633 lines
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633 lines
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<H1><A NAME="SECTION001130000000000000000"></A>
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<A NAME="sect7.network"></A>
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<BR>
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Delay networks
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</H1>
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.03"></A><A NAME="7867"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.3:</STRONG>
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A delay network. Here we add the incoming signal to a delayed
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copy of itself.</CAPTION>
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<TR><TD><IMG
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WIDTH="55" HEIGHT="219" BORDER="0"
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SRC="img679.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.03.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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If we consider our digital audio samples <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$"> to correspond to
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successive moments in time, then time shifting the signal by <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> samples
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corresponds to a
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<A NAME="7870"></A>
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<I>delay</I>
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of <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img680.png"
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ALT="$d/R$"> time units, where <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="$R$"> is the sample rate.
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Figure <A HREF="#fig07.03">7.3</A> shows one example of a
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<A NAME="7873"></A><I>linear delay network</I>:
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an assembly of delay units, possibly with amplitude
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scaling operations, combined using addition and subtraction. The output
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is a linear function of the input, in the sense that adding two signals at the
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input is the same as processing each one separately and adding the results.
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Moreover, linear delay networks create no new frequencies in the output that
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weren't present in the input, as long as the network remains time invariant,
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so that the gains and delay times do not change with time.
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<P>
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In general there are two ways of thinking about delay networks. We can think
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in the
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<A NAME="7875"></A><I>time domain</I>,
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in which we draw waveforms as functions of time (or of the index <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="$n$">), and
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consider delays as time shifts. Alternatively we may think in the
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<A NAME="7877"></A><I>frequency domain</I>,
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in which we dose the input with a complex sinusoid (so that its output is a
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sinusoid at the same frequency) and report the amplitude and/or phase change
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wrought by the network, as a function of the frequency. We'll now look at the
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delay network of Figure <A HREF="#fig07.03">7.3</A> in each of the two ways in turn.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.04"></A><A NAME="8340"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.4:</STRONG>
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The time domain view of the delay network of Figure <A HREF="#fig07.03">7.3</A>.
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The output is the sum of the input and its time shifted copy.</CAPTION>
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<TR><TD><IMG
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WIDTH="440" HEIGHT="257" BORDER="0"
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SRC="img681.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.04.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="#fig07.04">7.4</A> shows the network's behavior in the time domain. We
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invent some sort of suitable test function as input (it's a rectangular pulse
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eight samples wide in this example) and graph the input and output as functions
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of the 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="$n$">. This particular delay network adds the input to a
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delayed copy of itself.
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<P>
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A frequently used test function is an
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<A NAME="7886"></A><I>impulse</I>,
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which is a pulse lasting only one sample. The utility of this is that, if we
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know the output of the network for an impulse, we can find the output for
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any other digital audio signal--because any signal <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="$x[n]$"> is a sum of
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impulses, one of height <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img682.png"
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ALT="$x[0]$">, the next one occurring one sample later
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and having height <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img683.png"
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ALT="$x[1]$">, and so on. Later, when the networks get
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more complicated, we will move to using impulses as input signals to show
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their time-domain behavior.
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<P>
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On the other hand, we can analyze the same network in the frequency domain
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by considering a (complex-valued) test signal,
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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] = {Z^n}
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\end{displaymath}
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-->
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<IMG
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WIDTH="74" HEIGHT="28" BORDER="0"
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SRC="img684.png"
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ALT="\begin{displaymath}
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X[n] = {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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where Z has unit magnitude and argument <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">. We already know that the
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output is another complex sinusoid with the same frequency, that is,
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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H {Z^N}
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\end{displaymath}
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-->
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<IMG
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WIDTH="37" HEIGHT="24" BORDER="0"
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SRC="img685.png"
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ALT="\begin{displaymath}
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H {Z^N}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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for some complex number <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> (which we want to find). So we write the output
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directly as the sum of the input and its delayed copy:
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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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{Z^n} + {Z^{-d}} {Z^n} = (1 + {Z^{-d}}) {Z^n}
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\end{displaymath}
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-->
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<IMG
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WIDTH="200" HEIGHT="28" BORDER="0"
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SRC="img686.png"
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ALT="\begin{displaymath}
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{Z^n} + {Z^{-d}} {Z^n} = (1 + {Z^{-d}}) {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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and find by inspection 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 = 1 + {Z^{-d}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="92" HEIGHT="25" BORDER="0"
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SRC="img687.png"
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ALT="\begin{displaymath}
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H = 1 + {Z^{-d}}
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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 can understand the frequency-domain behavior of this delay network
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by studying how the complex number <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> varies as a function of the
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angluar frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">. We are especially interested in its argument and
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magnitude--which tell us the relative phase and amplitude of the sinusoid
