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<H1><A NAME="SECTION001150000000000000000">
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Power conservation and complex delay networks</A>
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</H1>
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<P>
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The same techniques will work to analyze any delay network, although for
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more complicated networks it becomes harder to characterize the results, or
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to design the network to have specific, desired properties. Another point
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of view can sometimes be usefully brought to the situation, particularly
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when flat frequency responses are needed, either in their own right or else
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to ensure that a complex, recirculating network remains stable at feedback
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gains close to one.
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<P>
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The central fact we will use is that if any delay network, with either one or
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many inputs and outputs, is constructed so that its output power
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(averaged over time) always
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equals its input power, that network has to have a flat frequency response.
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This is almost a tautology; if you put in a sinusoid at any frequency on one of
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the inputs, you will get sinusoids of the same frequency at the outputs, and
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the sum of the power on all the outputs will equal the power of the input, so
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the gain, suitably defined, is exactly one.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.11"></A><A NAME="8003"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.11:</STRONG>
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First fundamental building block for unitary delay networks:
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delay lines in parallel.</CAPTION>
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<TR><TD><IMG
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WIDTH="126" HEIGHT="170" BORDER="0"
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SRC="img737.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.11.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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In order to work with power-conserving delay networks we will need an
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explicit definition of "total average power".
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If there is only one signal (call it <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$">), the average power is
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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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P(x[n]) = \left [{{|x[0]|}^2} + {{|x[1]|}^2} + \cdots
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+ {{|x[N-1]|}^2} \right ] / N
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\end{displaymath}
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-->
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<IMG
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WIDTH="349" HEIGHT="34" BORDER="0"
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SRC="img738.png"
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ALT="\begin{displaymath}
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P(x[n]) = \left [{{\vert x[0]\vert}^2} + {{\vert x[1]\vert}^2} + \cdots
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+ {{\vert x[N-1]\vert}^2} \right ] / 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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where <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="$N$"> is a large enough number so that any fluctuations in amplitude get
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averaged out. This definition works as well for complex-valued signals
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as for real-valued ones. The average total power for several digital
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audio signals is
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just the sum of the individual signal's powers:
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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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P({x_1}[n] , \ldots , {x_r}[n]) = P({x_1}[n]) + \cdots + P({x_r}[n])
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\end{displaymath}
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-->
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<IMG
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WIDTH="326" HEIGHT="28" BORDER="0"
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SRC="img739.png"
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ALT="\begin{displaymath}
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P({x_1}[n] , \ldots , {x_r}[n]) = P({x_1}[n]) + \cdots + P({x_r}[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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where <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> is the number of signals to be combined.
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<P>
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It turns out that a wide range of interesting delay networks has the property
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that the total power output equals the total power input;
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they are called
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<A NAME="8013"></A><I>unitary</I>. To start with, we can put any number of delays in parallel, as
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shown in Figure <A HREF="#fig07.11">7.11</A>. Whatever the total power of the inputs,
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the total power of the outputs has to equal it.
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<P>
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A second family of power-preserving transformations is composed of rotations
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and reflections of the signals <IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img740.png"
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ALT="${x_1}[n]$">, ... , <IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img741.png"
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ALT="${x_r}[n]$">, considering them,
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at each fixed time point <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">, as the <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> coordinates of a point in
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<IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$">-dimensional space. The rotation or reflection must be one that leaves the
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origin <!-- MATH
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$(0, \ldots, 0)$
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-->
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<IMG
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WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img742.png"
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ALT="$(0, \ldots, 0)$"> fixed.
