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                <c val="#008080">SECTION 4 — DIGITAL FILTERING</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">Computes the zero-phase response of a filter, using a "forward-backward" filtering model. </p>
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              <c val="#008080">•</c>
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          <b>x</b>, complex-valued vector </p>
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          <b>C</b>, matrix of filter coefficients, in cascade form</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 x, containing the filtered response.<sp count="2" />The calculation corresponds to two sequential calls to the <b><u><link href="./Response.xmcd" title="Links to section" popup="false">response</link></u></b> function with the same filter coefficients, using the reversed output of the first call as input for the second call.<sp count="2" /></p>
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                <c val="#800000">Example - Zero-Phase Filtering</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">Here is a Butterworth lowpass IIR filter of order <region region-id="491" left="226.5" top="300.75" width="39.75" height="15.75" align-x="249.75" align-y="312" 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>2</ml:real></ml:define></math><rendering item-idref="6" /></region>.</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 phase response is not zero, or even linear. To correct this, we can use the forward-backward filtering function, corresponding to applying the Butterworth filter, then applying it again to the reversed signal.</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">As a sample input signal, consider an impulse at the midpoint of a length-N signal</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">The corresponding zero-phase output 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">To demonstrate the effect the filter has on the signal, compute the Fourier transforms of both and plot the magnitude and phase of Y.</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 transfer function, given by the ratio magnitudes <region region-id="663" left="240" top="1138.5" width="22.5" height="33.75" align-x="249.75" align-y="1158" 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:div /><ml:apply><ml:absval /><ml:id xml:space="preserve">X</ml:id></ml:apply><ml:apply><ml:absval /><ml:id xml:space="preserve">Y</ml:id></ml:apply></ml:apply></math><rendering item-idref="26" /></region> has the expected Butterworth cutoff shape:</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 phase change 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">The effect of <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">filtfilt</inlineAttr></b></f> is to double filter the signal magnitude, as shown in the plot, but the phase shift is zero, instead of the nonlinear phase of the original filter.</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">In interpreting the phase plot, it is important to remember that phases of<sp count="2" />-2<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"> and 0</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> are equivalent and that, when the magnitude of Y is very small, as it is for frequencies far above the cutoff</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, the phase can be meaningless due to numerical noise.</inlineAttr></f></p>
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