📄 part1_appendix.htm
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<P align=right><I><A
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<P align=center>Appendix</P></TD></TR>
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<P>The channel capacity of analog source was based on Euclid distance. But
in digital communication we must treat Humming distance. Let us start from
a proposition of mapping rule between Euclid and Humming distances, i.e.,
physical levels in voltage and code patterns. The widely used Gray code is
designed so that Humming distance between all pairs of code patterns
assigned to adjacent physical levels is <I>1</I>. The following shows Gray
code and its distance matrix for 4 bits case.</P>
<P align=center><IMG height=253 alt=img6.gif
src="Part1_appendix.files/img6.gif" width=82 border=0>
<IMG height=253 alt=img7.gif
src="Part1_appendix.files/img8.gif" width=286 border=0></P>
<P>The channel capacity of PAM system is calculated along the following
procedures.</P>
<OL>
<LI>Fix the value of SNR.
<LI>Let the number of levels be <IMG
src="Part1_appendix.files/appendix_htm_eqn4123.gif" border=0 NAMO_EQN__><!--NAMO_EQN__ 192 1L=2^{B}-->.
<LI>Calculate Humming distance matrix of Gray code.
<LI>Calculate error rate between physical levels.
<LI>Calculate average Hamming distance of error <IMG
src="Part1_appendix.files/appendix_htm_eqn4908.gif" border=0 NAMO_EQN__><!--NAMO_EQN__ 192 1p_{H}-->,where <IMG
src="Part1_appendix.files/appendix_htm_eqn5046.gif" border=0 NAMO_EQN__><!--NAMO_EQN__ 192 1p=p_{H}\slash B--> is bit error rate.
<LI>Get the
mutual entropy for <I>B</I> bit case by
<IMG src="Part1_appendix.files/appendix_htm_eqn8806.gif" border=0
NAMO_EQN__><!--NAMO_EQN__ 192 1B\cdot [1-\{ -p\, log_{2}p-(1-p)log_{2}(1-p)\} ]-->
<LI>Repeat procedures from<I> </I>2 to 6 for <I>B=1,2,••••</I>, we
have a curve of mutual entropy.
<LI>Maximum of the curve is the channel capacity for
given SNR. </LI></OL>
<P>Curves in the following figure
are <I>20dB, 25dB, •••, 40dB </I>from bottom to
top.</P>
<P align=center><IMG height=436 src="Part1_appendix.files/img50.gif"
width=705 border=0></P>
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