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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">WAVELETS EXTENSION PACK</p>
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          </f> These functions are defined in <b><u><link href="./Wavedef.xmcd" title="Link to wavedef.mcd" popup="false">wavedef.xmcd</link></u></b> and in this worksheet only. They are 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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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">These functions approximate or decompose a signal using the largest coefficients of a discrete wavelet transform. The arguments to the functions are:</p>
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                              <ml:apply>
                                <ml:minus />
                                <ml:id xml:space="preserve">J</ml:id>
                                <ml:real>1</ml:real>
                              </ml:apply>
                              <ml:id xml:space="preserve">k</ml:id>
                            </ml:sequence>
                          </ml:apply>
                        </ml:apply>
                        <ml:parens>
                          <ml:id xml:space="preserve">t</ml:id>
                        </ml:parens>
                      </ml:apply>
                    </ml:parens>
                  </ml:lambda>
                </ml:apply>
                <ml:id xml:space="preserve" font="1">...</ml:id>
              </ml:apply>
              <ml:apply>
                <ml:summation />
                <ml:lambda>
                  <ml:boundVars>
                    <ml:id xml:space="preserve">k</ml:id>
                  </ml:boundVars>
                  <ml:parens>
                    <ml:apply>
                      <ml:mult style="auto-select" />
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:apply>
                          <ml:indexer />
                          <ml:id xml:space="preserve">d</ml:id>
                          <ml:sequence>
                            <ml:real font="0">1</ml:real>
                            <ml:id xml:space="preserve">k</ml:id>
                          </ml:sequence>
                        </ml:apply>
                        <ml:apply>
                          <ml:indexer />
                          <ml:id xml:space="preserve">mother</ml:id>
                          <ml:sequence>
                            <ml:real>1</ml:real>
                            <ml:id xml:space="preserve">k</ml:id>
                          </ml:sequence>
                        </ml:apply>
                      </ml:apply>
                      <ml:parens>
                        <ml:id xml:space="preserve">t</ml:id>
                      </ml:parens>
                    </ml:apply>
                  </ml:parens>
                </ml:lambda>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:apply>
      </math>
      <rendering item-idref="6" />
    </region>
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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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">or, more concisely:</p>
      </text>
    </region>
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        <ml:apply>
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            <ml:id xml:space="preserve">f</ml:id>
            <ml:id xml:space="preserve">t</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:plus />
            <ml:apply>
              <ml:summation />
              <ml:lambda>
                <ml:boundVars>
                  <ml:id xml:space="preserve">k</ml:id>
                </ml:boundVars>
                <ml:parens>
                  <ml:apply>
                    <ml:mult style="auto-select" />
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">S</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:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">father</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:apply>
                    <ml:parens>
                      <ml:id xml:space="preserve">t</ml:id>
                    </ml:parens>
                  </ml:apply>
                </ml:parens>
              </ml:lambda>
            </ml:apply>
            <ml:apply>
              <ml:summation />
              <ml:lambda>
                <ml:boundVars>
                  <ml:id xml:space="preserve">j</ml:id>
                </ml:boundVars>
                <ml:apply>
                  <ml:summation />
                  <ml:lambda>
                    <ml:boundVars>
                      <ml:id xml:space="preserve">k</ml:id>
                    </ml:boundVars>
                    <ml:parens>
                      <ml:apply>
                        <ml:mult style="auto-select" />
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">d</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:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">mother</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:apply>
                        <ml:parens>
                          <ml:id xml:space="preserve">t</ml:id>
                        </ml:parens>
                      </ml:apply>
                    </ml:parens>
                  </ml:lambda>
                </ml:apply>
              </ml:lambda>
              <ml:bounds>
                <ml:real>1</ml:real>
                <ml:id xml:space="preserve">J</ml:id>
              </ml:bounds>
            </ml:apply>
          </ml:apply>
        </ml:apply>
      </math>
      <rendering item-idref="7" />
    </region>