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<P>
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For each sample number <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">, the total
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contribution to the average signal power is proportional 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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{|{x_1}|}^2 + \cdots + {|{x_r}|}^2
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\end{displaymath}
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-->
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<IMG
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WIDTH="119" HEIGHT="28" BORDER="0"
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SRC="img743.png"
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ALT="\begin{displaymath}
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{\vert{x_1}\vert}^2 + \cdots + {\vert{x_r}\vert}^2
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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 is just the Pythagorean distance from the origin to
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the point <!-- MATH
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$({x_1}, \ldots, {x_r})$
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-->
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<IMG
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WIDTH="83" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img744.png"
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ALT="$({x_1}, \ldots, {x_r})$">. Since rotations and reflections are
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distance-preserving transformations, the distance from the origin before
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transforming must
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equal the distance from the origin afterward. So the total power
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of a collection of signals must must be preserved by rotation.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.12"></A><A NAME="8024"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.12:</STRONG>
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Second fundamental building block for unitary delay networks:
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rotating two digital audio signals. Part (a) shows the transformation explicitly;
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(b) shows it as a matrix operation.</CAPTION>
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<TR><TD><IMG
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WIDTH="338" HEIGHT="291" BORDER="0"
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SRC="img745.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.12.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Figure <A HREF="#fig07.12">7.12</A> shows a rotation matrix operating on two signals. In
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part (a) the transformation is shown explicitly. If the input signals are
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<IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img740.png"
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ALT="${x_1}[n]$"> and <IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img746.png"
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ALT="${x_2}[n]$">, the outputs are:
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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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{y_1}[n] = c {x_1}[n] - s {x_2}[n]
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\end{displaymath}
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-->
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<IMG
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WIDTH="156" HEIGHT="28" BORDER="0"
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SRC="img747.png"
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ALT="\begin{displaymath}
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{y_1}[n] = c {x_1}[n] - s {x_2}[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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<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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{y_2}[n] = s {x_1}[n] + c {x_2}[n]
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\end{displaymath}
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-->
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<IMG
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WIDTH="156" HEIGHT="28" BORDER="0"
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SRC="img748.png"
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ALT="\begin{displaymath}
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{y_2}[n] = s {x_1}[n] + c {x_2}[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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where <IMG
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WIDTH="24" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img749.png"
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ALT="$c, s$"> are 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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c = \cos(\theta)
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\end{displaymath}
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-->
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<IMG
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WIDTH="67" HEIGHT="28" BORDER="0"
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SRC="img750.png"
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ALT="\begin{displaymath}
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c = \cos(\theta)
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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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s = \sin(\theta)
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\end{displaymath}
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-->
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<IMG
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WIDTH="67" HEIGHT="28" BORDER="0"
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SRC="img751.png"
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ALT="\begin{displaymath}
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s = \sin(\theta)
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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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for an
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<A NAME="8036"></A><I>angle of rotation</I> <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img634.png"
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ALT="$\theta$">.
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Considered as points on the Cartesian plane, the point <!-- MATH
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$({y_1}, {y_2})$
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-->
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<IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img752.png"
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ALT="$({y_1}, {y_2})$"> is
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just the point <!-- MATH
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$({x_1}, {x_2})$
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-->
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<IMG
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WIDTH="55" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img753.png"
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ALT="$({x_1}, {x_2})$"> rotated counter-clockwise by the angle
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<IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img634.png"
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ALT="$\theta$">. The two points are thus at the same distance from the origin:
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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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{|{y_1}|}^2 + {|{y_2}|}^2 = {|{x_1}|}^2 + {|{x_2}|}^2
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\end{displaymath}
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-->
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<IMG
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WIDTH="183" HEIGHT="28" BORDER="0"
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SRC="img754.png"
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ALT="\begin{displaymath}
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{\vert{y_1}\vert}^2 + {\vert{y_2}\vert}^2 = {\vert{x_1}\vert}^2 + {\vert{x_2}\vert}^2
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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 so the two output signals have the same total power as the two input
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signals.
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<P>
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For an alternative description of rotation in two dimensions,
|
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consider complex numbers <!-- MATH
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$X={x_1} + {x_2}i$
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-->
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<IMG
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WIDTH="96" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img755.png"
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ALT="$X={x_1} + {x_2}i$"> and
|
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<!-- MATH
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$Y={y_1} + {y_2}i$
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-->
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<IMG
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WIDTH="92" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img756.png"
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ALT="$Y={y_1} + {y_2}i$">. The above transformation amounts to setting
|
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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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Y = XZ
|
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|
\end{displaymath}
|
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-->
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<IMG
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WIDTH="60" HEIGHT="24" BORDER="0"
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SRC="img757.png"
|
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ALT="\begin{displaymath}
|
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Y = XZ
|
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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="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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|
SRC="img20.png"
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ALT="$Z$"> is a complex number with
|
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unit magnitude and argument <IMG
|
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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|
SRC="img634.png"
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ALT="$\theta$">. Since <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img22.png"
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ALT="$\vert Z\vert=1$">, it follows that
|
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<IMG
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WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img758.png"
|
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ALT="$\vert X\vert = \vert Y\vert$">.
