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<H2><A NAME="SECTION001237000000000000000">
Stretching the unit circle with rational functions</A>
</H2>

<P>
In Section <A HREF="node143.html#sect8.twopolebandpass">8.3.4</A> we saw a simple way to turn a low-pass
filter into a band-pass one.  It is tempting to apply the same method to turn
our Butterworth low-pass filter into a higher-quality band-pass filter; but if
we wish to preserve the high quality of the Butterworth filter we must be more
careful than before in the design of the transformation used.  In this section
we will prepare the way to making the Butterworth band-pass filter by
introducing a class of rational transformations of the complex plane which
preserve the unit circle.

<P>
This discussion is adapted from [<A
 HREF="node202.html#r-parks87">PB87</A>], pp. 201-206 (I'm grateful
to Julius Smith for this pointer).  There the transformation is carried out
in continuous time, but here we have adapted the method to operate in
discrete time, in order to make the discussion self-contained.

<P>
The idea is to start with any filter with a transfer function as before:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
H(Z) = {
        {
            (1 - {Q_1}{Z^{-1}}) \cdots (1 - {Q_j}{Z^{-1}})
        } \over {
            (1 - {P_1}{Z^{-1}}) \cdots (1 - {P_k}{Z^{-1}})
        }
    }
\end{displaymath}
 -->

<IMG
 WIDTH="263" HEIGHT="45" BORDER="0"
 SRC="img889.png"
 ALT="\begin{displaymath}
H(Z) = {
{
(1 - {Q_1}{Z^{-1}}) \cdots (1 - {Q_j}{Z^{-1}})
} \over {
(1 - {P_1}{Z^{-1}}) \cdots (1 - {P_k}{Z^{-1}})
}
}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
whose frequency response (the gain at a frequency <IMG
 WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img27.png"
 ALT="$\omega $">) is given by:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
| H(\cos(\omega) + i \sin(\omega)) |
\end{displaymath}
 -->

<IMG
 WIDTH="146" HEIGHT="28" BORDER="0"
 SRC="img938.png"
 ALT="\begin{displaymath}
\vert H(\cos(\omega) + i \sin(\omega)) \vert
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>

<P>
Now suppose we can find a rational function, <IMG
 WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img939.png"
 ALT="$R(Z)$">, which distorts the
unit circle in some desirable way.  For <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img36.png"
 ALT="$R$"> to be a rational function means
that it can be written as a quotient of two polynomials (for example, the
transfer function <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$H$"> is a rational function).  That <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img36.png"
 ALT="$R$"> sends points on the
unit circle to other points on the unit circle is just the condition that
<IMG
 WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img940.png"
 ALT="$\vert R(Z)\vert = 1$"> whenever <IMG
 WIDTH="44" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img941.png"
 ALT="$Z=1$">.  It can easily be checked that any function of
the form
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
R(Z) = U \cdot
    {{
        {A_n}{Z^n} + {A_{n-1}}{Z^{n-1}} + \cdots + {A_0}
    } \over {
        \overline{A_0}{Z^n} + \overline{A_1}{Z^{n-1}} + \cdots + \overline{A_n}
    }}
\end{displaymath}
 -->

<IMG
 WIDTH="295" HEIGHT="45" BORDER="0"
 SRC="img942.png"
 ALT="\begin{displaymath}
R(Z) = U \cdot
{{
{A_n}{Z^n} + {A_{n-1}}{Z^{n-1}} + \cdot...
...}{Z^n} + \overline{A_1}{Z^{n-1}} + \cdots + \overline{A_n}
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
(where <IMG
 WIDTH="54" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img943.png"
 ALT="$\vert U\vert=1$">) has this property.  The same reasoning as in Section
<A HREF="node134.html#sect8.secondform">8.2.2</A> confirms that <IMG
 WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img940.png"
 ALT="$\vert R(Z)\vert = 1$"> whenever <IMG
 WIDTH="44" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img941.png"
 ALT="$Z=1$">.

