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                <c val="#008080">SECTION 4 - DIGITAL FILTERING</c>
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        <phrase>digital filtering</phrase>
        <phrase>filtering, digital</phrase>
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        <phrase>IIR filter design</phrase>
        <phrase>filter design, IIR</phrase>
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                <c val="#008080">butter(N)</c>
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        <phrase>function, butter</phrase>
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                <c val="#008080">bessel(N, scale)</c>
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            </b>
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        <phrase>bessel function</phrase>
        <phrase>function, bessel</phrase>
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                <c val="#008080">)</c>
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        <phrase>cheby1 function</phrase>
        <phrase>function, cheby1</phrase>
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                <c val="#008080">cheby2(N, scale, atten)</c>
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        <phrase>cheby2 function</phrase>
        <phrase>function, cheby2</phrase>
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                <c val="#008080">iirlow(C, f)</c>
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        <phrase>iirlow function</phrase>
        <phrase>function, iirlow</phrase>
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                <c val="#008080">iirhigh(C, f)</c>
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        <phrase>iirhigh function</phrase>
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                <c val="#008080">iirpass(C, lowf, highf)</c>
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        <phrase>iirpass function</phrase>
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                <c val="#008080">iirstop(C, lowf, highf)</c>
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        <phrase>iirstop function</phrase>
        <phrase>function, iirstop</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">Generate coefficients for IIR filters. <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">butter</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false">, </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">bessel</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false">, </inlineAttr></f><b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">cheby1</f></inlineAttr></b>, and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">cheby2</inlineAttr></b></f> produce coefficients for an analog lowpass filter of the given type. These coefficients are used as an argument to one of the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">iir</inlineAttr></b></f> functions, which scale and perform bilinear transformation to produce the required frequency response. </p>
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      <indexes>
        <phrase>iir functions</phrase>
        <phrase>bilinear transformations</phrase>
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            <b>
              <c val="#008080">•</c>
            </b>
          </f>
          <f size="14">
            <b>
              <sp />
            </b>
          </f>
          <b>N</b>, integer filter order, </p>
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          <tab />
          <tab />for <b>bessel</b> filters, between 1 and 35, </p>
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          <tab />
          <tab />for all others, between 1 and 200</p>
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          <f size="14">
            <b>
              <c val="#008080">•</c>
            </b>
          </f>
          <f size="14">
            <b>
              <sp />
            </b>
          </f>
          <f family="Symbol" charset="2">
            <b>e</b>
          </f>, positive real number, allowable passband ripple</p>
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          <c val="#008080">
            <f size="14">
              <b>•</b>
            </f>
          </c>
          <f size="14">
            <b>
              <sp />
            </b>
          </f>
          <b>scale</b>, for <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">bessel</inlineAttr></b></f> filters, a real number greater than 0 which sets the lower edge of the stopband. </p>
        <p style="Normal" margin-left="7.5" margin-right="0" text-indent="-7.5" text-align="left" list-style-type="none" tabs="inherit">
          <b>
            <c val="#008080">
              <f size="14">•</f>
            </c>
          </b>
          <f size="14">
            <b>
              <sp />
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">scale</inlineAttr>
            </b>
          </f>, for <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">cheby2</inlineAttr></b></f> filters, a calculated real number greater than 1 setting the lower edge of the stopband. Calculations for each type of filter are shown in the examples.</p>
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            <c val="#008080">
              <b>•</b>
            </c>
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          <f size="14">
            <b>
              <sp />
            </b>
          </f>
          <b>atten</b>, real number greater than 1, the stopband attenuation factor</p>
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          <f size="14">
            <c val="#008080">
              <b>•</b>
            </c>
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          <f size="14">
            <b>
              <sp />
            </b>
          </f>
          <b>C</b>, coefficient array produced by one of the first four functions</p>
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          <c val="#008080">
            <b>
              <f size="14">•</f>
            </b>
          </c>
          <b>
            <f size="14">
              <sp />
            </f>
          </b>
          <b>f</b>,<b> lowf</b>,<b> highf</b>, real frequencies, normalized to the sampling frequency, between 0 and 0.5</p>
      </text>
      <indexes>
        <phrase>bessel filters</phrase>
        <phrase>filters, bessel</phrase>
