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It is clear, however, that by sending the
information in a redundant form theprobability of errors can be reduced.
This sentence contains 20 words and 115 characters, including the commas, period, and spaces.It can be “coded” into the 8-bit binary ASCII character set recognized by computers as the “text” format,which translates the character string (that is readable by humans) into a binary string containing 920 ( ) bits.
Suppose that Shannon's sentence is transmitted, but that errors occur so that of the bits are flipped from one to zero (or from zero to one).Then about of the letters have errors:
It is clea2, however, that by sendhng the
information in a redundaNt form theprobabilipy of errors can be reduced.
The message is comprehensible, although it appears to have been typed poorly.With bit error, about of the letters have errors:
It is clear, howaver, thad by sending the
information in a redundan4 form pheprkbability of errors cAf be reduced.
Still the underlying meaning is decipherable. A dedicated reader can often decipher text with up to about bit error ( symbol error). Thus, the message has been conveyed,despite the presence of the errors. The reader, with an extensive familiarity with Englishwords, sentences, and syntax, is able to recognize the presence of the errors and to correct them.
As the bit error rate grows to , about one third of the letters have errors,and many words have become incomprehensible. Because “space” is represented as an ASCII character just like all the othersymbols, errors can transform spaces into letters or letters into spaces, thus blurring the true boundaries between the words.
Wt is ahear, h/wav3p, dhat by sending phc
)hformatIon if a rEdundaft fnre thdprkba@)hity ob erropc can be reduaed.
With bit error, about half of the letters have errors and the message is completely illegible:
I4 "s C`d`rq h+Ae&d"( `(At by s`jdafd th$
hfFoPmati/. )f a p(d5jdan` fLbe thd`r'`ab!DITy o& dr`kp1 aa& bE rd@u!ed.
These examples were all generated using the following M
atlab program
redundant.m which takes the text
textm , translates
it into a binary string, and then causes
per percent of the
bits to be flipped. The program then gathers statistics onthe resulting numbers of bit errors and symbol errors (how many letterswere changed).
b=[0.5 1 -0.6]; % define channelm=1000; s=sign(randn(1,m)); % binary source of length m
r=filter(b,1,s); % output of channeln=3; % length of equalizer - 1
delta=3; % use delay <=n
p=length(r)-delta;R=toeplitz(r(n+1:p),r(n+1:-1:1)); % build matrix R
S=s(n+1-delta:p-delta)'; % and vector Sf=inv(R'*R)*R'*S % calculate equalizer f
Jmin=S'*S-S'*R*inv(R'*R)*R'*S % Jmin for this f and deltay=filter(f,1,r); % equalizer is a filter
dec=sign(y); % quantize and count errorserr=0.5*sum(abs(dec(delta+1:end)...
-s(1:end-delta)))
LSequalizer.m find a LS equalizer f for the channel b
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