|
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|
<P>
|
|
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|
|
If we perform a rotation on a pair of signals and then invert one (but not the
|
|
|
|
|
other) of them, the result is a
|
|
|
|
|
<A NAME="8050"></A>
|
|
|
|
|
<I>reflection</I>.
|
|
|
|
|
This also preserves total signal power, since we can invert any or all of a
|
|
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|
|
collection of signals without changing the total power. In two dimensions, a
|
|
|
|
|
reflection appears as a transformation of the form
|
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|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{y_1}[n] = c {x_1}[n] + s {x_2}[n]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
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|
|
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|
|
<IMG
|
|
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|
|
WIDTH="156" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img759.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{y_1}[n] = c {x_1}[n] + s {x_2}[n]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
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|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{y_2}[n] = s {x_1}[n] - c {x_2}[n]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="156" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img760.png"
|
|
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|
|
ALT="\begin{displaymath}
|
|
|
|
|
{y_2}[n] = s {x_1}[n] - c {x_2}[n]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
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|
|
|
|
|
|
<P>
|
|
|
|
|
A special and useful rotation matrix is obtained by setting
|
|
|
|
|
<!-- MATH
|
|
|
|
|
$\theta = \pi/4$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="58" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img34.png"
|
|
|
|
|
ALT="$\theta = \pi /4$">, so that <!-- MATH
|
|
|
|
|
$s = c = \sqrt{1/2}$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img761.png"
|
|
|
|
|
ALT="$s = c = \sqrt{1/2}$">. This
|
|
|
|
|
allows us to simplify the computation as shown in Figure <A HREF="#fig07.13">7.13</A> (part
|
|
|
|
|
a) because each signal need only be multiplied by the one quantity <IMG
|
|
|
|
|
WIDTH="39" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img762.png"
|
|
|
|
|
ALT="$c=s$">.
|
|
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|
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|
|
|
<P>
|
|
|
|
|
|
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig07.13"></A><A NAME="8389"></A>
|
|
|
|
|
<TABLE>
|
|
|
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.13:</STRONG>
|
|
|
|
|
Details about rotation (and reflection) matrix operations: (a)
|
|
|
|
|
rotation by the angle <!-- MATH
|
|
|
|
|
$\theta = \pi/4$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="58" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img34.png"
|
|
|
|
|
ALT="$\theta = \pi /4$">, so that
|
|
|
|
|
<!-- MATH
|
|
|
|
|
$a = \cos(\theta) = \sin(\theta) = \sqrt{1/2} \approx 0.7071$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="262" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img763.png"
|
|
|
|
|
ALT="$a = \cos(\theta) = \sin(\theta) = \sqrt{1/2} \approx 0.7071$">; (b) combining
|
|
|
|
|
two-dimensional rotations to make higher-dimensional ones.</CAPTION>
|
|
|
|
|
<TR><TD><IMG
|
|
|
|
|
WIDTH="298" HEIGHT="296" BORDER="0"
|
|
|
|
|
SRC="img764.png"
|
|
|
|
|
ALT="\begin{figure}\psfig{file=figs/fig07.13.ps}\end{figure}"></TD></TR>
|
|
|
|
|
</TABLE>
|
|
|
|
|
</DIV>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
More complicated rotations or reflections of more than two input signals may be
|
|
|
|
|
made by repeatedly rotating and/or reflecting them in pairs. For
|
|
|
|
|
example, in Figure <A HREF="#fig07.13">7.13</A> (part b), four signals are combined in
|
|
|
|
|
pairs, in two successive stages, so that in the end every signal input feeds
|
|
|
|
|
into all the outputs. We could do the same with eight signals (using three
|
|
|
|
|
stages) and so on. Furthermore, if we use the special angle <IMG
|
|
|
|
|
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img52.png"
|
|
|
|
|
ALT="$\pi /4$">, all the
|
|
|
|
|
input signals will contribute equally to each of the outputs.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Any combination of delays and rotation matrices, applied in succession to a
|
|
|
|
|
collection of audio signals, will result in a flat frequency response, since
|
|
|
|
|
each individual operation does.