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that comes out. We will work this example out in detail to show how the
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arithmetic of complex numbers can predict what happens when sinusoids are
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combined additively.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.05"></A><A NAME="8344"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.5:</STRONG>
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Analysis, in the complex plane, of the frequency-domain behavior of
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the delay network of Figure <A HREF="#fig07.03">7.3</A>. The complex number <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$"> encodes
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the frequency of the input. The delay line output is the input times
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<IMG
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WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img24.png"
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ALT="$Z^-d$">. The total (complex) gain is <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$">. We find the magnitude and argument
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of <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> by symmetrizing the sum, rotating it by <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img26.png"
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ALT="$d/2$"> times the
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angular frequency of the input.</CAPTION>
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<TR><TD><IMG
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WIDTH="355" HEIGHT="207" BORDER="0"
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SRC="img688.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.05.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="#fig07.05">7.5</A> shows the result, in the complex plane, when the
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quantities <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img262.png"
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ALT="$1$"> and <IMG
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WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img676.png"
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ALT="${Z^{-d}}$"> are combined additively. To add complex numbers
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we add their real and complex parts separately. So the complex number <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img262.png"
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ALT="$1$"> (real
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part <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img262.png"
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ALT="$1$">, imaginary part <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img179.png"
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ALT="$0$">) is added coordinate-wise to the complex
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number <IMG
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WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img676.png"
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ALT="${Z^{-d}}$"> (real part <!-- MATH
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$\cos(-d \omega)$
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-->
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<IMG
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WIDTH="68" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img689.png"
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ALT="$\cos(-d \omega)$">, imaginary part
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<!-- MATH
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$\sin(-d \omega)$
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-->
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<IMG
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WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img690.png"
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ALT="$\sin(-d \omega)$">). This is shown graphically by making a parallelogram,
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with corners at the origin and at the two points to be added, and whose
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fourth corner is the sum <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$">.
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<P>
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As the figure shows, the result can be understood by symmetrizing it about
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the real axis: instead of <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img262.png"
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ALT="$1$"> and <IMG
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WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img676.png"
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ALT="${Z^{-d}}$">, it's easier to sum the
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quantities <IMG
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WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img691.png"
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ALT="${Z^{d/2}}$"> and <IMG
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WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img692.png"
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ALT="${Z^{-d/2}}$">, because they are symmetric about
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the real (horizontal) axis. (Strictly speaking, we haven't properly defined
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the quantities <IMG
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WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img691.png"
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ALT="${Z^{d/2}}$"> and <IMG
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WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img692.png"
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ALT="${Z^{-d/2}}$">; we are using those expressions
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to denote unit
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complex numbers whose arguments are half those of <IMG
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WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img693.png"
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ALT="$Z^d$"> and <IMG
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WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img694.png"
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ALT="$Z^{-d}$">, so
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that squaring them would give <IMG
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WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img695.png"
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ALT="${Z^{d}}$"> and <IMG
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WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img676.png"
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ALT="${Z^{-d}}$">.)
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We rewrite the gain as:
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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^{-d/2}} ({Z^{d/2}} + {Z^{-d/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="img696.png"
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ALT="\begin{displaymath}
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H = {Z^{-d/2}} ({Z^{d/2}} + {Z^{-d/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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The first term is a phase shift of <IMG
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WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img697.png"
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ALT="$-d \omega / 2$">. The second term is
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best understood in rectangular form:
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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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{Z^{d/2}} + {Z^{-d/2}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="92" HEIGHT="25" BORDER="0"
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SRC="img698.png"
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ALT="\begin{displaymath}
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{Z^{d/2}} + {Z^{-d/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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<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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= (\cos(\omega d / 2) + i \sin(\omega d / 2)) +
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(\cos(\omega d / 2) - i \sin(\omega d / 2))
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\end{displaymath}
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-->
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<IMG
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WIDTH="384" HEIGHT="28" BORDER="0"
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SRC="img699.png"
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ALT="\begin{displaymath}
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= (\cos(\omega d / 2) + i \sin(\omega d / 2)) +
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(\cos(\omega d / 2) - i \sin(\omega d / 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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<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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= 2 \cos(\omega d / 2)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="93" HEIGHT="28" BORDER="0"
|
|
SRC="img700.png"
|
|
ALT="\begin{displaymath}
|
|
= 2 \cos(\omega d / 2)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
This real-valued quantity may be either positive or negative; its absolute value
|
|
gives the magnitude of the output:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
|H| = 2 |\cos(\omega d / 2) |
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="129" HEIGHT="28" BORDER="0"
|
|
SRC="img701.png"
|
|
ALT="\begin{displaymath}
|
|
\vert H\vert = 2 \vert\cos(\omega d / 2) \vert
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
The quantity <IMG
|
|
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img30.png"
|
|
ALT="$\vert H\vert$"> is called the
|
|
<A NAME="7917"></A><I>gain</I>
|
|
of the delay network at the angular frequency <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $">, and is graphed in
|
|
Figure <A HREF="#fig07.06">7.6</A>. The frequency-dependent gain of a delay network (that
|
|
is, the gain as a function of frequency) is called the network's
|
|
<A NAME="7920"></A><I>frequency response</I>.