    <region region-id="4676" left="6" top="594.75" width="334.5" height="86.25" align-x="6" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">where J is the number of multiresolution components (or scales) and k ranges from 1 to <region region-id="738" left="83.25" top="609" width="30.75" height="21" align-x="93.75" align-y="625.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">N</ml:id><ml:apply><ml:pow /><ml:real>2</ml:real><ml:apply><ml:neg /><ml:id xml:space="preserve">j</ml:id></ml:apply></ml:apply></ml:apply></math><rendering item-idref="8" /></region><f family="Arial" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000080"><sp /></c></inlineAttr></f>. The jth component has <region region-id="4710" left="228.75" top="609" width="30.75" height="21" align-x="239.25" align-y="625.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">N</ml:id><ml:apply><ml:pow /><ml:real>2</ml:real><ml:apply><ml:neg /><ml:id xml:space="preserve">j</ml:id></ml:apply></ml:apply></ml:apply></math><rendering item-idref="9" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> coefficients.</inlineAttr></f> The coefficients <region region-id="359" left="59.25" top="630.75" width="86.25" height="17.25" align-x="111.75" align-y="642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""><math optimize="false" disable-calc="false"><ml:sequence><ml:apply><ml:indexer /><ml:id xml:space="preserve">S</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:apply><ml:indexer /><ml:id xml:space="preserve">d</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:apply><ml:mult style="auto-select" /><ml:id xml:space="preserve" font="1">...</ml:id><ml:apply><ml:indexer /><ml:id xml:space="preserve">d</ml:id><ml:sequence><ml:real>1</ml:real><ml:id xml:space="preserve">k</ml:id></ml:sequence></ml:apply></ml:apply></ml:sequence></math><rendering item-idref="10" /></region>, , are the <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false"><c val="#000">wavelet transform coefficients</c></inlineAttr></i></f>. The functions <region region-id="360" left="69" top="648.75" width="54" height="17.25" align-x="109.5" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">father</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:parens><ml:id xml:space="preserve">t</ml:id></ml:parens></ml:apply></math><rendering item-idref="11" /></region> and <region region-id="361" left="145.5" top="648.75" width="58.5" height="17.25" align-x="190.5" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">mother</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:parens><ml:id xml:space="preserve">t</ml:id></ml:parens></ml:apply></math><rendering item-idref="12" /></region><c val="#000080"><f family="Arial" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sp /></inlineAttr></f></c>are the approximating wavelet functions. </p>
      </text>
      <indexes>
        <phrase>multiresolution component</phrase>
        <phrase>wavelet transform coefficients</phrase>
      </indexes>
    </region>
    <region region-id="4602" left="6" top="696.75" width="337.5" height="32.25" align-x="6" align-y="708" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The functions <region region-id="362" left="69" top="696.75" width="54" height="17.25" align-x="109.5" 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:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">father</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:parens><ml:id xml:space="preserve">t</ml:id></ml:parens></ml:apply></math><rendering item-idref="13" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sp /></inlineAttr></f>and <region region-id="363" left="145.5" top="696.75" width="58.5" height="17.25" align-x="190.5" 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:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">mother</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:parens><ml:id xml:space="preserve">t</ml:id></ml:parens></ml:apply></math><rendering item-idref="14" /></region> are generated from father(t) and mother(t) through scaling and translation as follows:</p>
      </text>
    </region>
    <region region-id="4645" left="24" top="735.75" width="296.25" height="51.75" align-x="207" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:apply>
          <ml:equal />
          <ml:apply>
            <ml:equal />
            <ml:apply>
              <ml:mult style="auto-select" />
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">father</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:parens>
                <ml:id xml:space="preserve">t</ml:id>
              </ml:parens>
            </ml:apply>
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:pow />
                <ml:real>2</ml:real>
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:neg />
                    <ml:id xml:space="preserve">J</ml:id>
                  </ml:apply>
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:id xml:space="preserve">father</ml:id>
                <ml:apply>
                  <ml:minus />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:pow />
                      <ml:real>2</ml:real>
                      <ml:apply>
                        <ml:neg />
                        <ml:id xml:space="preserve">J</ml:id>
                      </ml:apply>
                    </ml:apply>
                    <ml:id xml:space="preserve">t</ml:id>
                  </ml:apply>
                  <ml:id xml:space="preserve">k</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
          </ml:apply>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:pow />
              <ml:real>2</ml:real>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:neg />
                  <ml:id xml:space="preserve">J</ml:id>
                </ml:apply>
                <ml:real>2</ml:real>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">father</ml:id>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">t</ml:id>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:pow />
                      <ml:real>2</ml:real>
                      <ml:id xml:space="preserve">J</ml:id>
                    </ml:apply>
                    <ml:id xml:space="preserve">k</ml:id>
                  </ml:apply>
                </ml:apply>
                <ml:apply>
                  <ml:pow />
                  <ml:real>2</ml:real>
                  <ml:id xml:space="preserve">J</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:apply>
      </math>
      <rendering item-idref="15" />
    </region>
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        <ml:apply>
          <ml:equal />
          <ml:apply>
