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<H2><A NAME="SECTION001225000000000000000">
Real outputs from complex filters</A>
</H2>
<P>
In most applications, we start with a real-valued signal to filter and we need
a real-valued output, but in general, a compound filter with a transfer
function as above will give a complex-valued output. However, we can
construct filters with non-real-valued coefficients which nonetheless give
real-valued outputs, so that the analysis that we carry out using complex
numbers can be used to predict, explain, and control real-valued output
signals. We do this by pairing each elementary filter (with coefficient
<IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img880.png"
ALT="$P$"> or <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$">) with another having as its coefficient the complex conjugate
<IMG
WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img890.png"
ALT="$\overline{P}$"> or <IMG
WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img872.png"
ALT="$\overline{Q}$">.
<P>
For example, putting two non-recirculating filters, with coefficients <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> and
<IMG
WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img872.png"
ALT="$\overline{Q}$">, in series gives a transfer function equal to:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
H(Z) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})
\end{displaymath}
-->
<IMG
WIDTH="233" HEIGHT="28" BORDER="0"
SRC="img891.png"
ALT="\begin{displaymath}
H(Z) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
which has the property that:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
H(\overline{Z}) = \overline{H(Z)}
\end{displaymath}
-->
<IMG
WIDTH="99" HEIGHT="28" BORDER="0"
SRC="img892.png"
ALT="\begin{displaymath}
H(\overline{Z}) = \overline{H(Z)}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Now if we put any real-valued sinusoid:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{X_n} = 2 \, \mathrm{re}(A{Z^n}) = A{Z^n} + \overline{A}
{{\overline{Z}}^n}
\end{displaymath}
-->
<IMG
WIDTH="217" HEIGHT="28" BORDER="0"
SRC="img893.png"
ALT="\begin{displaymath}
{X_n} = 2 \, \mathrm{re}(A{Z^n}) = A{Z^n} + \overline{A}
{{\overline{Z}}^n}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
we get out:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A \cdot H(Z) \cdot {Z^n} +
\overline{A} \cdot \overline{H(Z)} \cdot {{\overline{Z}}^n}
\end{displaymath}
-->
<IMG
WIDTH="207" HEIGHT="28" BORDER="0"
SRC="img894.png"
ALT="\begin{displaymath}
A \cdot H(Z) \cdot {Z^n} +
\overline{A} \cdot \overline{H(Z)} \cdot {{\overline{Z}}^n}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
which, by inspection, is another real sinusoid.
Here we're using two properties of complex conjugates. First, you can
add and multiply them at will:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\overline{A+B} = \overline{A} + \overline{B}
\end{displaymath}
-->
<IMG
WIDTH="110" HEIGHT="25" BORDER="0"
SRC="img895.png"
ALT="\begin{displaymath}
\overline{A+B} = \overline{A} + \overline{B}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\overline{AB} = \overline{A} \cdot \overline{B}
\end{displaymath}
-->
<IMG
WIDTH="82" HEIGHT="24" BORDER="0"
SRC="img896.png"
ALT="\begin{displaymath}
\overline{AB} = \overline{A} \cdot \overline{B}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and second, anything plus its complex conjugate is real, and is in fact
twice its real part:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A + \overline{A} = 2 \, \mathrm{re} (A)
\end{displaymath}
-->
<IMG
WIDTH="112" HEIGHT="28" BORDER="0"
SRC="img897.png"
ALT="\begin{displaymath}
A + \overline{A} = 2 \, \mathrm{re} (A)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This result for two conjugate filters extends to any compound filter; in
general, we always get a real-valued output from a real-valued input if we
arrange that each coefficient <IMG
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img898.png"
ALT="$Q_i$"> and <IMG
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img899.png"
ALT="$P_i$"> in the compound filter is
either real-valued, or else appears in a pair with its complex conjugate.
<P>
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<ADDRESS>
Miller Puckette
2006-12-30
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