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                <c val="#008080">SECTION 4 - DIGITAL FILTERING</c>
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                <c val="#008080">Group Delay</c>
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        <phrase>grpdelay function</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">Calculates the group delay, the phase change of the input signal <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">envelope</inlineAttr></i></f>, imposed by a filter.</p>
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          <b>filter</b>, complex-valued matrix of filter coefficients in cascade form</p>
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          <b>N</b>
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          </f>number of frequencies at which to evaluate group delay </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">Returns a vector of length 2N containing the group delay at each frequency. The algorithm uses the Fourier transform of an FIR-only filter that that gives the same group delay as the original filter, with new coefficients c[n]. Group delay is given by Re[ j (1/C) dC/d<f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr></f>]. Group delay is one measure of the filter's impact on the phase of the input signal.</p>
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                <c val="#800000">Example - Group Delay of Various Filter Types </c>
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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">First, define several filters for which we can compare group delay. Consider a filter having a pair of complex conjugate zeros at radius r and angle theta, say</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Then, the corresponding filter vectors are the columns of</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Similarly, if the two points are poles instead of zeros, the coefficients a[n] and b[n] are interchanged and the matrix is</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">Here is an additional example that has one real zero, a pair of complex conjugate zeros, plus the same number of poles (See Oppenheim and Schafer, Example 5.9)</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">Finally, here is an example of a Butterworth lowpass IIR filter of order <region region-id="491" left="6" top="750.75" width="39.75" height="15.75" align-x="29.25" align-y="762" 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">Nb</ml:id><ml:real>6</ml:real></ml:define></math><rendering item-idref="11" /></region>. First, generate the filter:</p>
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        <phrase>Butterworth lowpass IIR filter</phrase>
        <phrase>IIR filter, Butterworth lowpass</phrase>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The resulting filter has three stages, each with a pair of poles:</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 group delay of these filters is evaluated evaluated at N frequencies. </p>
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                    <ml:id xml:space="preserve">qc</ml:id>
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                      <ml:real>1</ml:real>
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      <indexes>
        <phrase>grpdelay function</phrase>
        <phrase>function, grpdelay</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">Copy or type the other filter names in place of filt2zp to display other group delays.</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">Only half the frequency range has been plotted - the other half is symmetric, with the exception of the point <f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">= </inlineAttr></f><f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr></f> This particular filter has a zero on the unit circle at that frequency, so the actual group delay has a large negative spike (delta function) at that point. The numerical method approximates this spike with a very large negative value, as would be apparent in the plot if the point <f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr></f> =<f family="Times New Roman" charset="0"><sp /></f><f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">p</inlineAttr></f> were included.</p>
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