        <phrase>atten</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">Produce coefficient arrays with an even number of columns. If there are two columns, as in the output of <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">bessel</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false">,</inlineAttr></f> the filter consists of a single section: column 1 contains coefficients of the transfer function numerator, column 2 contains denominator coefficients. If there are more than two columns, each pair represents a section of a filter in cascade form. </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">For lowpass and highpass filters with even N, all sections are second-order, with three coefficients in each column. If the order is odd, the last section is first-order. For bandpass and bandstop filters, the conversion to a digital filter doubles the order of each section. </p>
      </text>
      <indexes>
        <phrase>bandpass filter conversion to digital</phrase>
        <phrase>bandstop filter conversion to digital</phrase>
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              <c val="#008080">•</c>
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          <f size="14">
            <b>
              <sp />
            </b>
          </f>These functions implement algorithms in Chapter 7 of <i>Signal Processing Algorithms</i> by Samuel Stearns and Ruth David (Prentice-Hall, Inc.).</p>
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              <c val="#008080">•</c>
            </b>
          </f>
          <f size="14">
            <b>
              <sp />
            </b>
          </f>The functions <b>gain</b> and <b>response</b> accept coefficient arrays in the form produced by these functions. Complex gain is found by multiplying section gains, and response by feeding each section's response into the next section. </p>
      </text>
      <indexes>
        <phrase>gain function</phrase>
        <phrase>response function</phrase>
        <phrase>function, gain</phrase>
        <phrase>function, response</phrase>
      </indexes>
    </region>
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      <pageBreak />
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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">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#800000">Example - Lowpass Chebyshev Type I Filter</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>filter, lowpass Chebyshev Type 1</phrase>
        <phrase>lowpass Chebyshev Type 1 filter</phrase>
        <phrase>Chebyshev Type 1 filter, lowpass, example</phrase>
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    </region>
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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">These examples are based on examples in Chapter 7 of Stearns and David. See this book for a clear and interesting treatment of the mathematics of digital filtering and IIR filter design in particular. </p>
      </text>
      <indexes>
        <phrase>digital filtering</phrase>
        <phrase>IIR filter design</phrase>
        <phrase>filter design, IIR</phrase>
      </indexes>
    </region>
    <region region-id="234" left="6" top="834.75" width="139.5" height="14.25" align-x="6" align-y="846" 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="0" text-align="left" list-style-type="none" tabs="inherit">For a Chebyshev Type I filter. </p>
      </text>
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    <region region-id="235" left="6" top="858.75" width="81.75" 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="0" text-align="left" list-style-type="none" tabs="inherit">Choose the order:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">N</ml:id>
          <ml:real>8</ml:real>
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    <region region-id="258" left="6" top="882.75" width="111" height="14.25" align-x="6" align-y="894" 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">Choose the parameter <f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">e</inlineAttr></f>:</p>
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        <ml:define warning="WarnRedefinedBIFunction">
          <ml:id xml:space="preserve">ε</ml:id>
          <ml:real>.5</ml:real>
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    <region region-id="259" left="6" top="906.75" width="117.75" height="14.25" align-x="6" align-y="918" 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">Choose cutoff frequency:</p>
      </text>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">f</ml:id>
          <ml:real>.25</ml:real>
        </ml:define>
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    <region region-id="241" left="6" top="930.75" width="135.75" height="14.25" align-x="6" align-y="942" 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">Generate the filter coefficients:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">A</ml:id>
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            <ml:id xml:space="preserve">iirlow</ml:id>
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              <ml:apply>
                <ml:id xml:space="preserve">cheby1</ml:id>
                <ml:sequence>
                  <ml:id xml:space="preserve">N</ml:id>
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                </ml:sequence>
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              <ml:id xml:space="preserve">f</ml:id>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">A</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="3" cols="8">
              <ml:real>0.43077387775477366</ml:real>
              <ml:real>0.86154775550954732</ml:real>
              <ml:real>0.43077387775477366</ml:real>
              <ml:real>1</ml:real>
              <ml:real>-0.0049780191270120322</ml:real>
              <ml:real>0.93145736630872911</ml:real>
              <ml:real>0.37607107573796122</ml:real>
              <ml:real>0.75214215147592245</ml:real>
              <ml:real>0.37607107573796122</ml:real>
              <ml:real>1</ml:real>
              <ml:real>-0.28635453045028447</ml:real>
              <ml:real>0.79063883340212948</ml:real>
              <ml:real>0.20786132492338239</ml:real>
              <ml:real>0.41572264984676477</ml:real>
              <ml:real>0.20786132492338239</ml:real>
              <ml:real>1</ml:real>
              <ml:real>-0.80134564746642778</ml:real>
              <ml:real>0.63279094715995732</ml:real>
              <ml:real>0.049744317123995518</ml:real>
              <ml:real>0.099488634247991037</ml:real>
              <ml:real>0.049744317123995518</ml:real>
              <ml:real>1</ml:real>
              <ml:real>-1.3021734824709859</ml:real>
              <ml:real>0.501150750966968</ml:real>
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    </region>
