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481 lines
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<H2><A NAME="SECTION001237000000000000000">
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Stretching the unit circle with rational functions</A>
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</H2>
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<P>
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In Section <A HREF="node143.html#sect8.twopolebandpass">8.3.4</A> we saw a simple way to turn a low-pass
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filter into a band-pass one. It is tempting to apply the same method to turn
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our Butterworth low-pass filter into a higher-quality band-pass filter; but if
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we wish to preserve the high quality of the Butterworth filter we must be more
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careful than before in the design of the transformation used. In this section
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we will prepare the way to making the Butterworth band-pass filter by
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introducing a class of rational transformations of the complex plane which
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preserve the unit circle.
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<P>
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This discussion is adapted from [<A
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HREF="node202.html#r-parks87">PB87</A>], pp. 201-206 (I'm grateful
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to Julius Smith for this pointer). There the transformation is carried out
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in continuous time, but here we have adapted the method to operate in
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discrete time, in order to make the discussion self-contained.
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<P>
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The idea is to start with any filter with a transfer function as before:
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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) = {
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{
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(1 - {Q_1}{Z^{-1}}) \cdots (1 - {Q_j}{Z^{-1}})
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} \over {
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(1 - {P_1}{Z^{-1}}) \cdots (1 - {P_k}{Z^{-1}})
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}
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}
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\end{displaymath}
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-->
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<IMG
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WIDTH="263" HEIGHT="45" BORDER="0"
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SRC="img889.png"
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ALT="\begin{displaymath}
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H(Z) = {
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{
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(1 - {Q_1}{Z^{-1}}) \cdots (1 - {Q_j}{Z^{-1}})
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} \over {
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(1 - {P_1}{Z^{-1}}) \cdots (1 - {P_k}{Z^{-1}})
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}
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}
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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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whose frequency response (the gain at a 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 $">) is given by:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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| H(\cos(\omega) + i \sin(\omega)) |
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\end{displaymath}
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-->
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<IMG
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WIDTH="146" HEIGHT="28" BORDER="0"
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SRC="img938.png"
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ALT="\begin{displaymath}
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\vert H(\cos(\omega) + i \sin(\omega)) \vert
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<P>
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Now suppose we can find a rational function, <IMG
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WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img939.png"
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ALT="$R(Z)$">, which distorts the
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unit circle in some desirable way. For <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> to be a rational function means
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that it can be written as a quotient of two polynomials (for example, the
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transfer function <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> is a rational function). That <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> sends points on the
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unit circle to other points on the unit circle is just the condition that
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<IMG
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WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img940.png"
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ALT="$\vert R(Z)\vert = 1$"> whenever <IMG
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WIDTH="44" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img941.png"
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ALT="$Z=1$">. It can easily be checked that any function of
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the form
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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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R(Z) = U \cdot
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{{
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{A_n}{Z^n} + {A_{n-1}}{Z^{n-1}} + \cdots + {A_0}
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} \over {
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\overline{A_0}{Z^n} + \overline{A_1}{Z^{n-1}} + \cdots + \overline{A_n}
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="295" HEIGHT="45" BORDER="0"
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SRC="img942.png"
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ALT="\begin{displaymath}
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R(Z) = U \cdot
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{{
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{A_n}{Z^n} + {A_{n-1}}{Z^{n-1}} + \cdot...
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...}{Z^n} + \overline{A_1}{Z^{n-1}} + \cdots + \overline{A_n}
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}}
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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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(where <IMG
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WIDTH="54" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img943.png"
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ALT="$\vert U\vert=1$">) has this property. The same reasoning as in Section
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<A HREF="node134.html#sect8.secondform">8.2.2</A> confirms that <IMG
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WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img940.png"
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ALT="$\vert R(Z)\vert = 1$"> whenever <IMG
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WIDTH="44" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img941.png"
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ALT="$Z=1$">.
