of the wavelet packet basis matrix </f></inlineAttr><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">b</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false"> for levels 0, 1, .. </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">n</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr line-through="false">. The arguments to these functions are:</inlineAttr></f></p>
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          <b>n</b>
          <b>,</b> the number of levels in the basis.</p>
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          <b>d</b>
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          <b>b</b>
          <b>,</b> the wavelet packet basis matrix.</p>
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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">Creating a Basis for One Level of a Wavelet Packet Table</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The function <b>create_level</b> creates a <b>d-</b>dimensional wavelet packet basis comprising the collection of all wavelet packets at the level <b>n</b>. Recall that in one dimension, the wavelet packets at level <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n</inlineAttr></b></f> have oscillation parameters ranging over 0,1...2<f family="Times New Roman" charset="0" size="12"><b><inlineAttr font-style="normal" underline="false" line-through="false"><sup>n</sup></inlineAttr></b></f>-1. The result returned by <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">cr</inlineAttr></b></f><b>eate_level </b>specifies these packets, where column 0 indicates levels and column 1 indicates oscillation parameters. </p>
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        <phrase>wavelet packets' oscillation parameter</phrase>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">In two dimensions, the wavelet packets at level <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n</inlineAttr></b></f> are the products of the one-dimensional (1D) wavelet packets in each dimension. The result returned by <b>create_level </b>specifies this set of two-dimensional (2D) </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">wavelet packets, where column 0 indicates levels, column 1 indicates horizontal oscillation parameters, and column 2 indicates vertical oscillation parameters.</p>
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              <ml:real>2</ml:real>
              <ml:real>3</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="6" />
        </resultFormat>
      </math>
      <rendering item-idref="7" />
    </region>
    <region region-id="4600" left="0" top="1050" width="6000" height="6" align-x="0" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4664" left="6" top="1068.75" width="277.5" height="14.25" align-x="6" align-y="1080" 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">Wavelet Transform Basis from a Wavelet Packet Table</p>
      </text>
      <indexes>
        <phrase>wavelet transform basis from a wavelet packet table</phrase>
      </indexes>
    </region>
    <region region-id="4668" left="24" top="1098.75" width="59.25" height="14.25" align-x="24" align-y="1110" 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" size="12">
            <b>
              <inlineAttr line-through="false">wavebs(n,d)</inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>wavebs function</phrase>
      </indexes>
    </region>
    <region region-id="4663" left="6" top="1128.75" width="333" height="71.25" align-x="6" align-y="1140" 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">The function <b>wavebs</b> creates the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">d</inlineAttr></b></f>-dimensional wavelet packet basis with <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n</inlineAttr></b></f> levels corresponding to the wavelet transform. Since the mother wavelet is the wavelet packet of index 1, the wavelet basis comprises the index 1 packet at every level in addition to the father wavelet (packet index 0) at the last level. For example:</p>
      </text>
    </region>
    <region region-id="4535" left="150" top="1248.75" width="177.75" height="42.75" align-x="150" align-y="1260" 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">The first entry is the father wavelet. The remaining entries are mother wavelets at each level.</p>
      </text>
    </region>
    <region region-id="4669" left="24" top="1218.75" width="122.25" height="99.75" align-x="93.75" align-y="1272" 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">wavebs</ml:id>
            <ml:sequence>
              <ml:real>4</ml:real>
              <ml:real>1</ml:real>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="5" cols="2">
              <ml:real>4</ml:real>
              <ml:real>4</ml:real>
              <ml:real>3</ml:real>
              <ml:real>2</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="8" />
    </region>
    <region region-id="4680" left="6" top="1326.75" width="338.25" height="85.5" align-x="6" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">In two dimensions, the wavelet basis at each level is composed of three basis functions: the product of the mother in x and the mother in y; the product of the father in x and the mother in y; and the product of the mother in x and the father in y. At the final level we must also add the product of the father in x and the father in y to get a complete basis. For example:</p>
      </text>
    </region>
    <region region-id="4670" left="24" top="1419.75" width="136.5" height="141.75" align-x="93.75" align-y="1494" 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">wavebs</ml:id>
            <ml:sequence>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="7" cols="3">
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="9" />
    </region>
    <region region-id="30" left="300" top="1578" width="6000" height="6" align-x="300" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4223" left="6" top="1590.75" width="227.25" height="14.25" align-x="6" align-y="1602" 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">Plotting a Basis from a Wavelet Packet Table</p>
      </text>
      <indexes>
        <phrase>plotting a basis from a wavelet packet table</phrase>
      </indexes>
    </region>
    <region region-id="4638" left="30" top="1620.75" width="66.75" height="14.25" align-x="30" align-y="1632" 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" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#008080">pgridplot(b,n)</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>pgridplot function</phrase>
      </indexes>
    </region>
    <region region-id="4639" left="6" top="1644.75" width="335.25" height="88.5" align-x="6" align-y="1656" 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" size="12">
            <inlineAttr font-weight="normal" line-through="false">This function creates a </inlineAttr>
