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                <c val="#008080">SECTION 4 - DIGITAL FILTERING</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">Computes the complex gain of a digital FIR or IIR filter. </p>
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          <b>C</b>, a real or complex-valued vector or array of filter coefficients. </p>
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          <tab />col. 1: numerator coefficients</p>
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          <tab />col. 2: denominator coefficients</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">The <b>gain</b> function returns the gain at the single frequency, or a vector of gains - the transfer function - if a vector of frequencies is used. This is useful for plotting. See <b><u><link href="./Filters.xmcd" title="Links to section" popup="false">FIR Filter Design</link></u></b> and <b><u><link href="./Iir.xmcd" title="Links to section" popup="false">IIR Filter Design</link></u></b>, also in section 4, for more on these filter-generating functions.</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">so the transfer function 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">The magnitude of the gain over frequencies ranging from 0 to half the sampling frequency is shown.</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">If you prefer to scale the frequency so that the sampling frequency is represented as <inlineAttr font-weight="normal" line-through="false">2</inlineAttr><f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" line-through="false">p</inlineAttr></f>, you can define your range variable accordingly and then divide the frequency argument to the gain function by<b><sp /></b><inlineAttr font-weight="normal" line-through="false">2</inlineAttr><f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" line-through="false">p</inlineAttr></f>:</p>
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                <c val="#800000">Example - Higher Order Lowpass Filters</c>
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        <phrase>lowpass filters, higher order</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">The next example is a sixth-order lowpass filter with three sections corresponding to three quadratic factors of the numerator and denominator of the transfer function:</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">IIR filter cutoff frequency:</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 sixth-order filter the response falls off much more rapidly than for the second-order filter defined by the array <f family="MS Serif" charset="0"><inlineAttr font-weight="normal" line-through="false">A</inlineAttr></f>. Here's a plot of the gain of both filters compared.</p>
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            <ml:id xml:space="preserve">G2</ml:id>
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              <ml:id xml:space="preserve">x</ml:id>
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            <ml:id xml:space="preserve">gain</ml:id>
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              <ml:id xml:space="preserve">C</ml:id>
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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">Finally we compute the gain for an FIR filter designed using the function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">bandpass</inlineAttr></b></f>.</p>
      </text>
      <indexes>
        <phrase>bandpass function</phrase>
        <phrase>function, bandpass</phrase>
        <phrase>FIR filter, gain for</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">Compute coefficients for a length 51 bandpass filter using a Blackman window with passband between f<f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sub>low</sub></inlineAttr></f> and f<f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sub>high</sub></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">:</inlineAttr></f></p>
      </text>
      <indexes>
        <phrase>Blackman window</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">Plot the gain in dB:</p>
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          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#800000">Function Definition</c>
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            </b>
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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">You can compute gains directly from the definition of the transfer function, using the summation operator. For example, the gain of the filter <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" font-style="normal" line-through="false">F</inlineAttr></f> at a frequency <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">x</inlineAttr></f> is given by </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">This is the transfer function evaluated at <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">z = 2</inlineAttr></f><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">p</inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">jx</inlineAttr></f>. </p>
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              <ml:real>0.50001379918631239</ml:real>
              <ml:imag>-4.0694003884938351E-16</ml:imag>
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            <ml:id xml:space="preserve">gain</ml:id>
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              <ml:id xml:space="preserve">F</ml:id>
              <ml:real>.2</ml:real>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:complex>
              <ml:real>0.50001379918631161</ml:real>
              <ml:imag>4.7184478546569153E-16</ml:imag>
            </ml:complex>
          </result>
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    <region region-id="231" left="6" top="2040.75" width="303" height="71.25" align-x="6" align-y="2052" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">gain</inlineAttr>
            </b>
          </f> is faster than the summation operator, so it's convenient when you want to compute the gain at a large number of frequencies, for example when plotting the frequency response. For very long filters even the computation with <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">gain</inlineAttr></b></f> will take a while, so choose a fairly coarse grid (say .01) for plotting.</p>
      </text>
      <indexes>
        <phrase>gain function</phrase>
        <phrase>function, gain</phrase>
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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 - FIR Filter Phase</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>FIR filter phase</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">The phase of the complex gain represents the phase shift of the 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">FIR filter cutoff frequency:</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 coefficients for a lowpass filter, using a Hanning window (last argument of 4) in the <b><u><link href="./Filters.xmcd" title="Links to section" popup="false">lowpass</link></u></b> function:</p>
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        <phrase>Hanning window</phrase>
        <phrase>lowpass function</phrase>
        <phrase>function, lowpass</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">Plot the phase change using the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">arg</inlineAttr></b></f> function:</p>
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        <phrase>arg function</phrase>
        <phrase>function, arg</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">The phase is linear in the passband, as we'd expect, but we're seeing the effect of the filter delay. We can compute the complex gain relative to the delayed signal by dividing by the transfer function of the delay <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">D</inlineAttr></f>, which 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">where</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 length 37 filter the delay<f family="MS Serif" charset="0"><sp /></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">D</inlineAttr></f> is 18, so the relative phase shift 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">When the gain is real and negative, roundoff errors in the gain computation can leave a tiny positive or negative imaginary part, so the phase may jump arbitrarily between <f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">p</inlineAttr></f><b><sp /></b>and <inlineAttr font-weight="normal" line-through="false">-</inlineAttr><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">p</inlineAttr></f>. To make sure these values get handled uniformly, redefine the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">arg</inlineAttr></b></f> as follows:</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">Now plot the magnitude of the gain in decibels and the phase shift relative to the delayed signal. </p>
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