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<!DOCTYPE document PUBLIC "-//CNX//DTD CNXML 0.5 plus MathML//EN" "http://cnx.rice.edu/cnxml/0.5/DTD/cnxml_mathml.dtd">
<document xmlns="http://cnx.rice.edu/cnxml" xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="new0">
  
  <name>Uniform Modulated Filterbank</name>
  
  <metadata>
  <md:version>2.6</md:version>
  <md:created>2002/10/24</md:created>
  <md:revised>2005/10/06 11:14:49.323 GMT-5</md:revised>
  <md:authorlist>
      <md:author id="schniter">
      <md:firstname>Phil</md:firstname>
      
      <md:surname>Schniter</md:surname>
      <md:email>schniter@ee.eng.ohio-state.edu</md:email>
    </md:author>
  </md:authorlist>

  <md:maintainerlist>
    <md:maintainer id="mjhaag">
      <md:firstname>Michael</md:firstname>
      
      <md:surname>Haag</md:surname>
      <md:email>mjhaag@rice.edu</md:email>
    </md:maintainer>
    <md:maintainer id="schniter">
      <md:firstname>Phil</md:firstname>
      
      <md:surname>Schniter</md:surname>
      <md:email>schniter@ee.eng.ohio-state.edu</md:email>
    </md:maintainer>
  </md:maintainerlist>
  
  

  <md:abstract/>
</metadata>


  <content>
    <para id="p1">
      A modulated <cnxn document="m10423" strength="7">filterbank</cnxn>
      is composed of analysis branches which:

      <list id="list1" type="enumerated">
	<item>
	  Modulate the input to center the desired sub-band at DC
	</item>
	<item>
	  Lowpass filter the modulated signal to isolate the
	  desired sub-band
	</item>
	<item>
	  Downsample the lowpass signal
	</item>
      </list>
      The synthesis branches interpolate the sub-band signals by
      upsampling and lowpass filtering, then modulate each sub-band
      back to its original spectral location (see <cnxn strength="8" target="fig1"/>).
    </para>
    
    <figure id="fig1">
      <media type="image/png" src="uni_f1.png"/>
    </figure>

    <para id="p2">
      In <m:math><m:ci>M</m:ci></m:math>-branch critically-sampled
      uniformly-modulated filterbanks, the
      <m:math><m:ci>k</m:ci></m:math>th branch has a modulation
      frequency of
      <m:math>
	<m:apply>
	  <m:eq/>
	  <m:apply>
	    <m:ci><m:mo>±</m:mo></m:ci>
	    <m:ci><m:msub>
	      <m:mi>ω</m:mi>
		<m:mi>k</m:mi>
	      </m:msub></m:ci>
	  </m:apply>
	  <m:apply>
	    <m:times/>
	    <m:apply>
	      <m:divide/>
	      <m:apply>
		<m:times/>
		<m:cn>2</m:cn>
		<m:pi/>
	      </m:apply>
	      <m:ci>M</m:ci>
	    </m:apply>
	    <m:ci>k</m:ci>
	    <m:ci>radians</m:ci>
	  </m:apply>
	</m:apply>
      </m:math>, a lowpass cutoff frequency of 
      <m:math>
	<m:apply>
	  <m:divide/>
	  <m:pi/>
	  <m:ci>M</m:ci>
	</m:apply>
      </m:math> radians, and a downsampling/upsampling factor of 
      <m:math><m:ci>M</m:ci></m:math>.  
    </para>

    <para id="p3">
      The most common way to implement this type of uniform modulated
      filterbank is through the use of <cnxn document="m10424" strength="9">polyphase filterbanks and DFTs</cnxn>.
    </para>

  </content>
</document>