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<P>
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Once we have a suitable rational function <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$">, we can simply compose it with the
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original transfer function <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> to fabricate a new rational function,
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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J(Z) = H(R(Z))
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\end{displaymath}
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-->
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<IMG
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WIDTH="117" HEIGHT="28" BORDER="0"
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SRC="img944.png"
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ALT="\begin{displaymath}
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J(Z) = H(R(Z))
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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The gain of the new filter <IMG
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WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img945.png"
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ALT="$J$"> at the frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> is then equal to
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that of <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> at a different frequency <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">, chosen so that:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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\cos(\phi) + i \sin(\phi) = R(\cos(\omega) + i \sin(\omega))
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\end{displaymath}
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-->
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<IMG
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WIDTH="270" HEIGHT="28" BORDER="0"
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SRC="img946.png"
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ALT="\begin{displaymath}
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\cos(\phi) + i \sin(\phi) = R(\cos(\omega) + i \sin(\omega))
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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The function <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> moves points around on the unit
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circle; <IMG
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WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img945.png"
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ALT="$J$"> at any point equals <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> on the point <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> moves it to.
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<P>
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For example, suppose we start with a one-zero, one-pole low-pass filter:
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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) =
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{{
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1 + {Z^{-1}}
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} \over {
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1 - g{Z^{-1}}
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="127" HEIGHT="44" BORDER="0"
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SRC="img947.png"
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ALT="\begin{displaymath}
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H(Z) =
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{{
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1 + {Z^{-1}}
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} \over {
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1 - g{Z^{-1}}
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}}
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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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and apply the function
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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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R(Z) = -{Z^2} = -
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{{
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1 \cdot {Z^2} + 0 \cdot Z + 0
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} \over {
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0 \cdot {Z^2} + 0 \cdot Z + 1
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="242" HEIGHT="42" BORDER="0"
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SRC="img948.png"
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ALT="\begin{displaymath}
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R(Z) = -{Z^2} = -
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{{
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1 \cdot {Z^2} + 0 \cdot Z + 0
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} \over {
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0 \cdot {Z^2} + 0 \cdot Z + 1
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}}
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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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Geometrically, this choice of <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> stretches the unit circle uniformly to twice
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its circumference and wraps it around itself twice. 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
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<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="$-1$"> are both sent to the point <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="$-1$">, and the points <IMG
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WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img646.png"
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ALT="$i$"> and <IMG
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WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img949.png"
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ALT="$-i$"> are sent
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to the point <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="$1$">. The resulting transfer
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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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J(Z) =
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{{
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1 - {Z^{-2}}
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} \over {
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1 + g{Z^{-2}}
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}}
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=
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{{
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(1 - {Z^{-1}})(1 + {Z^{-1}})
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} \over {
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(1 - i\sqrt{g} {Z^{-1}})(1 + i\sqrt{g} {Z^{-1}})
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="339" HEIGHT="46" BORDER="0"
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SRC="img950.png"
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ALT="\begin{displaymath}
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J(Z) =
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{{
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1 - {Z^{-2}}
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} \over {
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1 + g{Z^{-2}}
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}}
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=
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...
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... \over {
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(1 - i\sqrt{g} {Z^{-1}})(1 + i\sqrt{g} {Z^{-1}})
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}}
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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 pole-zero plots of <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> and <IMG
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WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img945.png"
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ALT="$J$"> are shown in Figure <A HREF="#fig08.19">8.19</A>. From
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a low-pass filter we ended up with a band-pass filter. The points <IMG
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WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img646.png"
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ALT="$i$"> and <IMG
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WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img949.png"
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ALT="$-i$">
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which <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> sends 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="$1$"> (where the original filter's gain is highest) become
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points of highest gain for the new filter.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig08.19"></A><A NAME="10765"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.19:</STRONG>
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One-pole, one-zero low-pass filter: (a) pole-zero plot; (b)
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plot for the resulting filter after the transformation <IMG
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WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img51.png"
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ALT="$R(Z) = -{Z^2}$">. The
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result is a band-pass filter with center frequency <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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</CAPTION>
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<TR><TD><IMG
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WIDTH="467" HEIGHT="247" BORDER="0"
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SRC="img951.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.19.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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<HR>
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HREF="node147.html">Butterworth band-pass filter</A>
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Miller Puckette
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2006-12-30
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