            <ml:equal />
            <ml:apply>
              <ml:mult style="auto-select" />
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">mother</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:parens>
                <ml:id xml:space="preserve">t</ml:id>
              </ml:parens>
            </ml:apply>
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:pow />
                <ml:real>2</ml:real>
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:neg />
                    <ml:id xml:space="preserve">j</ml:id>
                  </ml:apply>
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:id xml:space="preserve">mother</ml:id>
                <ml:apply>
                  <ml:minus />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:pow />
                      <ml:real>2</ml:real>
                      <ml:apply>
                        <ml:neg />
                        <ml:id xml:space="preserve">j</ml:id>
                      </ml:apply>
                    </ml:apply>
                    <ml:id xml:space="preserve">t</ml:id>
                  </ml:apply>
                  <ml:id xml:space="preserve">k</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
          </ml:apply>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:pow />
              <ml:real>2</ml:real>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:neg />
                  <ml:id xml:space="preserve">j</ml:id>
                </ml:apply>
                <ml:real>2</ml:real>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">mother</ml:id>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">t</ml:id>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:pow />
                      <ml:real>2</ml:real>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:apply>
                    <ml:id xml:space="preserve">k</ml:id>
                  </ml:apply>
                </ml:apply>
                <ml:apply>
                  <ml:pow />
                  <ml:real>2</ml:real>
                  <ml:id xml:space="preserve">j</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:apply>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="4334" left="0" top="852" width="6000" height="6" align-x="0" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="4319" left="6" top="858.75" width="230.25" height="14.25" align-x="6" align-y="870" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
              <c val="#000">The wavelet coefficients are given by the integrals: </c>
            </inlineAttr>
          </f>
        </p>
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      <math optimize="false" disable-calc="false">
        <ml:apply>
          <ml:equal />
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">S</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:apply>
            <ml:integral auto-algorithm="true" />
            <ml:lambda>
              <ml:boundVars>
                <ml:id xml:space="preserve">t</ml:id>
              </ml:boundVars>
              <ml:apply>
                <ml:mult style="default" />
                <ml:apply>
                  <ml:mult style="auto-select" />
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">father</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:parens>
                    <ml:id xml:space="preserve">t</ml:id>
                  </ml:parens>
                </ml:apply>
                <ml:apply>
                  <ml:id xml:space="preserve">f</ml:id>
                  <ml:id xml:space="preserve">t</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:lambda>
            <ml:bounds>
              <ml:apply>
                <ml:neg />
                <ml:id xml:space="preserve">∞</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">∞</ml:id>
            </ml:bounds>
          </ml:apply>
        </ml:apply>
      </math>
      <rendering item-idref="17" />
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    <region region-id="4323" left="6" top="936.75" width="16.5" height="14.25" align-x="6" align-y="948" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">and</p>
      </text>
    </region>
    <region region-id="4654" left="24" top="957" width="152.25" height="45.75" align-x="51.75" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:apply>
          <ml:equal />
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">d</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:apply>
            <ml:integral auto-algorithm="true" />
            <ml:lambda>
              <ml:boundVars>
                <ml:id xml:space="preserve">t</ml:id>
              </ml:boundVars>
              <ml:apply>
                <ml:mult style="default" />
                <ml:apply>
                  <ml:mult style="auto-select" />
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">mother</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:parens>
                    <ml:id xml:space="preserve">t</ml:id>
                  </ml:parens>
                </ml:apply>
                <ml:apply>
                  <ml:id xml:space="preserve">f</ml:id>
                  <ml:id xml:space="preserve">t</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:lambda>
            <ml:bounds>
              <ml:apply>
                <ml:neg />