<P>
Once we have a suitable rational function <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img36.png"
 ALT="$R$">, we can simply compose it with the
original transfer function <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$H$"> to fabricate a new rational function,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
J(Z) = H(R(Z))
\end{displaymath}
 -->

<IMG
 WIDTH="117" HEIGHT="28" BORDER="0"
 SRC="img944.png"
 ALT="\begin{displaymath}
J(Z) = H(R(Z))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The gain of the new filter <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img945.png"
 ALT="$J$"> at the frequency <IMG
 WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img27.png"
 ALT="$\omega $"> is then equal to
that of <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$H$"> at a different frequency <IMG
 WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img77.png"
 ALT="$\phi$">, chosen so that:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
\cos(\phi) + i \sin(\phi) = R(\cos(\omega) + i \sin(\omega))
\end{displaymath}
 -->

<IMG
 WIDTH="270" HEIGHT="28" BORDER="0"
 SRC="img946.png"
 ALT="\begin{displaymath}
\cos(\phi) + i \sin(\phi) = R(\cos(\omega) + i \sin(\omega))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The function <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img36.png"
 ALT="$R$"> moves points around on the unit
circle; <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img945.png"
 ALT="$J$"> at any point equals <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$H$"> on the point <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img36.png"
 ALT="$R$"> moves it to.

<P>
For example, suppose we start with a one-zero, one-pole low-pass filter:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
H(Z) =
    {{
        1 + {Z^{-1}}
    } \over {
        1 - g{Z^{-1}}
    }}
\end{displaymath}
 -->

<IMG
 WIDTH="127" HEIGHT="44" BORDER="0"
 SRC="img947.png"
 ALT="\begin{displaymath}
H(Z) =
{{
1 + {Z^{-1}}
} \over {
1 - g{Z^{-1}}
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and apply the function 
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
R(Z) = -{Z^2} = -
    {{
        1 \cdot {Z^2} + 0 \cdot Z + 0
    } \over {
        0 \cdot {Z^2} + 0 \cdot Z + 1
    }}
\end{displaymath}
 -->

<IMG
 WIDTH="242" HEIGHT="42" BORDER="0"
 SRC="img948.png"
 ALT="\begin{displaymath}
R(Z) = -{Z^2} = -
{{
1 \cdot {Z^2} + 0 \cdot Z + 0
} \over {
0 \cdot {Z^2} + 0 \cdot Z + 1
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Geometrically, this choice of <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img36.png"
 ALT="$R$"> stretches the unit circle uniformly to twice
its circumference and wraps it around itself twice.  The points <IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img262.png"
 ALT="$1$"> and
<IMG
 WIDTH="23" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img401.png"
 ALT="$-1$"> are both sent to the point <IMG
 WIDTH="23" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img401.png"
 ALT="$-1$">, and the points <IMG
 WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img646.png"
 ALT="$i$"> and <IMG
 WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img949.png"
 ALT="$-i$"> are sent
to the point <IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img262.png"
 ALT="$1$">.  The resulting transfer
function is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
J(Z) = 
    {{
        1 - {Z^{-2}}
    } \over {
        1 + g{Z^{-2}}
    }}
    =
    {{
        (1 - {Z^{-1}})(1  + {Z^{-1}})
    } \over {
        (1 - i\sqrt{g} {Z^{-1}})(1 + i\sqrt{g}  {Z^{-1}})
    }}
\end{displaymath}
 -->

<IMG
 WIDTH="339" HEIGHT="46" BORDER="0"
 SRC="img950.png"
 ALT="\begin{displaymath}
J(Z) =
{{
1 - {Z^{-2}}
} \over {
1 + g{Z^{-2}}
}}
=
...
... \over {
(1 - i\sqrt{g} {Z^{-1}})(1 + i\sqrt{g} {Z^{-1}})
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The pole-zero plots of <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img25.png"
 ALT="$H$"> and <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img945.png"
 ALT="$J$"> are shown in Figure <A HREF="#fig08.19">8.19</A>.  From
a low-pass filter we ended up with a band-pass filter.  The points <IMG
 WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img646.png"
 ALT="$i$"> and <IMG
 WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img949.png"
 ALT="$-i$">
which <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img36.png"
 ALT="$R$"> sends to <IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img262.png"
 ALT="$1$"> (where the original filter's gain is highest) become
points of highest gain for the new filter.

<P>

<DIV ALIGN="CENTER"><A NAME="fig08.19"></A><A NAME="10765"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.19:</STRONG>
One-pole, one-zero low-pass filter: (a) pole-zero plot; (b)
plot for the resulting filter after the transformation <IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img51.png"
 ALT="$R(Z) = -{Z^2}$">.  The
result is a band-pass filter with center frequency <IMG
 WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.png"
 ALT="$\pi /2$">.
</CAPTION>
<TR><TD><IMG
 WIDTH="467" HEIGHT="247" BORDER="0"
 SRC="img951.png"
 ALT="\begin{figure}\psfig{file=figs/fig08.19.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>

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