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<H1><A NAME="SECTION001230000000000000000">
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Designing filters</A>
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</H1>
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<P>
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The frequency response of a series of elementary recirculating and
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non-recirculating filters can be estimated graphically by plotting all the
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coefficients <!-- MATH
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${Q_1}, \ldots, {Q_j}$
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-->
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<IMG
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SRC="img887.png"
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ALT="${Q_1}, \ldots, {Q_j}$"> and <!-- MATH
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${P_1}, \ldots, {P_k}$
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-->
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<IMG
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WIDTH="74" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img888.png"
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ALT="${P_1}, \ldots, {P_k}$"> on the
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complex plane and reasoning as in Figure <A HREF="node133.html#fig08.08">8.8</A>. The overall
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frequency response is the product of all the distances from 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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to each of the <IMG
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WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img898.png"
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ALT="$Q_i$">, divided by the product of the distances to each of
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the <IMG
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WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img899.png"
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ALT="$P_i$">.
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<P>
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One customarily marks each of the <IMG
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SRC="img898.png"
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ALT="$Q_i$"> with an "o" (calling it a "zero")
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and each of the <IMG
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WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img899.png"
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ALT="$P_i$"> with an "x" (a "pole"); their names are borrowed
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from the field of complex analysis. A plot showing the poles and zeroes
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associated with a filter is unimaginatively called a <A NAME="10312"></A><I>pole-zero plot</I>.
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<P>
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When <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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close to a zero the frequency response tends to dip, and when it is close to
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a pole, the frequency response tends to rise. The effect of a pole or a zero
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is more pronounced, and also more local, if it is close to the unit circle
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that <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 constrained to lie on.
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Poles must lie within the unit circle for a stable filter. Zeros may lie on
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or outside it, but any zero <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="$Q$"> outside the unit circle may be replaced by one
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within it, at the point <!-- MATH
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$1/\overline{Q}$
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-->
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<IMG
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WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img907.png"
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ALT="$1/\overline{Q}$">, to give a constant multiple of the
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same frequency response. Except in special cases we will keep the zeros
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inside the circle as well as the poles.
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<P>
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In the rest of this section we will show how to construct several of the
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filter types most widely used in electronic music. The theory of digital
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filter design is vast, and we will only give an introduction
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here. A deeper treatment is available online from Julius Smith at
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ccrma.stanford.edu. See also [<A
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HREF="node202.html#r-steiglitz96">Ste96</A>] for an introduction
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to filter design from the more general viewpoint of digital signal
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processing.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html2651"
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HREF="node140.html">One-pole low-pass filter</A>
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<LI><A NAME="tex2html2652"
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HREF="node141.html">One-pole, one-zero high-pass filter</A>
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<LI><A NAME="tex2html2653"
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HREF="node142.html">Shelving filter</A>
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<LI><A NAME="tex2html2654"
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HREF="node143.html">Band-pass filter</A>
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<LI><A NAME="tex2html2655"
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HREF="node144.html">Peaking and stop-band filter</A>
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<LI><A NAME="tex2html2656"
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HREF="node145.html">Butterworth filters</A>
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<LI><A NAME="tex2html2657"
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HREF="node146.html">Stretching the unit circle with rational functions</A>
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<LI><A NAME="tex2html2658"
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HREF="node147.html">Butterworth band-pass filter</A>
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<LI><A NAME="tex2html2659"
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HREF="node148.html">Time-varying coefficients</A>
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<LI><A NAME="tex2html2660"
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HREF="node149.html">Impulse responses of recirculating filters</A>
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<LI><A NAME="tex2html2661"
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HREF="node150.html">All-pass filters</A>
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</UL>
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HREF="node127.html">Filters</A>
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<B> <A NAME="tex2html2646"
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Miller Puckette
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2006-12-30
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