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A binary symmetric channel with p as transition probability, users repetition code ‘n’ times, with n = 2m + 1 being an odd integer. In a block of n bits, if the number of zeros exceed number of decoder decodes it as ‘0” otherwise as “ 1”. An error occurs when m + 1 or more transmissions out 3 are incorrect. What is the probability of error?

A binary symmetric channel with p as transition probability, users repetition code ‘n’ times, with n = 2m + 1 being an odd integer. In a block of n bits, if the number of zeros exceed number of decoder decodes it as ‘0” otherwise as “ 1”. An error occurs when m + 1 or more transmissions out 3 are incorrect. What is the probability of error? Correct Answer 3p<sup style="">2</sup> (1 –p) +p<sup style="">3</sup>

Given:

Probability of error when  0 is received = p

Probability of error when  1 is received = 1 - p

If the number of zeros exceeds the number, the decoder decodes it as ‘0” otherwise as “ 1”. 

For n = 3 bits

Bits received by the decoder

Probability of Error

0 0 0

p3

0 0 1

p2(1 - p)

0 1 0

p2(1 - p)

0 1 1

 x 

1 0 0

p2(1 - p)

1 0 1

 x

1 1 0

  x 

1 1 1

 x 

Hence the probability of error is,

Pe = p3 + p2(1 - p) + p2(1 - p) + p2(1 - p) 

Pe = p3 + 3p2(1 - p)

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