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<TD><P CLASS="Table"><A NAME="pgfId=183711"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183713"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183715"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183717"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183719"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183721"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183723"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183725"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183727"></A>S<SUB CLASS="Subscript">2</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183729"></A>S<SUB CLASS="Subscript">1</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183731"></A>250</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183733"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183735"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183737"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183739"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183741"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183743"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183745"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183747"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183749"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183751"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183753"></A>S<SUB CLASS="Subscript">1</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183755"></A>S<SUB CLASS="Subscript">2</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183757"></A>350</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183759"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183761"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183763"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183765"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183767"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183769"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183771"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183773"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183775"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183777"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183779"></A>S<SUB CLASS="Subscript">2</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183781"></A>S<SUB CLASS="Subscript">1</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183783"></A>450</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183785"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183787"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183789"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183791"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183793"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183795"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183797"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183799"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183801"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183803"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183805"></A>S<SUB CLASS="Subscript">1</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183807"></A>S<SUB CLASS="Subscript">2</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183809"></A>550</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183811"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183813"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183815"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183817"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183819"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183821"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183823"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183825"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183827"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183829"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183831"></A>S<SUB CLASS="Subscript">2</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183833"></A>S<SUB CLASS="Subscript">1</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183835"></A>650</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183837"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183839"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183841"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183843"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183845"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183847"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183849"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183851"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183853"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183855"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183857"></A>S<SUB CLASS="Subscript">1</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183859"></A>S<SUB CLASS="Subscript">2</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183861"></A>750</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183863"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183865"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183867"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183869"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183871"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183873"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183875"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183877"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183879"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183881"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183883"></A>S<SUB CLASS="Subscript">2</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183885"></A>S<SUB CLASS="Subscript">1</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183887"></A>850</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183889"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183891"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183893"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183895"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183897"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183899"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183901"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183903"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183905"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183907"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183909"></A>S<SUB CLASS="Subscript">1</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183911"></A>S<SUB CLASS="Subscript">2</SUB></TD></TR>
<TR>
<TD><P CLASS="TableLeft"><A NAME="pgfId=183913"></A>950</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183915"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183917"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183919"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183921"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183923"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183925"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183927"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183929"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183931"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183933"></A>&nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=183935"></A>S<SUB CLASS="Subscript">2</SUB></TD>
<TD><P CLASS="Table"><A NAME="pgfId=183937"></A>S<SUB CLASS="Subscript">1</SUB></TD></TR>
</TABLE>
<P CLASS="Body"><A NAME="pgfId=133168"></A>Next we transmit the eight possible
encoder outputs (Y<SUB CLASS="Subscript">n</SUB> = 0-7 ) as <B>signals</B>
over our noisy communications channel (perhaps a microwave signal to a satellite)
using the <B>signal constellation</B> shown in Figure&nbsp;11.4. Typically
this is done using <B>phase-shift keying</B> (<B> PSK</B>) with each signal
position corresponding to a different phase shift in the transmitted carrier
signal.</P>
<H2><A NAME="pgfId=133382"></A>11.12.2&nbsp;&nbsp;&nbsp;The Received Signal</H2>
<P><P CLASS="BodyAfterHead"><A NAME="pgfId=133433"></A>The noisy signal
enters the receiver. It is now our task to discover which of the eight possible
signals were transmitted at each time step. First we calculate the distance
of each received signal from each of the known eight positions in the signal
constellation. Table&nbsp;11.8 shows the distances between signals in the
8PSK constellation. We are going to assume that there is no noise in the
channel to illustrate the operation of the Viterbi decoder, so that the
distances in Table&nbsp;11.8 represent the possible distance measures of
our received signal from the 8PSK signals.</P>
<P><P CLASS="Body"><A NAME="pgfId=134700"></A>The distances, <B>X</B>, in
the first column of Table&nbsp;11.8 are the geometric or algebraic distances.
We measure the <B>Euclidean distance</B>, <B>E</B> = <B>X</B><SUP CLASS="Superscript">2</SUP>
shown as <B>B</B> (the binary quantized value of <B>E</B>) in Table&nbsp;11.8.
The rounding errors that result from conversion to fixed-width binary are
<B>quantization errors</B> and are important in any practical implementation
of the Viterbi decoder. The effect of the quantization error is to add a
form of noise to the received signal.</P>
<P><P CLASS="Body"><A NAME="pgfId=133166"></A>The following code models
the receiver section that digitizes the noisy analog received signal and
computes the binary distance measures. Eight binary-distance measures, <CODE>in0-in7</CODE>
, are generated each time a signal is received. Since each of the distance
measures is 3 bits wide, there are a total of 24 bits (8 <FONT FACE="Symbol">&yen;</FONT>
3) that form the digital inputs to the Viterbi decoder.</P>
<P><TABLE BORDER="1" CELLSPACING="2" CELLPADDING="2">
<TR>
<TD COLSPAN="7"><P CLASS="TableTitle"><A NAME="pgfId=184230"></A>TABLE&nbsp;11.8&nbsp;&nbsp;&nbsp;&nbsp;Distance
measures for Viterbi encoding (8PSK).</TD></TR>
<TR>
<TD><P CLASS="TableFirst"><A NAME="pgfId=184244"></A>Signal</TD>
<TD><P CLASS="TableFirst"><A NAME="pgfId=184246"></A>Algebraic distance from
signal 0</TD>
<TD><P CLASS="TableFirst"><A NAME="pgfId=184249"></A><B>X</B> = Distance from
signal 0</TD>
<TD><P><P CLASS="TableFirst"><A NAME="pgfId=184251"></A>Euclidean distance</P>
<P><P CLASS="TableFirst"><A NAME="pgfId=184252"></A><B>E = X<SUP>2</SUP></B></TD>
<TD><P CLASS="TableFirst"><A NAME="pgfId=184254"></A><B>B</B> = binary quantized
value of <B>E</B></TD>
<TD><P CLASS="TableFirst"><A NAME="pgfId=184256"></A><B>D</B> = decimal value
of B</TD>
<TD><P><P CLASS="TableFirst"><A NAME="pgfId=184258"></A>Quantization error</P>
<P><P CLASS="TableFirst"><A NAME="pgfId=184259"></A><B>Q</B> = <B>D</B>
- 1.75<B> E</B></TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=184261"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184266"></A>2 sin (0 &pi; / 8) &nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184268"></A>0.00</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184270"></A>0.00</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184272"></A>000</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184274"></A>0</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184279"></A>0 &nbsp;</TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=184281"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184286"></A>2 sin (1 &pi; / 8) &nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184288"></A>0.77</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184290"></A>0.59</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184292"></A>001</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184294"></A>1</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184299"></A>-0.0325 &nbsp;</TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=184301"></A>2</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184306"></A>2 sin (2 &pi; / 8) &nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184308"></A>1.41</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184310"></A>2.00</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184312"></A>100</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184314"></A>4</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184319"></A>0.5 &nbsp;</TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=184321"></A>3</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184326"></A>2 sin (3 &pi; / 8) &nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184328"></A>1.85</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184330"></A>3.41</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184332"></A>110</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184334"></A>6</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184339"></A>0.0325 &nbsp;</TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=184341"></A>4</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184346"></A>2 sin (4 &pi; / 8) &nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184348"></A>2.00</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184350"></A>4.00</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184352"></A>111</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184354"></A>7</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184359"></A>0 &nbsp;</TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=184361"></A>5</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184366"></A>2 sin (5 &pi; / 8) &nbsp;</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184368"></A>1.85</TD>
<TD><P CLASS="Table"><A NAME="pgfId=184370"></A>3.41</TD>