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2022-04-12 21:54:18 -03:00
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<B> Next:</B> <A NAME="tex2html2624"
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HREF="node138.html">Two recirculating filters for</A>
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<B> Up:</B> <A NAME="tex2html2618"
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HREF="node132.html">Elementary filters</A>
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HREF="node136.html">Compound filters</A>
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<B> <A NAME="tex2html2620"
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001225000000000000000">
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Real outputs from complex filters</A>
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</H2>
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<P>
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In most applications, we start with a real-valued signal to filter and we need
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a real-valued output, but in general, a compound filter with a transfer
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function as above will give a complex-valued output. However, we can
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construct filters with non-real-valued coefficients which nonetheless give
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real-valued outputs, so that the analysis that we carry out using complex
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numbers can be used to predict, explain, and control real-valued output
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signals. We do this by pairing each elementary filter (with coefficient
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<IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img880.png"
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ALT="$P$"> or <IMG
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$Q$">) with another having as its coefficient the complex conjugate
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<IMG
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WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img890.png"
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ALT="$\overline{P}$"> or <IMG
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WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img872.png"
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ALT="$\overline{Q}$">.
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<P>
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For example, putting two non-recirculating filters, with coefficients <IMG
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$Q$"> and
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<IMG
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WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img872.png"
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ALT="$\overline{Q}$">, in series gives a transfer function equal to:
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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) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})
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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="img891.png"
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ALT="\begin{displaymath}
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H(Z) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})
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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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which has the property 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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H(\overline{Z}) = \overline{H(Z)}
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\end{displaymath}
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-->
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<IMG
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WIDTH="99" HEIGHT="28" BORDER="0"
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SRC="img892.png"
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ALT="\begin{displaymath}
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H(\overline{Z}) = \overline{H(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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Now if we put any real-valued sinusoid:
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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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{X_n} = 2 \, \mathrm{re}(A{Z^n}) = A{Z^n} + \overline{A}
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{{\overline{Z}}^n}
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\end{displaymath}
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-->
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<IMG
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WIDTH="217" HEIGHT="28" BORDER="0"
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SRC="img893.png"
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ALT="\begin{displaymath}
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{X_n} = 2 \, \mathrm{re}(A{Z^n}) = A{Z^n} + \overline{A}
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{{\overline{Z}}^n}
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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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we get out:
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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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A \cdot H(Z) \cdot {Z^n} +
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\overline{A} \cdot \overline{H(Z)} \cdot {{\overline{Z}}^n}
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\end{displaymath}
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-->
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<IMG
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WIDTH="207" HEIGHT="28" BORDER="0"
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SRC="img894.png"
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ALT="\begin{displaymath}
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A \cdot H(Z) \cdot {Z^n} +
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\overline{A} \cdot \overline{H(Z)} \cdot {{\overline{Z}}^n}
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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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which, by inspection, is another real sinusoid.
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Here we're using two properties of complex conjugates. First, you can
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add and multiply them at will:
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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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\overline{A+B} = \overline{A} + \overline{B}
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\end{displaymath}
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-->
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<IMG
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WIDTH="110" HEIGHT="25" BORDER="0"
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SRC="img895.png"
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ALT="\begin{displaymath}
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\overline{A+B} = \overline{A} + \overline{B}
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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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<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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\overline{AB} = \overline{A} \cdot \overline{B}
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\end{displaymath}
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-->
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<IMG
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WIDTH="82" HEIGHT="24" BORDER="0"
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SRC="img896.png"
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ALT="\begin{displaymath}
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\overline{AB} = \overline{A} \cdot \overline{B}
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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 second, anything plus its complex conjugate is real, and is in fact
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twice its real part:
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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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A + \overline{A} = 2 \, \mathrm{re} (A)
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\end{displaymath}
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-->
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<IMG
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WIDTH="112" HEIGHT="28" BORDER="0"
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SRC="img897.png"
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ALT="\begin{displaymath}
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A + \overline{A} = 2 \, \mathrm{re} (A)
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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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This result for two conjugate filters extends to any compound filter; in
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general, we always get a real-valued output from a real-valued input if we
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arrange that each coefficient <IMG
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WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img898.png"
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ALT="$Q_i$"> and <IMG
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WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img899.png"
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ALT="$P_i$"> in the compound filter is
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either real-valued, or else appears in a pair with its complex conjugate.
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<P>
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<B> Next:</B> <A NAME="tex2html2624"
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HREF="node138.html">Two recirculating filters for</A>
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<B> Up:</B> <A NAME="tex2html2618"
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HREF="node132.html">Elementary filters</A>
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<B> Previous:</B> <A NAME="tex2html2612"
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HREF="node136.html">Compound filters</A>
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<B> <A NAME="tex2html2620"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2622"
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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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