|
|
|
|
|
This already allows us to generate an infinitude of
|
|
|
|
|
flat-response delay networks, but so far, none of them are recirculating. A
|
|
|
|
|
third operation, shown in Figure <A HREF="#fig07.14">7.14</A>, allows us to make
|
|
|
|
|
recirculating networks that still enjoy flat frequency responses.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
|
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig07.14"></A><A NAME="8069"></A>
|
|
|
|
|
<TABLE>
|
|
|
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.14:</STRONG>
|
|
|
|
|
Flat frequency response in recirculating networks: (a) in general,
|
|
|
|
|
using a rotation matrix <IMG
|
|
|
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img36.png"
|
2022-04-12 23:17:03 -03:00
|
|
|
ALT="$R$">; (b) the "all-pass" configuration.</CAPTION>
|
2022-04-12 21:54:18 -03:00
|
|
|
<TR><TD><IMG
|
|
|
|
|
WIDTH="381" HEIGHT="287" BORDER="0"
|
|
|
|
|
SRC="img765.png"
|
|
|
|
|
ALT="\begin{figure}\psfig{file=figs/fig07.14.ps}\end{figure}"></TD></TR>
|
|
|
|
|
</TABLE>
|
|
|
|
|
</DIV>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Part (a) of the figure shows the general layout. The transformation <IMG
|
|
|
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img36.png"
|
|
|
|
|
ALT="$R$"> is
|
|
|
|
|
assumed to be any combination of delays and mixing matrices that preserves
|
|
|
|
|
total power. The signals <!-- MATH
|
|
|
|
|
${x_1}, \ldots {x_k}$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="64" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img766.png"
|
|
|
|
|
ALT="${x_1}, \ldots {x_k}$"> go into a
|
|
|
|
|
unitary delay network, and the output signals <!-- MATH
|
|
|
|
|
${y_1}, \ldots {y_k}$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="62" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img767.png"
|
|
|
|
|
ALT="${y_1}, \ldots {y_k}$"> emerge.
|
|
|
|
|
Some other signals <!-- MATH
|
|
|
|
|
${w_1}, \ldots {w_j}$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="68" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img768.png"
|
|
|
|
|
ALT="${w_1}, \ldots {w_j}$"> (where <IMG
|
|
|
|
|
WIDTH="10" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img769.png"
|
|
|
|
|
ALT="$j$"> is not necessarily equal to
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$k$">) appear at the output of the transformation <IMG
|
|
|
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img36.png"
|
|
|
|
|
ALT="$R$"> and are fed back to its
|
|
|
|
|
input.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
If <IMG
|
|
|
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img36.png"
|
|
|
|
|
ALT="$R$"> is indeed power preserving, the total input power (the power of the
|
|
|
|
|
signals <!-- MATH
|
|
|
|
|
${x_1}, \ldots {x_k}$
|
|
|
|
|
-->
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="64" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img766.png"
|
|
|
|
|
ALT="${x_1}, \ldots {x_k}$"> plus that of the signals <!-- MATH
|
|
|
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${w_1}, \ldots {w_j}$
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-->
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<IMG
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ALT="${w_1}, \ldots {w_j}$">)
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must equal the output power (the power of the signals <!-- MATH
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${y_1}, \ldots {y_k}$
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-->
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<IMG
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ALT="${y_1}, \ldots {y_k}$">
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plus <!-- MATH
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${w_1}, \ldots {w_j}$
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-->
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<IMG
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ALT="${w_1}, \ldots {w_j}$">), and subtracting all the <IMG
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ALT="$w$"> from the
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equality, we find that the total input and output power are equal.
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<P>
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If we let <IMG
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ALT="$j=k=1$"> so that there is one <IMG
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SRC="img243.png"
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ALT="$x$">, <IMG
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ALT="$y$">, and <IMG
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ALT="$w$">,
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and
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let the transformation <IMG
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ALT="$R$"> be a rotation by <IMG
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ALT="$\theta$"> followed by a delay of
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<IMG
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ALT="$d$"> samples on the <IMG
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ALT="$W$"> output, the result is the well-known
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<A NAME="8086"></A><A NAME="8087"></A><I>all-pass filter</I>.
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With some juggling, and letting <!-- MATH
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$c = \cos(\theta)$
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-->
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<IMG
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ALT="$c = \cos(\theta)$">, we can show it is
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equivalent to the network
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shown in part (b) of the figure. All-pass filters have many applications, some
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of which we will visit later in this book.
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<P>
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<HR>
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HREF="node104.html">Time shifts and delays</A>
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HREF="node109.html">Recirculating delay networks</A>
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<B> <A NAME="tex2html2196"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2198"
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