                <ml:id xml:space="preserve">∞</ml:id>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The magnitude of the integrals gives a measure of the contribution of the corresponding wavelet function to the approximating sum. The above wavelet approximation is an orthogonal series approximation since the basis functions are by construction <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false"><c val="#000">orthogonal</c></inlineAttr></i></f>: </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">There are many other kinds of orthogonal series representations and approximations of a signal. The Fourier series expansion, which uses the sine and cosine orthogonal basis functions, is the most important such representation in engineering and science applications. Wavelet series approximations are used increasingly often in a wide range of fields.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">In practice, we never actually use these formulae. Instead, we use the discrete wavelet transform (DWT), which performs these calculations much faster.</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">These methods are similar to those for signal soft denoising discussed in <b><u><link href="./denoise.xmcd" title="Links to section" popup="false">Nonparametric Signal Denoising</link></u></b>. </p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">waveapprox(y, J, K, filter)</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">In the wavelet approximation, we do the wavelet transform, sort the coefficients by magnitude, and then add them into the inverse wavelet transform in order of decreasing magnitude. To illustrate the accuracy of the approximation, the reconstruction error is plotted as additional coefficients are used. In this example, the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">ramp</inlineAttr></b></f> function is defined.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The ramp is then run through the wavelet transform, the largest coefficient is found, and the inverse wavelet transform is performed with just that single largest coefficient. This is all accomplished using the function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">waveapprox</c></inlineAttr></b></f>, defined below and in the file <b><u><link href="./Wavedef.xmcd" title="Links to additional wavelet functions" popup="false">wavedef.xmcd</link></u></b>:</p>
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                <ml:id xml:space="preserve">reverse</ml:id>
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                <ml:real>0</ml:real>
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                    <ml:id xml:space="preserve">rows</ml:id>
                    <ml:id xml:space="preserve">w</ml:id>
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                  <ml:real>1</ml:real>
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                <ml:apply>
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                  <ml:id xml:space="preserve">i</ml:id>
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                  <ml:mult style="default" />
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                    <ml:apply>
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                  <ml:apply>
                    <ml:indexer />
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                    <ml:id xml:space="preserve">i</ml:id>
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                </ml:apply>
              </ml:localDefine>
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            <ml:localDefine>
              <ml:id xml:space="preserve">R</ml:id>
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                <ml:id xml:space="preserve">idwt</ml:id>
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                  <ml:id xml:space="preserve">wt</ml:id>
                  <ml:id xml:space="preserve">J</ml:id>
                  <ml:id xml:space="preserve">filter</ml:id>
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              </ml:apply>
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              <ml:real>1</ml:real>
              <ml:id xml:space="preserve">filter</ml:id>
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          <ml:id xml:space="preserve">ε</ml:id>
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        <ml:eval>
          <ml:apply>
            <ml:absval />
            <ml:id xml:space="preserve">ε</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2.8420473853219628</ml:real>
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      <plot disable-calc="false" item-idref="38" />
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    </region>
    <region region-id="4507" left="174" top="2190" width="179.25" height="145.5" align-x="174" align-y="2190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="40" />
      <rendering item-idref="41" />
    </region>
    <region region-id="4167" left="0" top="2340" width="6000" height="6" align-x="0" align-y="2340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Repeat the procedure using only 5 coefficients:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">R</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">waveapprox</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">y</ml:id>
              <ml:id xml:space="preserve">J</ml:id>
              <ml:real>5</ml:real>
              <ml:id xml:space="preserve">filter</ml:id>
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          </ml:apply>
        </ml:define>
      </math>
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    <region region-id="4447" left="210" top="2370" width="51" height="16.5" align-x="225" align-y="2382" 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">ε</ml:id>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:absval />
            <ml:id xml:space="preserve">ε</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>1.510015179644963</ml:real>
          </result>
        </ml:eval>
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      <rendering item-idref="44" />
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      <plot disable-calc="false" item-idref="45" />
      <rendering item-idref="46" />
    </region>
    <region region-id="4669" left="180" top="2406" width="179.25" height="145.5" align-x="180" align-y="2406" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="47" />
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    <region region-id="4437" left="6" top="2556.75" width="183" height="14.25" align-x="6" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Repeat the process using 10 coefficients:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">R</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">waveapprox</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">y</ml:id>
