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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<H2><A NAME="SECTION001221000000000000000"></A>
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<A NAME="sect8.nonrecirculating"></A>
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<BR>
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Elementary non-recirculating filter
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</H2>
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<P>
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The non-recirculating comb
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filter may be generalized to yield the design shown in Figure <A HREF="#fig08.07">8.7</A>.
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This is the
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<A NAME="10133"></A><A NAME="10134"></A><I>elementary non-recirculating filter</I>,
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of the first form. Its single, complex-valued parameter <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="$Q$"> controls the
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complex gain of the delayed signal subtracted from the original one.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig08.07"></A><A NAME="10680"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.7:</STRONG>
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A delay network with a single-sample delay and a complex
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gain <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="$Q$">. This is the non-recirculating elementary filter, first form. Compare
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the non-recirculating comb filter shown in Figure <A HREF="node108.html#fig07.03">7.3</A>,
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which corresponds to choosing <IMG
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WIDTH="57" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img43.png"
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ALT="$Q=-1$"> here.</CAPTION>
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<TR><TD><IMG
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WIDTH="90" HEIGHT="210" BORDER="0"
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SRC="img860.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.07.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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To find its frequency response, as in Chapter 7 we feed the delay network
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a complex sinusoid <!-- MATH
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$1, Z, {Z^2}, \ldots$
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-->
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<IMG
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WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img664.png"
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ALT="$1, Z, {Z^2}, \ldots$"> whose frequency is <!-- MATH
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$\omega=\arg(Z)$
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-->
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<IMG
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WIDTH="81" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img861.png"
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ALT="$\omega=\arg(Z)$">.
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The <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 of the input 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="$Z^n$"> and that of the output
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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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(1 - Q{Z^{-1}}){Z^n}
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\end{displaymath}
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-->
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<IMG
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WIDTH="99" HEIGHT="28" BORDER="0"
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SRC="img862.png"
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ALT="\begin{displaymath}
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(1 - Q{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 - Q{Z^{-1}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="128" HEIGHT="28" BORDER="0"
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SRC="img863.png"
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ALT="\begin{displaymath}
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H(Z) = 1 - 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 can be analyzed graphically as shown in Figure <A HREF="#fig08.08">8.8</A>.
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The real numbers
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<IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> and <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$\alpha $"> are the magnitude and argument of the complex number <IMG
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$Q$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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Q = r \cdot (\cos(\alpha) + i \sin(\alpha))
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\end{displaymath}
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-->
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<IMG
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WIDTH="178" HEIGHT="28" BORDER="0"
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SRC="img864.png"
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ALT="\begin{displaymath}
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Q = r \cdot (\cos(\alpha) + i \sin(\alpha))
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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The gain of the filter is the distance from the point <IMG
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$Q$"> to the point <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$">
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in the complex plane. Analytically we can see this because
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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|1 - Q{Z^{-1}}| = |Z||1 - Q{Z^{-1}}| = |Q - Z|
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\end{displaymath}
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-->
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<IMG
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WIDTH="268" HEIGHT="28" BORDER="0"
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SRC="img865.png"
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ALT="\begin{displaymath}
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\vert 1 - Q{Z^{-1}}\vert = \vert Z\vert\vert 1 - Q{Z^{-1}}\vert = \vert Q - Z\vert
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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Graphically, the number <IMG
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WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img866.png"
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ALT="$Q{Z^{-1}}$"> is just the number <IMG
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$Q$"> rotated backwards
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(clockwise) by the angular frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> of the incoming sinusoid. The
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value <!-- MATH
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$|1 - Q{Z^{-1}}|$
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-->
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<IMG
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WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img867.png"
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ALT="$\vert 1 - Q{Z^{-1}}\vert$"> is the distance from <IMG
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WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img866.png"
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ALT="$Q{Z^{-1}}$"> to <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img262.png"
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ALT="$1$"> in the complex
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plane, which is equal to the distance from <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="$Q$"> to <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="$Z$">.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig08.08"></A><A NAME="10688"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.8:</STRONG>
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Diagram for calculating the frequency response of the
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non-recirculating elementary filter
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(Figure <A HREF="#fig08.07">8.7</A>). The frequency response is given by the length of the
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segment connecting <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="$Z$"> to <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="$Q$"> in the complex plane.</CAPTION>
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<TR><TD><IMG
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WIDTH="341" HEIGHT="379" BORDER="0"
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SRC="img868.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.08.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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As the frequency of the input sweeps from 0 to <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $">, the point <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$"> travels
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couterclockwise around the unit circle. At the point where <!-- MATH
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$\omega = \alpha$
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-->
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<IMG
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WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img869.png"
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ALT="$\omega = \alpha$">,
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the distance is at a minimum, equal to <IMG
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WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img870.png"
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ALT="$1-r$">. The maximum occurs which <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="$Z$"> is
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at the opposite point of the circle. Figure <A HREF="#fig08.09">8.9</A> shows the transfer
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function for three different values of <IMG
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SRC="img44.png"
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ALT="$r=\vert Q\vert$">.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig08.09"></A><A NAME="10689"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.9:</STRONG>
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Frequency response of the elementary non-recirculating filter
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Figure <A HREF="#fig08.07">8.7</A>. Three values of <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="$Q$"> are used, all with the
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same argument (-2 radians), but with varying absolute value (magnitude) <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img44.png"
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ALT="$r=\vert Q\vert$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="323" HEIGHT="212" BORDER="0"
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SRC="img871.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.09.ps}\end{figure}"></TD></TR>
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</TABLE>
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