243 lines
7.4 KiB
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243 lines
7.4 KiB
HTML
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original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION001234000000000000000"></A>
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<A NAME="sect8.twopolebandpass"></A>
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<BR>
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Band-pass filter
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</H2>
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<P>
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Starting with the three filter types shown above, which all have real-valued
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poles and zeros, we now transform them to operate on bands located off the real
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axis. The low-pass, high-pass, and shelving filters will then become
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band-pass, stop-band, and peaking filters. First we develop the band-pass
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filter. Suppose we want a center frequency at <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 $"> radians and a
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bandwidth of <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $">. We take the low-pass filter with cutoff frequency
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<IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $">; its pole is located, for small values of <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $">, roughly at
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<IMG
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WIDTH="69" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img919.png"
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ALT="$p = 1 - \beta$">.
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Now rotate this value by <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> radians in the complex plane, i.e.,
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multiply by the complex number <!-- MATH
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$\cos \omega + i \sin \omega$
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-->
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<IMG
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WIDTH="98" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img920.png"
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ALT="$\cos \omega + i \sin \omega$">. The new pole
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is at:
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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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{P_1} = (1 - \beta) (\cos \omega + i \sin \omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="194" HEIGHT="28" BORDER="0"
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SRC="img921.png"
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ALT="\begin{displaymath}
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{P_1} = (1 - \beta) (\cos \omega + i \sin \omega)
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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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To get a real-valued output, this must be paired with another pole:
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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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{P_2} = \overline{P_1} = (1 - \beta) (\cos \omega - i \sin \omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="233" HEIGHT="28" BORDER="0"
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SRC="img922.png"
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ALT="\begin{displaymath}
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{P_2} = \overline{P_1} = (1 - \beta) (\cos \omega - i \sin \omega)
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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 resulting pole-zero plot is as shown in Figure <A HREF="#fig08.15">8.15</A>.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig08.15"></A><A NAME="10361"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.15:</STRONG>
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Two-pole band-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="537" HEIGHT="244" BORDER="0"
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SRC="img923.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.15.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 peak is approximately (not exactly) at the desired center 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 $">, and the frequency response drops by 3 decibels approximately <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $">
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radians above and below it. It is often desirable to normalize the filter to
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have a peak gain near unity; this is done by multiplying the input or output by
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the product of the distances of the two poles to the peak on the circle, or
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(very approximately):
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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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\beta * (\beta + 2 \omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="84" HEIGHT="28" BORDER="0"
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SRC="img924.png"
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ALT="\begin{displaymath}
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\beta * (\beta + 2 \omega)
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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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For some applications it is desirable to add a zero at the points <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="$1$"> and <IMG
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WIDTH="23" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img401.png"
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ALT="$-1$">,
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so that the gain drops to zero at angular frequencies <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img179.png"
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ALT="$0$"> and <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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HREF="node144.html">Peaking and stop-band filter</A>
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<B> Up:</B> <A NAME="tex2html2711"
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HREF="node139.html">Designing filters</A>
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<B> Previous:</B> <A NAME="tex2html2705"
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HREF="node142.html">Shelving filter</A>
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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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