    <region region-id="260" left="6" top="1050.75" width="291.75" height="42.75" align-x="6" align-y="1062" 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 filter<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> A</inlineAttr></f> has the expected four quadratic sections. Plot the magnitude of the gain, and put a marker at the ripple specified by the choice of <f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">e</inlineAttr></f>, which is </p>
      </text>
    </region>
    <region region-id="261" left="36" top="1102.5" width="87.75" height="42" align-x="69.75" align-y="1122" 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">ripple</ml:id>
          <ml:apply>
            <ml:div />
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            <ml:apply>
              <ml:sqrt />
              <ml:apply>
                <ml:plus />
                <ml:real>1</ml:real>
                <ml:apply>
                  <ml:pow />
                  <ml:id xml:space="preserve">ε</ml:id>
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="11" />
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    <region region-id="262" left="36" top="1158.75" width="75.75" height="15.75" align-x="49.5" align-y="1170" 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">x</ml:id>
          <ml:range>
            <ml:sequence>
              <ml:real>0</ml:real>
              <ml:real>.001</ml:real>
            </ml:sequence>
            <ml:real>.5</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="708" left="24" top="1188" width="268.5" height="133.5" align-x="24" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="13" />
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    <region region-id="721" left="6" top="1338.75" width="224.25" height="14.25" align-x="6" align-y="1350" 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">(For more on the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">gain</inlineAttr></b></f> function, see <b><u><link href="./Gain.xmcd" title="Links to section" popup="false">Filter Gain</link></u></b>.)</p>
      </text>
      <indexes>
        <phrase>gain function</phrase>
        <phrase>filter gain</phrase>
        <phrase>function, gain</phrase>
      </indexes>
    </region>
    <region region-id="720" left="0" top="1368" width="6000" height="6" align-x="0" align-y="1368" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="265" left="6" top="1380.75" width="308.25" height="42.75" align-x="6" align-y="1392" 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">Since the transfer function is given as a product of quadratic factors, it's straightforward to plot the poles on the complex plane. To find the complex roots of each factor of the denominator, define</p>
      </text>
    </region>
    <region region-id="267" left="36" top="1429.5" width="142.5" height="42.75" align-x="66.75" align-y="1458" 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="WarnRedefinedBIUnit">
          <ml:function>
            <ml:id xml:space="preserve">R</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">v</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:minus />
              <ml:apply>
                <ml:neg />
                <ml:apply>
                  <ml:indexer />
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                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:sqrt />
                <ml:apply>
                  <ml:minus />
                  <ml:apply>
                    <ml:pow />
                    <ml:parens>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">v</ml:id>
                        <ml:real>1</ml:real>
                      </ml:apply>
                    </ml:parens>
                    <ml:real>2</ml:real>
                  </ml:apply>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:real>4</ml:real>
                    <ml:apply>
                      <ml:indexer />
                      <ml:id xml:space="preserve">v</ml:id>
                      <ml:real>2</ml:real>
                    </ml:apply>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
            </ml:apply>
            <ml:real>2</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="268" left="6" top="1482.75" width="291.75" height="42.75" align-x="6" align-y="1494" 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 poles of the transfer function of the filter <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">A</inlineAttr></f> are the roots of quadratics with coefficients given by the odd-numbered columns of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">A</inlineAttr></f>, with the coefficient of the quadratic term first. </p>
      </text>
    </region>
    <region region-id="280" left="6" top="1530.75" width="220.5" height="14.25" align-x="6" align-y="1542" 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">Find one root for each factor of the denominator:</p>
      </text>
    </region>
    <region region-id="281" left="36" top="1560.75" width="42.75" height="15.75" align-x="46.5" align-y="1572" 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">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>3</ml:real>
          </ml:range>
        </ml:define>
      </math>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
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            <ml:id xml:space="preserve">i</ml:id>
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          <ml:apply>
            <ml:id xml:space="preserve">R</ml:id>
            <ml:apply>
              <ml:matcol />
              <ml:id xml:space="preserve">A</ml:id>
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                <ml:real>1</ml:real>
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            </ml:apply>
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    <region region-id="283" left="6" top="1584.75" width="115.5" height="14.25" align-x="6" align-y="1596" 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">Fill in the conjugate roots:</p>
      </text>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
            <ml:indexer />
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            <ml:apply>
              <ml:plus />
              <ml:id xml:space="preserve">i</ml:id>
              <ml:real>4</ml:real>
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          <ml:apply>
            <ml:conjugate />
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              <ml:indexer />
              <ml:id xml:space="preserve">r</ml:id>
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    <region region-id="285" left="6" top="1632.75" width="297" height="28.5" align-x="6" align-y="1644" 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">Below we display the transfer function poles and plot them on the complex plane.</p>
