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<H2><A ID="SECTION001351000000000000000"></A>
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<A ID="sect9.phaserelationship"></A>
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<BR>
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Phase relationships between channels
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</H2>
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<P>
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In the scheme above, the phase of each <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$"> depends only on the previous
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value for the same
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channel. The phase relationships between neighboring channels are left to
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chance. This sometimes works fine, but sometimes the incoherence
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of neighboring channels gives rise to an unintended chorus effect. We
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would ideally wish for <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$"> and <IMG
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WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1195.png"
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ALT="$S[m,k+1]$"> to have the same phase
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relationship as for <IMG
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WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1196.png"
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ALT="$T'[k]$"> and <IMG
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WIDTH="64" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1197.png"
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ALT="$T'[k+1]$">, but also for the phase relationship
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between <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$"> and <IMG
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WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1198.png"
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ALT="$S[m-1,k]$"> to be the same as between <IMG
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WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1196.png"
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ALT="$T'[k]$"> and <IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1199.png"
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ALT="$T[k]$">.
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<P>
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These <IMG
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WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1200.png"
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ALT="$2N$"> equations for <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> phases in general will have no solution, but we
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can alter the equation for <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$"> above so that whenever there happens to be
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a solution to the over-constrained system of equations, the reconstruction
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algorithm homes in on the solution. This approach is called
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<A ID="12657"></A><I>phase locking</I>
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[<A
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HREF="node202.html#r-puckette95b">Puc95b</A>], and has the virtue of simplicity although more
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sophisticated techniques are available [<A
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HREF="node202.html#r-dolson97">DL97</A>]).
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<P>
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The desired output phase relation, at the frame <IMG
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WIDTH="44" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1185.png"
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ALT="$m-1$">, 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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\angle T[k+1] - \angle T[k] = \angle S[m-1, k+1] - \angle S[m-1, k]
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\end{displaymath}
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-->
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<IMG
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WIDTH="364" HEIGHT="28" BORDER="0"
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SRC="img1201.png"
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ALT="\begin{displaymath}
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\angle T[k+1] - \angle T[k] = \angle S[m-1, k+1] - \angle S[m-1, k]
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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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or, rearranging:
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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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\angle \left \{ {{S[m-1, k+1]} \over {T[k+1]}} \right \} =
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\angle \left \{ {{S[m-1, k]} \over {T[k]}} \right \}
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\end{displaymath}
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-->
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<IMG
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WIDTH="279" HEIGHT="45" BORDER="0"
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SRC="img1202.png"
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ALT="\begin{displaymath}
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\angle \left \{ {{S[m-1, k+1]} \over {T[k+1]}} \right \} =
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\angle \left \{ {{S[m-1, k]} \over {T[k]}} \right \}
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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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In other words, the phase of the quotient <IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1203.png"
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ALT="$S/T$"> should not depend on <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">.
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With this in mind, we can rewrite the recursion formula for <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$">:
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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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S[m, k] \; = \;
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{
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{ \left |
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R[k]
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\right |}
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^
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{-1}
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}
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\cdot
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{
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{R[k] T'[k]}
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}
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\end{displaymath}
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-->
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<IMG
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WIDTH="208" HEIGHT="28" BORDER="0"
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SRC="img1204.png"
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ALT="\begin{displaymath}
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S[m, k] \; = \;
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{
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{ \left \vert
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R[k]
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\right \vert}
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^
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{-1}
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}
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\cdot
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{
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{R[k] T'[k]}
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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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with
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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[k] \; = \;
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{
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{
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\overline {T[k]} \; \cdot \; {S[m-1, k]}
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} \over {
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\left |
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{S[m-1, k]}
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\right |
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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="188" HEIGHT="47" BORDER="0"
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SRC="img1205.png"
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ALT="\begin{displaymath}
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R[k] \; = \;
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{
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{
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\overline {T[k]} \; \cdot \; {S[m-1, k]}
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} \over {
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\left \vert
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{S[m-1, k]}
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\right \vert
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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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and because of the previous equation, the <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1206.png"
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ALT="$R[k]$"> should all be in phase. The
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trick is now to replace <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1206.png"
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ALT="$R[k]$">
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for each <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> with the sum of three neighboring ones. The computation is then:
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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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S[m, k] \; = \;
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{
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{ \left |
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R'[k]
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\right |}
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^
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{-1}
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}
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\; \cdot \;
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{
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{R'[k] T'[k]}
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}
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\end{displaymath}
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-->
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<IMG
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WIDTH="226" HEIGHT="28" BORDER="0"
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SRC="img1207.png"
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ALT="\begin{displaymath}
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S[m, k] \; = \;
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{
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{ \left \vert
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R'[k]
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\right \vert}
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^
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{-1}
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}
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\; \cdot \;
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{
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{R'[k] T'[k]}
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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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with
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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'[k] = R[k+1] + R[k] + R[k-1]
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\end{displaymath}
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-->
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<IMG
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WIDTH="238" HEIGHT="28" BORDER="0"
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SRC="img1208.png"
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ALT="\begin{displaymath}
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R'[k] = R[k+1] + R[k] + R[k-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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If the channels are already in the correct phase relationship, this has
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no effect (the resulting phase will be the same as if only <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1206.png"
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ALT="$R[k]$"> were
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used.) But in general the sum will share two terms in common with its
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neighbor at <IMG
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WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img602.png"
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ALT="$k+1$">:
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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'[k+1] = R[k+2] + R[k+1] + R[k]
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\end{displaymath}
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-->
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<IMG
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WIDTH="265" HEIGHT="28" BORDER="0"
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SRC="img1209.png"
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ALT="\begin{displaymath}
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R'[k+1] = R[k+2] + R[k+1] + R[k]
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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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so that the <IMG
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WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img1210.png"
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ALT="$R'$"> will tend to point more in the same direction than the <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="$R$">
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do. Applying this iteratively will eventually line all the <IMG
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WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img1210.png"
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ALT="$R'$"> up to the
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same phase, as long as the phase relationships between the measured spectra
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<IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img557.png"
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ALT="$T$"> and <IMG
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WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img1211.png"
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ALT="$T'$"> allow it.
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Miller Puckette
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2006-12-30
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