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<document xmlns="http://cnx.rice.edu/cnxml" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:bib="http://bibtexml.sf.net/" id="m10142">

  <name>Chirp-Z Transform</name>


  <metadata>
  <md:version>2.8</md:version>
  <md:created>2001/12/31</md:created>
  <md:revised>2005/09/20 15:20:30.690 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="jgrab">
      <md:firstname>Jacob</md:firstname>
      
      <md:surname>Grabczewski</md:surname>
      <md:email>jgrab@owlnet.rice.edu</md:email>
    </md:maintainer>
  </md:maintainerlist>
  
  <md:keywordlist>
    <md:keyword>CZT</md:keyword>
  </md:keywordlist>

  <md:abstract>This module covers the fundamentals of Chirp Z-Transform.</md:abstract>
</metadata>


  <content>
    <section id="sec1">
      <name>Chirp-Z Transform</name>

      <para id="sec1para1">
	Note: There are other ways to sample circles, segments, or
	spirals in z-domain that have fast (FFT-like) transforms.
      </para>


      <figure orient="horizontal" id="fig1">
	<subfigure id="fig1sub1">
	  <media type="image/png" src="spiral.png"/>
	</subfigure>
	<subfigure id="fig1sub2">
	  <media type="image/png" src="circle.png"/>
	</subfigure>
	<subfigure id="fig1sub3">
	  <media type="image/png" src="segment.png"/>
	</subfigure>
      </figure>


      <para id="sec1para2">
	Perhaps the most famous is the CZT
	<equation id="eq01">
	  <m:math>
	    <m:apply>
	      <m:eq/>
	      <m:apply>
		<m:ci type="fn">X</m:ci>
		<m:ci>
		  <m:msub>
		    <m:mi>z</m:mi>
		    <m:mi>l</m:mi>
		  </m:msub>
		</m:ci>
	      </m:apply>
	      <m:apply>
		<m:sum/>
		<m:bvar><m:ci>n</m:ci></m:bvar>
		<m:lowlimit><m:cn>0</m:cn></m:lowlimit>
		<m:uplimit>
		  <m:apply>
		    <m:minus/>
		    <m:ci>N</m:ci>
		    <m:cn>1</m:cn>
		  </m:apply>
		</m:uplimit>
		<m:apply>
		  <m:times/>
		  <m:apply>
		    <m:ci type="fn" class="discrete">x</m:ci>
		    <m:ci>n</m:ci>
		  </m:apply>
		  <m:apply>
		    <m:power/>
		    <m:apply>
		      <m:times/>
		      <m:ci>A</m:ci>
		      <m:apply>
			<m:power/>
			<m:ci>V</m:ci>
			<m:apply>
			  <m:minus/>
			  <m:ci>l</m:ci>
			</m:apply>
		      </m:apply>
		    </m:apply>
                    <m:ci>n</m:ci>
		  </m:apply>
		</m:apply>
	      </m:apply>
	    </m:apply>
	    <m:apply>
	      <m:forall/>
	      <m:bvar><m:ci>l</m:ci></m:bvar>
	      <m:condition>
		<m:apply>
		  <m:eq/>
		  <m:ci>l</m:ci>
		  <m:set>
		    <m:cn>0</m:cn>
		    <m:mo>…</m:mo>
		    <m:apply>
		      <m:minus/>
		      <m:ci>N</m:ci>
		      <m:cn>1</m:cn>
		    </m:apply>
		  </m:set>
		</m:apply>
	      </m:condition>
	    </m:apply>
	  </m:math>
	</equation>
      </para>

      <para id="sec1para3">  
	<m:math>
	  <m:apply>
	    <m:eq/>
	    <m:ci>A</m:ci>
	    <m:apply>
	      <m:times/>
	      <m:ci>
		<m:msub>
		  <m:mi>A</m:mi>
		  <m:mn>0</m:mn>
		</m:msub>
	      </m:ci>
	      <m:apply>
		<m:exp/>
                <m:apply>
                  <m:times/>
		  <m:imaginaryi/>
		  <m:ci>
		    <m:msub>
		      <m:mi>θ</m:mi>
		      <m:mn>0</m:mn>
		    </m:msub>
		  </m:ci>
                </m:apply>
	      </m:apply>
	    </m:apply>
	  </m:apply>
	</m:math> is the starting point.
      </para>

      <para id="sec1para4">
	<m:math>
	  <m:apply>
	    <m:eq/>
	    <m:ci>V</m:ci>
	    <m:apply>
	      <m:times/>
	      <m:ci>
		<m:msub>
		  <m:mi>V</m:mi>
		  <m:mn>0</m:mn>
		</m:msub>
	      </m:ci>
	      <m:apply>
		<m:exp/>
                <m:apply>
                  <m:times/>
		  <m:imaginaryi/>
		  <m:ci>
		    <m:msub>
		      <m:mi>φ</m:mi>
		      <m:mn>0</m:mn>
		    </m:msub>
		  </m:ci>
                </m:apply>
	      </m:apply>
	    </m:apply>
	  </m:apply>
	</m:math> spirals in if 
	<m:math>
	  <m:apply>
	    <m:gt/>
	    <m:ci>
	      <m:msub>
		<m:mi>V</m:mi>
		<m:mn>0</m:mn>
	      </m:msub>
	    </m:ci>
	    <m:cn>1</m:cn>
	  </m:apply>
	</m:math>,
      </para>

      <para id="sec1para5">
	spirals out if 
	<m:math>
	  <m:apply>
	    <m:lt/>
	    <m:ci>
	      <m:msub>
		<m:mi>V</m:mi>
		<m:mn>0</m:mn>
	      </m:msub>
	    </m:ci>
	    <m:cn>1</m:cn>
	  </m:apply>
	</m:math>,
      </para>

      <para id="sec1para6">
	and circles if 
	<m:math>
	  <m:apply>
	    <m:eq/>
	    <m:ci>
	      <m:msub>
		<m:mi>V</m:mi>
		<m:mn>0</m:mn>
	      </m:msub>
	    </m:ci>
	    <m:cn>1</m:cn>
	  </m:apply>
	</m:math>.
      </para>
    </section>
  </content>
</document>