      </text>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">k</ml:id>
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            <ml:real>0</ml:real>
            <ml:real>7</ml:real>
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      <plot disable-calc="false" item-idref="20" />
      <rendering item-idref="21" />
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    <region region-id="289" left="36" top="1698" width="151.5" height="173.25" align-x="47.25" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">r</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="8" cols="1">
              <ml:complex>
                <ml:real>0.0024890095635060161</ml:real>
                <ml:imag>-0.96511718000464686</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.14317726522514224</ml:real>
                <ml:imag>-0.8775756970910138</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.40067282373321389</ml:real>
                <ml:imag>-0.68720610844317309</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.65108674123549293</ml:real>
                <ml:imag>-0.27791510638019357</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.0024890095635060161</ml:real>
                <ml:imag>0.96511718000464686</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.14317726522514224</ml:real>
                <ml:imag>0.8775756970910138</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.40067282373321389</ml:real>
                <ml:imag>0.68720610844317309</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.65108674123549293</ml:real>
                <ml:imag>0.27791510638019357</ml:imag>
              </ml:complex>
            </ml:matrix>
          </result>
        </ml:eval>
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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 pole plot shows that all the poles are inside the unit circle, so the filter is stable, as required. We can also verify this numerically by computing</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">To plot the impulse response of the filter<f family="Times New Roman" charset="0"><sp /></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">A</inlineAttr></f>, use the <b><u><link href="./Response.xmcd" title="Links to section" popup="false">response function</link></u></b>, giving it a vector of length 1 representing a unit impulse as the first argument. Here's a plot of the first 100 terms of the impulse response:</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">Length of the response signal to be computed:</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">Although the response is infinite in length, it has dropped to nearly 0 after the first 80 terms. The stability condition guarantees that </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">is bounded.</p>
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                <c val="#800000">Example - Lowpass Chebyshev Type II Filter</c>
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        <phrase>filter, lowpass Chebyshev Type II</phrase>
        <phrase>lowpass Chebyshev Type II filter</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">Set the parameters for a Type II Chebyshev filter.</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">Compute the scale (see Stearns and David for the theory behind this computation; this number will be greater than 1):</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">We try a fifth-order filter, which has two quadratic sections and one linear section:</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">A plot of the response shows how well the design criteria are met. By setting the axis markers to <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">pass</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false">, </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">stop</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">1/</inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">atten</inlineAttr></b></f> we can see that the stopband begins at the required location (the intersection of the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">stop</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">1/</inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">atten</inlineAttr></b></f> lines) and that the passband ends to the right of the cutoff .1. </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">For a larger attenuation, say 1000, a higher order would be needed to get the transition band down to the required width. We plot the gains in dB, marking the desired 30 dB attenuation:</p>
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                <c val="#800000">Computing the Scale Factor</c>
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        <phrase>scale factor, computing the</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">Here's how to compute the scale factor for the other three filter types. Only the first equation below is activated. If you want to activate the second or third equation, then right-click on the equations and select <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">E</inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">nable Evaluation</inlineAttr></b></f>.</p>
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    </region>
    <region region-id="340" left="6" top="3474.75" width="100.5" height="14.25" align-x="6" align-y="3486" 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">For a <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">highpass filter</inlineAttr></b></f>:</p>
      </text>
      <indexes>
        <phrase>filter, highpass</phrase>
        <phrase>highpass filter</phrase>
      </indexes>
    </region>
    <region region-id="346" left="36" top="3495.75" width="101.25" height="35.25" align-x="66.75" align-y="3516" 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">scale</ml:id>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:id xml:space="preserve">tan</ml:id>
              <ml:apply>
                <ml:mult style="default" />
                <ml:id xml:space="preserve">π</ml:id>
                <ml:id xml:space="preserve">pass</ml:id>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">tan</ml:id>
              <ml:apply>
                <ml:mult style="default" />
                <ml:id xml:space="preserve">π</ml:id>
                <ml:id xml:space="preserve">stop</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="47" />
    </region>