|
|
|
|
<P>
|
|
Since the network has greater gain at some frequencies
|
|
than at others, it may be considered as a
|
|
<A NAME="7922"></A><I>filter</I>
|
|
that can be used to separate certain components of a sound from others.
|
|
Because of the shape of this particular gain expression as a function of
|
|
<IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $">, this kind of delay network
|
|
is called a (non-recirculating)
|
|
<A NAME="7924"></A><I>comb filter</I>.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig07.06"></A><A NAME="8359"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.6:</STRONG>
|
|
Gain of the delay network of Figure <A HREF="#fig07.03">7.3</A>, shown as a function
|
|
of angular frequency <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $">.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="326" HEIGHT="137" BORDER="0"
|
|
SRC="img702.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig07.06.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
The output of the network is a sum of two sinusoids of equal amplitude, and
|
|
whose phases differ by <IMG
|
|
WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img703.png"
|
|
ALT="$\omega d $">. The resulting frequency response agrees
|
|
with common sense: if the angular frequency <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $"> is set so that an integer
|
|
number of periods fit into <IMG
|
|
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img28.png"
|
|
ALT="$d$"> samples, i.e., if <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $"> is a multiple of
|
|
<IMG
|
|
WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img704.png"
|
|
ALT="$2\pi/d$">, the output of the delay is exactly the same as the original signal,
|
|
and so the two combine to make an output with twice the original amplitude. On
|
|
the other hand, if for example we take <!-- MATH
|
|
$\omega = \pi/d$
|
|
-->
|
|
<IMG
|
|
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img705.png"
|
|
ALT="$\omega = \pi/d$"> so that the delay is
|
|
half the period, then the delay output is out of phase and cancels the input
|
|
exactly.
|
|
|
|
<P>
|
|
This particular delay network has an interesting application: if we have a
|
|
periodic (or nearly periodic) incoming signal, whose fundamental frequency is
|
|
<IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $"> radians per sample, we can tune the comb filter so that the peaks in
|
|
the gain are aligned at even harmonics and the odd ones fall where the gain is
|
|
zero. To do this we choose <IMG
|
|
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img706.png"
|
|
ALT="$d=\pi/\omega$">, i.e., set the delay time to exactly
|
|
one half period of the incoming signal. In this way we get a new signal whose
|
|
harmonics are <!-- MATH
|
|
$2\omega, 4\omega, 6\omega, \ldots$
|
|
-->
|
|
<IMG
|
|
WIDTH="98" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img707.png"
|
|
ALT="$2\omega, 4\omega, 6\omega, \ldots$">, and so it now has a new
|
|
fundamental frequency at twice the original one. Except for a factor of two,
|
|
the amplitudes of the remaining harmonics still follow the spectral envelope of
|
|
the original sound. So we have a tool now for raising the pitch of an incoming
|
|
sound by an octave without changing its spectral envelope. This octave doubler
|
|
is the reverse of the octave divider introduced back in Chapter
|
|
<A HREF="node75.html#chapter-modulation">5</A>.
|
|
|
|
<P>
|
|
The time and frequency domains offer complementary ways of looking
|
|
at the same delay network. When the delays inside the network are smaller than
|
|
the ear's ability to resolve events in time--less than about 20
|
|
milliseconds--the time domain picture becomes less relevant to our
|
|
understanding of the delay network, and we turn mostly to the frequency-domain
|
|
picture. On the other hand, when delays are greater than about 50
|
|
msec, the peaks and valleys of plots showing gain versus frequency (such as
|
|
that of Figure <A HREF="#fig07.06">7.6</A>) crowd so closely together that
|
|
the frequency-domain view becomes less important. Both are nonetheless valid
|
|
over the entire range of possible delay times.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
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<BR>
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<B> Next:</B> <A NAME="tex2html2172"
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HREF="node109.html">Recirculating delay networks</A>
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<B> Up:</B> <A NAME="tex2html2166"
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HREF="node104.html">Time shifts and delays</A>
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<B> Previous:</B> <A NAME="tex2html2160"
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HREF="node107.html">Time shifts and phase</A>
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<B> <A NAME="tex2html2168"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2170"
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
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</ADDRESS>
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</HTML>
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