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                <c val="#008080">SECTION 4 - DIGITAL FILTERING</c>
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        <phrase>filtering, digital</phrase>
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                <c val="#008080">multirate(x, n, m, [f])</c>
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        <phrase>multirate function</phrase>
        <phrase>function, multirate</phrase>
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                <c val="#008080">resample(x, m, n)</c>
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        <phrase>interpolate function</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">These functions change the sampling rate of a signal in different ways and under different assumptions. </p>
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        <phrase>signal, sampling rate of a</phrase>
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            </b>
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          <b>x</b>, a vector or matrix of real or complex numbers representing the samples of a signal. Each column of matrix input is processed individually. For <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">resample</inlineAttr></b></f>, data must be real.</p>
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              <b>•</b>
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          <b>n</b>, the integer-valued upsampling/interpolation factor (for <b>interpolate</b> and <b>resample</b>, <f family="Symbol" charset="2">³ </f>2)</p>
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              <b>•</b>
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          <b>m</b>, the integer-valued downsampling/decimation factor (for <b>resample</b>, <f family="Symbol" charset="2">³ </f>2)</p>
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          <b>f</b>, [optional] vector of FIR lowpass filter coefficients used between upsampling and downsampling. Default filter has length 32, gain n, windowed with a Hanning taper, with cutoff</p>
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        <phrase>FIR lowpass filter coefficients</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">Return the resampled signal. For <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">multirate </inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">and</inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"> resample</inlineAttr></b></f>, a vector approximately of length n/m times the original data length. <b>interpolate</b> returns an even number of points, either n times the input length or n times the input length plus 1. </p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">We recommend that you use<b> multirate</b>, because it works in the most general case and avoids the aliasing effect that might arise when downsampling a signal. </p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Notes</p>
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          </f> filters in the time domain, and requires only that the signal be bandlimited. <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">interpolate</inlineAttr></b></f> and <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">resample</f></inlineAttr></b> assume periodicity of the signal, and perform operations in the frequency domain after taking an FFT.</p>
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          </f> Multirate uses the group delay of a linear-phase FIR filter to estimate where the initial output transient ends, and truncates. For filter length L, the group delay is L/2, so multirate skips the first L/2m samples of signal output. To observe the transient, zero-pad the signal. If the provided filter does not have linear-phase, some data at the beginning may be missing, so zero-padding the signal in this case is also advised.</p>
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          </f>iltering with multirate is carried out using a polyphase representation of the filter coefficients.</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - multirate 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">The function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">multirate</inlineAttr></b></f> is the most flexible and general of all the resampling routines. The only requirement is that the signal be bandlimited. To resample, <f family="Times New Roman" charset="0" size="12"><b><inlineAttr font-style="normal" underline="false" line-through="false">x</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">(n)</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> is upsampled, inserting n-1 zero samples between original data points. The upsampled vector is lowpass-filtered</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> in the time domain</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, effectively interpolating the zero samples. This also avoids aliasing that may be caused by downsampling. The </inlineAttr></f>resultant sequence is downsampled by picking every m<sup>th</sup> sample.</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">Using the default FIR filter and a sum-of-sines signal, </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 above plot shows that the resampled signal is shifted slightly, and contains n/m times the number of original data points. The shift is introduced by the FIR filter (it is equal to its group delay).</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - multirate with Optional Filter</p>
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          <f family="Times New Roman" charset="0">
            <inlineAttr font-weight="normal" line-through="false">The next example uses a specified</inlineAttr>
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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">In this case, resampling also caused signal attenuation, because the FIR filter did not have a gain equal to the upsampling rate, n. There is no scaling effect when we adjust the FIR filter coefficients to the correct gain.</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - Pure Upsampling</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">You can use multirate to upsample or downsample only. For example, to upsample only,</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - Interpolate and Resample</p>
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          <b>interpolate</b> performs an FFT on the original data, adding zeros in the middle of the coefficient range to create a transform that has the required interpolated length, then inverse transforming the extended spectrum. There is no unique method for interpolating data (for example, one might also use a specially designed FIR filter), but for regularly behaved data this function returns a reasonable interpolation. </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">Change n and examine the plot 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">The function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">resample</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">(x, m, n)</inlineAttr></f> operates by interpolating, as above, to produce<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> n times as many points, and then taking every m</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sup>th</sup></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> point to produce the output. </inlineAttr></f></p>
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      <indexes>
        <phrase>resample function</phrase>
        <phrase>function, resample</phrase>
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          <ml:id xml:space="preserve">m</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">Change the decimation factor m and the interpolation factor n, and observe the plot below.</p>
      </text>
      <indexes>
        <phrase>decimation factor</phrase>
      </indexes>
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        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">y</ml:id>
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            <ml:id xml:space="preserve">resample</ml:id>
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              <ml:id xml:space="preserve">x</ml:id>
              <ml:id xml:space="preserve">m</ml:id>
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            <ml:real>9</ml:real>
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          <ml:id xml:space="preserve">k</ml:id>
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                <ml:id xml:space="preserve">length</ml:id>
                <ml:id xml:space="preserve">y</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">Here the result has 3/5 as many points, so the spacing between the samples is 5/3 of the original spacing. </p>
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      <plot disable-calc="false" item-idref="44" />
      <rendering item-idref="45" />
    </region>
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      <pageBreak />
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - aperiodic input</p>
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      <indexes>
        <phrase>input, aperiodic, example</phrase>
        <phrase>aperiodic input, 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">Finally, here's an example of what happens when the assumption of a periodic input with an integral number of periods is not met:</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">Interpolation factor:</p>
      </text>
    </region>
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        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">n</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">The interpolation routine assumes that<f family="Times New Roman" charset="0"><inlineAttr line-through="false"><sp /></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">X</inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"><sp /></inlineAttr></f>is periodic, and therefore tries to interpolate points near the end of the graph that will go back up to the initial height of 1. However, the function will give sensible results if "wrapping" the input sequence gives a reasonably smooth 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">Interpolation factor:</p>
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