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198 lines
5.7 KiB
198 lines
5.7 KiB
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original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A ID="SECTION001223000000000000000"></A>
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<A ID="sect8.recirculating"></A>
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<BR>
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Elementary recirculating filter
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</H2>
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<P>
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The
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<A ID="10185"></A><A ID="10186"></A><I>elementary recirculating filter</I> is the recirculating comb filter of
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Figure <A HREF="node109.html#fig07.07">7.7</A> with a
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complex-valued feedback gain <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img880.png"
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ALT="$P$"> as shown in Figure <A HREF="#fig08.11">8.11</A> (part
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a). By the same analysis as before, feeding this network a sinusoid whose
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<IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">th sample is <IMG
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WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img653.png"
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ALT="$Z^n$"> gives an output of:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{{1} \over {1 - P {Z^{-1}}}} {Z^n}
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\end{displaymath}
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-->
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<IMG
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WIDTH="91" HEIGHT="40" BORDER="0"
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SRC="img881.png"
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ALT="\begin{displaymath}
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{{1} \over {1 - P {Z^{-1}}}} {Z^n}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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so the transfer function is
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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H(Z) = {{1} \over {1 - P {Z^{-1}}}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="131" HEIGHT="40" BORDER="0"
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SRC="img882.png"
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ALT="\begin{displaymath}
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H(Z) = {{1} \over {1 - P {Z^{-1}}}}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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The recirculating filter is stable when <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img883.png"
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ALT="$\vert P\vert < 1$">; when, instead, <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img884.png"
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ALT="$\vert P\vert> 1$"> the
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output grows exponentially as the delayed sample recirculates.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig08.11"></A><A ID="10197"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.11:</STRONG>
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The elementary recirculating filter: (a) block diagram; (b)
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frequency response.</CAPTION>
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<TR><TD><IMG
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WIDTH="454" HEIGHT="259" BORDER="0"
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SRC="img885.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.11.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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The transfer function is thus just the inverse of that of the non-recirculating
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filter (first form). If you put the two in series with <IMG
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WIDTH="49" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img886.png"
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ALT="$P=Q$">, the output
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theoretically equals the input. (This analysis only demonstrates it for
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sinusoidal inputs; that it follows for other signals as well can be verified by
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working out the impulse response of the combined network).
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HREF="node134.html">Non-recirculating filter, second form</A>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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