    <region region-id="347" left="6" top="3540.75" width="297" height="42.75" align-x="6" align-y="3552" 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">For a <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">bandpass filter</inlineAttr></b></f> the parameters<f family="Times New Roman" charset="0"><inlineAttr line-through="false"><sp /></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">pass1</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">pass2</inlineAttr></b></f><inlineAttr line-through="false"><f family="Times New Roman" charset="0"> are the lower and upper edges of the passband and the parameters </f></inlineAttr><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">stop1</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">stop2</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> are the edges of the two stopbands. </inlineAttr></f></p>
      </text>
      <indexes>
        <phrase>bandpass filter</phrase>
        <phrase>filter, bandpass</phrase>
      </indexes>
    </region>
    <region region-id="348" left="36" top="3595.5" width="315" height="86.25" align-x="66.75" align-y="3642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="true">
        <ml:define>
          <ml:id xml:space="preserve">scale</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">min</ml:id>
            <ml:matrix rows="2" cols="1">
              <ml:apply>
                <ml:absval />
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:pow />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">stop1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:real>2</ml:real>
                    </ml:apply>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass2</ml:id>
                        </ml:apply>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:parens>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass2</ml:id>
                          </ml:apply>
                        </ml:apply>
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass1</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                    </ml:parens>
                    <ml:apply>
                      <ml:id xml:space="preserve">tan</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">π</ml:id>
                        <ml:id xml:space="preserve">stop1</ml:id>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:absval />
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:pow />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">stop2</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:real>2</ml:real>
                    </ml:apply>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass2</ml:id>
                        </ml:apply>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:parens>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass2</ml:id>
                          </ml:apply>
                        </ml:apply>
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass1</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                    </ml:parens>
                    <ml:apply>
                      <ml:id xml:space="preserve">tan</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">π</ml:id>
                        <ml:id xml:space="preserve">stop2</ml:id>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
            </ml:matrix>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="48" />
    </region>
    <region region-id="352" left="6" top="3690.75" width="294" height="42.75" align-x="6" align-y="3702" 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">For a <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">bandstop filter</inlineAttr></b></f> the parameters<f family="Times New Roman" charset="0"><inlineAttr line-through="false"><sp /></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">pass1</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"><b>pass2</b></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> are the edges of the two passbands and the parameters </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">stop1</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">stop2</inlineAttr></b></f> are the lower and upper edges of the stopband. </p>
      </text>
      <indexes>
        <phrase>bandstop filter</phrase>
        <phrase>filter, bandstop</phrase>
      </indexes>
    </region>
    <region region-id="355" left="36" top="3745.5" width="315" height="86.25" align-x="66.75" align-y="3792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="true">
        <ml:define>
          <ml:id xml:space="preserve">scale</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">min</ml:id>
            <ml:matrix rows="2" cols="1">
              <ml:apply>
                <ml:absval />
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:parens>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass2</ml:id>
                          </ml:apply>
                        </ml:apply>
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass1</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                    </ml:parens>
                    <ml:apply>
                      <ml:id xml:space="preserve">tan</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">π</ml:id>
                        <ml:id xml:space="preserve">stop1</ml:id>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:pow />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">stop1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:real>2</ml:real>
                    </ml:apply>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass2</ml:id>
                        </ml:apply>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:absval />
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:parens>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass2</ml:id>
                          </ml:apply>
                        </ml:apply>
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass1</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                    </ml:parens>
                    <ml:apply>
                      <ml:id xml:space="preserve">tan</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">π</ml:id>
                        <ml:id xml:space="preserve">stop2</ml:id>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:pow />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">stop2</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:real>2</ml:real>
                    </ml:apply>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass2</ml:id>
                        </ml:apply>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
            </ml:matrix>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="49" />
    </region>
    <region region-id="96" left="0" top="3840" width="6000" height="6" align-x="0" align-y="3840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="356" left="6" top="3852.75" width="302.25" height="28.5" align-x="6" align-y="3864" 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">As an example, compute coefficients for a bandpass filter with the following parameters:</p>