          </f>plot of the wavelet packet basis matrix <b>b</b> for levels 0, 1, .. <b>n</b><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">. </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">b</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"> is a 1D wavelet packet basis matrix with 2 columns</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">. For each row in </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">b</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">, column 0 indicates levels, and column 1 indicates oscillation parameters that may vary from </inlineAttr></f>0 to 2<b><sup>n</sup></b>-1. <f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">To view the pgrid plot, use the picture operator by pressing </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">[Ctrl] T</inlineAttr></b></f><inlineAttr font-weight="normal" line-through="false"><f family="Times New Roman" charset="0" size="12"> on the keyboard</f></inlineAttr><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false"><sp /></inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">or choosing </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">Picture</inlineAttr></b></f><inlineAttr font-weight="normal" line-through="false"><f family="Times New Roman" charset="0" size="12"> from the </f></inlineAttr><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false">Insert </inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">menu</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false">.</inlineAttr></f></p>
      </text>
    </region>
    <region region-id="4687" left="6" top="1740.75" width="340.5" height="28.5" align-x="6" align-y="1752" 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="Times New Roman" charset="0">
            <b>
              <inlineAttr font-style="normal" line-through="false">Note:</inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0">
            <b>
              <i>
                <inlineAttr line-through="false">
                  <sp />
                </inlineAttr>
              </i>
            </b>
          </f>The function <b><u><link href="./best.xmcd" title="Links to Best Basis" popup="false">basis_display2d</link></u></b> in the <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">Wavelets Extension Pack</inlineAttr></i></f> DLL is used to display a 2D wavelet packet basis matrix.</p>
      </text>
    </region>
    <region region-id="4641" left="0" top="1788" width="6000" height="6" align-x="0" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4640" left="6" top="1800.75" width="356.25" height="555" align-x="101.25" align-y="1812" 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">pgridplot</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">b</ml:id>
              <ml:id xml:space="preserve">levels</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:program>
            <ml:localDefine>
              <ml:id xml:space="preserve">levels</ml:id>
              <ml:apply>
                <ml:plus />
                <ml:id xml:space="preserve">levels</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:localDefine>
            <ml:localDefine>
              <ml:id xml:space="preserve">cols</ml:id>
              <ml:apply>
                <ml:pow />
                <ml:real>2</ml:real>
                <ml:id xml:space="preserve">levels</ml:id>
              </ml:apply>
            </ml:localDefine>
            <ml:localDefine>
              <ml:id xml:space="preserve">hsize</ml:id>
              <ml:real>4</ml:real>
            </ml:localDefine>
            <ml:localDefine>
              <ml:id xml:space="preserve">vsize</ml:id>
              <ml:real>16</ml:real>
            </ml:localDefine>
            <ml:for>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:range>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:id xml:space="preserve">levels</ml:id>
                  <ml:id xml:space="preserve">vsize</ml:id>
                </ml:apply>
              </ml:range>
              <ml:for>
                <ml:id xml:space="preserve">j</ml:id>
                <ml:range>
                  <ml:real>0</ml:real>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:id xml:space="preserve">cols</ml:id>
                    <ml:id xml:space="preserve">hsize</ml:id>
                  </ml:apply>
                </ml:range>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">Q</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>220</ml:real>
                </ml:localDefine>
              </ml:for>
            </ml:for>
            <ml:for>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:range>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:apply>
                    <ml:id xml:space="preserve">rows</ml:id>
                    <ml:id xml:space="preserve">b</ml:id>
                  </ml:apply>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:range>
              <ml:for>
                <ml:id xml:space="preserve">j</ml:id>
                <ml:range>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:id xml:space="preserve">vsize</ml:id>
                    <ml:apply>
                      <ml:indexer />
                      <ml:id xml:space="preserve">b</ml:id>
                      <ml:sequence>
                        <ml:id xml:space="preserve">i</ml:id>
                        <ml:real>0</ml:real>
                      </ml:sequence>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:id xml:space="preserve">vsize</ml:id>
                    <ml:parens>
                      <ml:apply>
                        <ml:plus />
                        <ml:apply>
                          <ml:indexer />
                          <ml:id xml:space="preserve">b</ml:id>
                          <ml:sequence>
                            <ml:id xml:space="preserve">i</ml:id>
                            <ml:real>0</ml:real>
                          </ml:sequence>
                        </ml:apply>
                        <ml:real>1</ml:real>
                      </ml:apply>
                    </ml:parens>
                  </ml:apply>
                </ml:range>
                <ml:program>
                  <ml:localDefine>
                    <ml:id xml:space="preserve">offset</ml:id>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:pow />
                        <ml:real>2</ml:real>
                        <ml:apply>
                          <ml:minus />
                          <ml:id xml:space="preserve">levels</ml:id>
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">b</ml:id>
                            <ml:sequence>
                              <ml:id xml:space="preserve">i</ml:id>
                              <ml:real>0</ml:real>
                            </ml:sequence>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                      <ml:id xml:space="preserve">hsize</ml:id>
                    </ml:apply>
                  </ml:localDefine>
                  <ml:for>
                    <ml:id xml:space="preserve">k</ml:id>
                    <ml:range>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">offset</ml:id>
