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HREF="node141.html">One-pole, one-zero high-pass filter</A>
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<B> <A ID="tex2html2673"
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<H2><A ID="SECTION001231000000000000000"></A>
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<A ID="sect08.low-pass"></A>
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<BR>
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One-pole low-pass filter
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</H2>
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<P>
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<DIV ALIGN="CENTER"><A ID="fig08.12"></A><A ID="10320"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.12:</STRONG>
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One-pole low-pass filter: (a) pole-zero diagram; (b)
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frequency response.</CAPTION>
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<TR><TD><IMG
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WIDTH="546" HEIGHT="256" BORDER="0"
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SRC="img908.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.12.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 one-pole low-pass filter has a single pole located at a positive real
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number <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">, as pictured in Figure <A HREF="#fig08.12">8.12</A>. This is just a recirculating
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comb filter with delay length <IMG
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WIDTH="40" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img736.png"
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ALT="$d=1$">, and the analysis of Section
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<A HREF="node109.html#sect7.recirculatingcomb">7.4</A> applies. The maximum gain occurs at a
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frequency of zero, corresponding to the point on the circle closest to the
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point <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">. The gain there is <IMG
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WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img909.png"
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ALT="$1/(1-p)$">. Assuming <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$"> is close
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to one, if we move a distance of <IMG
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WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img910.png"
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ALT="$1-p$"> units
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up or down from the real (horizontal) axis, the distance increases by a
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factor of about <IMG
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WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img732.png"
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ALT="$\sqrt{2}$">, and so we expect the half-power point to occur at
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an angular frequency of about <IMG
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WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img910.png"
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ALT="$1-p$">.
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<P>
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This calculation is often made in reverse: if we wish the half-power point to
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lie at a given 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 $">, we set <IMG
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WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img911.png"
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ALT="$p = 1-\omega$">. This
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approximation only works well if the value of <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> is well under <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$\pi /2$">,
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as it often is in practice.
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It is customary to normalize the one-pole low-pass filter, multiplying it by
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the constant factor <IMG
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WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img910.png"
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ALT="$1-p$"> in order to give a gain of 1 at zero frequency;
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nonzero frequencies will then get a gain less than one.
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<P>
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The frequency response is graphed in Figure <A HREF="#fig08.12">8.12</A> (part b). The
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audible frequencies only reach to the middle of the graph; the right-hand
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side of the frequency response curve all lies above the Nyquist frequency
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<IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">.
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<P>
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The one-pole low-pass filter is often used to seek trends in noisy signals.
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For instance, if you use a physical controller and only
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care about changes on the order of 1/10 second or so, you can smooth the
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values with a low-pass filter whose half-power point is 20 or 30
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cycles per second.
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