      </text>
    </region>
    <region region-id="365" left="36" top="3888.75" width="59.25" height="15.75" align-x="69.75" align-y="3900" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">stop1</ml:id>
          <ml:real>0.1</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="50" />
    </region>
    <region region-id="372" left="120" top="3888.75" width="66" height="15.75" align-x="154.5" align-y="3900" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">pass1</ml:id>
          <ml:real>0.15</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="51" />
    </region>
    <region region-id="529" left="210" top="3888.75" width="58.5" height="15.75" align-x="240" align-y="3900" show-border="false" show-highlight="true" 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">atten</ml:id>
          <ml:real>100</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="52" />
    </region>
    <region region-id="368" left="36" top="3912.75" width="65.25" height="15.75" align-x="69.75" align-y="3924" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">stop2</ml:id>
          <ml:real>0.35</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="53" />
    </region>
    <region region-id="528" left="120" top="3912.75" width="60" height="15.75" align-x="154.5" align-y="3924" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">pass2</ml:id>
          <ml:real>0.3</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="54" />
    </region>
    <region region-id="362" left="36" top="3937.5" width="310.5" height="86.25" align-x="66.75" align-y="3984" 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">scale</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">min</ml:id>
            <ml:matrix rows="2" cols="1">
              <ml:apply>
                <ml:absval />
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:pow />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">stop1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:real>2</ml:real>
                    </ml:apply>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass2</ml:id>
                        </ml:apply>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:parens>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass2</ml:id>
                          </ml:apply>
                        </ml:apply>
                        <ml:apply>
                          <ml:id xml:space="preserve">tan</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">π</ml:id>
                            <ml:id xml:space="preserve">pass1</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                    </ml:parens>
                    <ml:apply>
                      <ml:id xml:space="preserve">tan</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">π</ml:id>
                        <ml:id xml:space="preserve">stop1</ml:id>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:absval />
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:pow />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">stop2</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:real>2</ml:real>
                    </ml:apply>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass1</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">tan</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">π</ml:id>
                          <ml:id xml:space="preserve">pass2</ml:id>
                        </ml:apply>
                      </ml:apply>
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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">Plot the response in dB.</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 right stopband edge falls exactly at the correct location. The left stopband edge is just above the target value, so this transition band is narrower than required.</p>
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          <f family="Times New Roman" charset="0">
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                <c val="#800000">Example - Lowpass Bessel Filter</c>
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        <phrase>filter, lowpass Bessel</phrase>
        <phrase>lowpass Bessel filter</phrase>
        <phrase>Bessel filter, lowpass, example</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">A <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Bessel filter</inlineAttr></b></f> is returned as a single section. Set the parameters. The scale parameter can be used to control the gain at the cutoff, as we see below.</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">For this Bessel filter the gain at the cutoff is </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">Trade a lower gain at the cutoff for increased attenuation in the stopband using the scaling factor. The gain of the analog transfer function <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">R</inlineAttr></f> corresponding to the analog coefficients is</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">To get a gain at <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">f</inlineAttr></f> of 0.8 for our digital filter, we find the point where the analog gain is 0.8, using<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> Mathcad</inlineAttr></f>'s <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">root</inlineAttr></b></f> function. We start with a guess value of 1:</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 plot shows that we've found the right value. Now we recompute our digital filter coefficients using this scale factor.</p>
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                <c val="#800000">Example - Bandpass, Bandstop, and Highpass Butterworth filters</c>
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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">Note that the order of the input coefficients is doubled by th<f family="Times New Roman" charset="0"><inlineAttr line-through="false">e </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">iirpass</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">iirstop</inlineAttr></b></f> functions, so fourth-order analog coefficients result in an eighth-order filter. </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">Set the parameters:</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">Compute the coefficients:</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 next example is a 16th-order Butterworth bandstop filter, with the same band edges. The filter has four 4th-degree sections.</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 last example is a 10th order Butterworth highpass filter, with 5 quadratic sections.</p>
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