                        <ml:apply>
                          <ml:indexer />
                          <ml:id xml:space="preserve">b</ml:id>
                          <ml:sequence>
                            <ml:id xml:space="preserve">i</ml:id>
                            <ml:real>1</ml:real>
                          </ml:sequence>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">offset</ml:id>
                        <ml:parens>
                          <ml:apply>
                            <ml:plus />
                            <ml:apply>
                              <ml:indexer />
                              <ml:id xml:space="preserve">b</ml:id>
                              <ml:sequence>
                                <ml:id xml:space="preserve">i</ml:id>
                                <ml:real>1</ml:real>
                              </ml:sequence>
                            </ml:apply>
                            <ml:real>1</ml:real>
                          </ml:apply>
                        </ml:parens>
                      </ml:apply>
                    </ml:range>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">Q</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">j</ml:id>
                          <ml:id xml:space="preserve">k</ml:id>
                        </ml:sequence>
                      </ml:apply>
                      <ml:real>127</ml:real>
                    </ml:localDefine>
                  </ml:for>
                </ml:program>
              </ml:for>
            </ml:for>
            <ml:for>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:range>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">levels</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:range>
              <ml:program>
                <ml:localDefine>
                  <ml:id xml:space="preserve">divisions</ml:id>
                  <ml:apply>
                    <ml:pow />
                    <ml:real>2</ml:real>
                    <ml:id xml:space="preserve">i</ml:id>
                  </ml:apply>
                </ml:localDefine>
                <ml:localDefine>
                  <ml:id xml:space="preserve">offset</ml:id>
                  <ml:apply>
                    <ml:pow />
                    <ml:real>2</ml:real>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">levels</ml:id>
                      <ml:id xml:space="preserve">i</ml:id>
                    </ml:apply>
                  </ml:apply>
                </ml:localDefine>
                <ml:for>
                  <ml:id xml:space="preserve">j</ml:id>
                  <ml:range>
                    <ml:real>0</ml:real>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">divisions</ml:id>
                      <ml:real>1</ml:real>
                    </ml:apply>
                  </ml:range>
                  <ml:for>
                    <ml:id xml:space="preserve">k</ml:id>
                    <ml:range>
                      <ml:sequence>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:id xml:space="preserve">vsize</ml:id>
                        </ml:apply>
                        <ml:parens>
                          <ml:apply>
                            <ml:plus />
                            <ml:apply>
                              <ml:mult style="default" />
                              <ml:id xml:space="preserve">i</ml:id>
                              <ml:id xml:space="preserve">vsize</ml:id>
                            </ml:apply>
                            <ml:real>2</ml:real>
                          </ml:apply>
                        </ml:parens>
                      </ml:sequence>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:parens>
                          <ml:apply>
                            <ml:plus />
                            <ml:id xml:space="preserve">i</ml:id>
                            <ml:real>1</ml:real>
                          </ml:apply>
                        </ml:parens>
                        <ml:id xml:space="preserve">vsize</ml:id>
                      </ml:apply>
                    </ml:range>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">Q</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">k</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:apply>
                              <ml:mult style="default" />
                              <ml:id xml:space="preserve">offset</ml:id>
                              <ml:id xml:space="preserve">j</ml:id>
                            </ml:apply>
                            <ml:id xml:space="preserve">hsize</ml:id>
                          </ml:apply>
                        </ml:sequence>
                      </ml:apply>
                      <ml:real>0</ml:real>
                    </ml:localDefine>
                  </ml:for>
                </ml:for>
                <ml:for>
                  <ml:id xml:space="preserve">j</ml:id>
                  <ml:range>
                    <ml:sequence>
                      <ml:real>0</ml:real>
                      <ml:real>2</ml:real>
                    </ml:sequence>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:id xml:space="preserve">cols</ml:id>
                      <ml:id xml:space="preserve">hsize</ml:id>
                    </ml:apply>
                  </ml:range>
                  <ml:localDefine>
                    <ml:apply>
                      <ml:indexer />
                      <ml:id xml:space="preserve">Q</ml:id>
                      <ml:sequence>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:id xml:space="preserve">vsize</ml:id>
                        </ml:apply>
                        <ml:id xml:space="preserve">j</ml:id>
                      </ml:sequence>
                    </ml:apply>
                    <ml:real>0</ml:real>
                  </ml:localDefine>
                </ml:for>
              </ml:program>
            </ml:for>
            <ml:for>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:range>
                <ml:sequence>
                  <ml:real>0</ml:real>
                  <ml:real>2</ml:real>
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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">Earlier, you created a wavelet packet table T for a signal. Now extract elements of that table for further processing. The set of elements to extract is called a <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">basis</inlineAttr></i></f>. For a basis to be valid, it must cover every column of the table once and only once. Use a pgrid plot to view the basis and verify that the basis is valid. Once you have a valid basis, extract the basis functions from the wavelet packet table and perform the inverse wavelet packet 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">Two simple bases are the DWT basis and the single-level basis. The single-level basis extracts all of the wavelet packets at a single level. An example of the level 3 basis is shown below. </p>
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