              <ml:id xml:space="preserve">J</ml:id>
              <ml:real>10</ml:real>
              <ml:id xml:space="preserve">filter</ml:id>
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        </ml:define>
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      <math optimize="false" disable-calc="false">
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            <ml:real>0.82023836824623053</ml:real>
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      <plot disable-calc="false" item-idref="52" />
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    </region>
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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">As shown above, using more coefficients better approximates the original</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">function and thus reduces the approximation error.</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">wavedecomp(y, J, K, filter)</p>
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      <indexes>
        <phrase>wavedecomp function</phrase>
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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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
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              <c val="#000">The function </c>
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          <b>
            <c val="#008080">wavedecomp(y, J, K, filter) </c>
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          <f family="Times New Roman" charset="0" size="12">
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              <c val="#000">decomposes the vector </c>
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            <b>
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                <c val="#000">y</c>
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            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000"> using wavelet </c>
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              <b>
                <c val="#000">filter</c>
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          <f family="Times New Roman" charset="0" size="12">
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              <c val="#000"> according to the </c>
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            <b>
              <inlineAttr line-through="false">
                <c val="#000">K</c>
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          <f family="Times New Roman" charset="0" size="12">
            <c val="#000">
              <inlineAttr font-weight="normal" line-through="false"> largest wavelet coefficients of a level </inlineAttr>
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                <c val="#000">J</c>
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          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">
                <sp />
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            <inlineAttr line-through="false">
              <b>
                <c val="#000">dwt</c>
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          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000"> with periodic boundary conditions. It returns a vector of </c>
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            <b>
              <inlineAttr line-through="false">
                <c val="#000">K</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <c val="#000">
              <inlineAttr font-weight="normal" line-through="false"> nested vectors. Each of the individual vectors is one level of the wavelet approximation to the original function </inlineAttr>
            </c>
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            <b>
              <inlineAttr line-through="false">
                <c val="#000">y</c>
              </inlineAttr>
            </b>
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          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">.</c>
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          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000"> For example, element 0 is the IDWT of the largest wavelet coefficient. Summing the elements in order gives successively better approximations to the original function </c>
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          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">y</c>
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            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">. </c>
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            <b>
              <inlineAttr line-through="false">
                <c val="#008080">wavedecomp</c>
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            </b>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <c val="#000">
              <f family="Times New Roman" charset="0" size="12"> is </f>
            </c>
          </inlineAttr>defined below and in the file <b><u><link href="./Wavedef.xmcd" title="Links to additional wavelet functions" popup="false">wavedef.xmcd</link></u></b>:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define>
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              <ml:id xml:space="preserve">x</ml:id>
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                  <ml:real>0</ml:real>
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                    <ml:real>1</ml:real>
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                      <ml:real>1</ml:real>
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                      <ml:apply>
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                      <ml:real>0</ml:real>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Plotting the log of the RMS error as a function of the number of coefficients included in the approximation shows a steady improvement. As shown above, there is essentially perfect reconstruction after about fifteen coefficients.</p>
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