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<H1><A ID="SECTION001200000000000000000"></A>
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<A ID="chapter-filter"></A>
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<BR>
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Filters
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</H1>
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<P>
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In the previous chapter we saw that a delay network can have a non-uniform
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frequency response--a gain that varies as a function of
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frequency. Delay networks also typically change the phase of incoming signals
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variably depending on frequency. When the delay times used are very short,
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the most important properties of a delay network become its frequency and
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phase response. A delay network that is designed specifically for its
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frequency or phase response is called a
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<A ID="10048"></A><I>filter</I>.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig08.01"></A><A ID="10052"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.1:</STRONG>
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Representations of a filter: (a) in a block diagram; (b) a graph of its
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frequency response.</CAPTION>
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<TR><TD><IMG
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WIDTH="357" HEIGHT="200" BORDER="0"
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SRC="img852.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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In block diagrams, filters are shown as in Figure <A HREF="#fig08.01">8.1</A> (part a). The
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curve shown within the block gives a qualitative representation of the filter's
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frequency response. The frequency response may vary with time, and depending
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on the design of the filter, one or more controls (or additional audio inputs)
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might be used to change it.
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<P>
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Suppose, following the procedure of Section <A HREF="node108.html#sect7.network">7.3</A>, we put
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a unit-amplitude, complex-valued sinusoid with 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 $"> into a filter. We
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expect to get out a sinusoid of the same frequency and some amplitude, which
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depends on <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">. This gives us a complex-valued function <IMG
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WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img853.png"
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ALT="$H(\omega)$">, which is called the
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<A ID="10057"></A><I>transfer function</I>
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of the filter.
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<P>
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The frequency response is the gain as a function of the frequency
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<IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">. It is is equal to the magnitude of the transfer function. A
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filter's frequency response is customarily graphed as in Figure
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<A HREF="#fig08.01">8.1</A> (part b).
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An incoming unit-amplitude sinusoid of frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> comes out
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of the filter with magnitude <IMG
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WIDTH="49" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img725.png"
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ALT="$\vert H(\omega)\vert$">.
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<P>
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It is sometimes also useful to know the phase response of the filter, equal
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to <!-- MATH
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$\angle( H(\omega))$
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-->
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<IMG
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WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img854.png"
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ALT="$\angle( H(\omega))$">. For a fixed 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 $">, the filter's
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output phase will be <!-- MATH
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$\angle (H(\omega))$
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-->
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<IMG
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WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img854.png"
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ALT="$\angle( H(\omega))$"> radians ahead of its input
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phase.
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<P>
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The design and use of filters is a huge subject, because the wide range of uses
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a filter might be put to suggests a wide variety of filter design processes.
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In some applications a filter must exactly follow a prescribed frequency
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response, in others it is important to minimize computation time, in others the
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phase response is important, and in still others the filter must behave well
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when its parameters change quickly with time.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A ID="tex2html2443"
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HREF="node128.html">Taxonomy of filters</A>
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<UL>
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<LI><A ID="tex2html2444"
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HREF="node129.html">Low-pass and high-pass filters</A>
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<LI><A ID="tex2html2445"
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HREF="node130.html">Band-pass and stop-band filters</A>
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<LI><A ID="tex2html2446"
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HREF="node131.html">Equalizing filters</A>
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</UL>
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<BR>
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<LI><A ID="tex2html2447"
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HREF="node132.html">Elementary filters</A>
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<UL>
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<LI><A ID="tex2html2448"
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HREF="node133.html">Elementary non-recirculating filter</A>
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<LI><A ID="tex2html2449"
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HREF="node134.html">Non-recirculating filter, second form</A>
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<LI><A ID="tex2html2450"
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HREF="node135.html">Elementary recirculating filter</A>
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<LI><A ID="tex2html2451"
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HREF="node136.html">Compound filters</A>
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<LI><A ID="tex2html2452"
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HREF="node137.html">Real outputs from complex filters</A>
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<LI><A ID="tex2html2453"
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HREF="node138.html">Two recirculating filters for the price of one</A>
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</UL>
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<BR>
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<LI><A ID="tex2html2454"
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HREF="node139.html">Designing filters</A>
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<UL>
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<LI><A ID="tex2html2455"
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HREF="node140.html">One-pole low-pass filter</A>
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<LI><A ID="tex2html2456"
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HREF="node141.html">One-pole, one-zero high-pass filter</A>
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<LI><A ID="tex2html2457"
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HREF="node142.html">Shelving filter</A>
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<LI><A ID="tex2html2458"
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HREF="node143.html">Band-pass filter</A>
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<LI><A ID="tex2html2459"
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HREF="node144.html">Peaking and stop-band filter</A>
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<LI><A ID="tex2html2460"
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HREF="node145.html">Butterworth filters</A>
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<LI><A ID="tex2html2461"
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HREF="node146.html">Stretching the unit circle with rational functions</A>
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<LI><A ID="tex2html2462"
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HREF="node147.html">Butterworth band-pass filter</A>
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<LI><A ID="tex2html2463"
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HREF="node148.html">Time-varying coefficients</A>
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<LI><A ID="tex2html2464"
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HREF="node149.html">Impulse responses of recirculating filters</A>
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<LI><A ID="tex2html2465"
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HREF="node150.html">All-pass filters</A>
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</UL>
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<BR>
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<LI><A ID="tex2html2466"
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HREF="node151.html">Applications</A>
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<UL>
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<LI><A ID="tex2html2467"
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HREF="node152.html">Subtractive synthesis</A>
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<LI><A ID="tex2html2468"
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HREF="node153.html">Envelope following</A>
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<LI><A ID="tex2html2469"
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HREF="node154.html">Single Sideband Modulation</A>
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</UL>
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<BR>
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<LI><A ID="tex2html2470"
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HREF="node155.html">Examples</A>
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<UL>
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<LI><A ID="tex2html2471"
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HREF="node156.html">Prefabricated low-, high-, and band-pass filters</A>
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<LI><A ID="tex2html2472"
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HREF="node157.html">Prefabricated time-varying band-pass filter</A>
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<LI><A ID="tex2html2473"
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HREF="node158.html">Envelope followers</A>
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<LI><A ID="tex2html2474"
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HREF="node159.html">Single sideband modulation</A>
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<LI><A ID="tex2html2475"
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HREF="node160.html">Using elementary filters directly: shelving and peaking</A>
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<LI><A ID="tex2html2476"
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HREF="node161.html">Making and using all-pass filters</A>
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</UL>
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<BR>
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<LI><A ID="tex2html2477"
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HREF="node162.html">Exercises</A>
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</UL>
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<B> Up:</B> <A ID="tex2html2436"
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HREF="book.html">book</A>
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<B> Previous:</B> <A ID="tex2html2430"
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HREF="node126.html">Exercises</A>
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<B> <A ID="tex2html2438"
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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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