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234 lines
6.4 KiB
234 lines
6.4 KiB
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<H2><A ID="SECTION001222000000000000000"></A>
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<A ID="sect8.secondform"></A>
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<BR>
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Non-recirculating filter, second form
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</H2>
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<P>
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Sometimes we will need a variant of the filter above, shown in Figure
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<A HREF="#fig08.10">8.10</A>, called the
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<I>elementary non-recirculating filter, second form</I>.
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Instead of multiplying the delay output by <IMG
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$Q$"> we multiply the direct signal
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by its
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<A ID="10166"></A><I>complex conjugate</I>
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<IMG
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WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img872.png"
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ALT="$\overline{Q}$">. If
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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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A = a+bi = r \cdot (\cos(\alpha) + i \sin(\alpha))
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\end{displaymath}
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-->
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<IMG
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WIDTH="238" HEIGHT="28" BORDER="0"
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SRC="img873.png"
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ALT="\begin{displaymath}
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A = a+bi = r \cdot (\cos(\alpha) + i \sin(\alpha))
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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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is any complex number, its complex conjugate is defined as:
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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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\overline{A} = a-bi = r \cdot (\cos(\alpha) - i \sin(\alpha))
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\end{displaymath}
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-->
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<IMG
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WIDTH="238" HEIGHT="28" BORDER="0"
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SRC="img874.png"
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ALT="\begin{displaymath}
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\overline{A} = a-bi = r \cdot (\cos(\alpha) - i \sin(\alpha))
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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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Graphically this reflects points of the complex plane up and down across the
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real axis. The transfer function of the new filter 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) = \overline{Q} - {Z^{-1}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="120" HEIGHT="28" BORDER="0"
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SRC="img875.png"
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ALT="\begin{displaymath}
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H(Z) = \overline{Q} - {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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This gives rise to the same frequency response as before since
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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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|\overline{Q} - {Z^{-1}}| = |Q - \overline{Z^{-1}}| = |Q- Z|
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\end{displaymath}
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-->
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<IMG
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WIDTH="232" HEIGHT="28" BORDER="0"
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SRC="img876.png"
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ALT="\begin{displaymath}
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\vert\overline{Q} - {Z^{-1}}\vert = \vert Q - \overline{Z^{-1}}\vert = \vert Q- Z\vert
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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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Here we use the fact that <!-- MATH
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$\overline{Z} = {Z^{-1}}$
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-->
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<IMG
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WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img877.png"
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ALT="$\overline{Z} = {Z^{-1}}$">, for any unit complex
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number <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$">, as can be verified by writing out <IMG
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WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img878.png"
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ALT="$Z\overline{Z}$"> in either
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polar or rectangular form.
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<P>
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Although the two forms of the elementary non-recirculating filter have the same
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frequency response, their phase responses are different; this will
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occasionally lead us to prefer the second form.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig08.10"></A><A ID="10180"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.10:</STRONG>
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The elementary non-recirculating filter, second form.</CAPTION>
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<TR><TD><IMG
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WIDTH="121" HEIGHT="168" BORDER="0"
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SRC="img879.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.10.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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HREF="node135.html">Elementary recirculating filter</A>
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<B> Up:</B> <A ID="tex2html2576"
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HREF="node132.html">Elementary filters</A>
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<B> Previous:</B> <A ID="tex2html2570"
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HREF="node133.html">Elementary non-recirculating filter</A>
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<B> <A ID="tex2html2578"
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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