PS06 - University of Illinois Spring 2011 ECE 361: Problem...

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Unformatted text preview: University of Illinois Spring 2011 ECE 361: Problem Set 6 Erasure Channels; Gaussian Channel Capacity Due: Thursday March 17 at 9:30 a.m. Reading: Lecture Notes: Lectures 12-14 1. [Single erasure correcting code] Consider a binary erasure channel (inputs 0 and 1; outputs 0, 1, ?) with erasure probability p = 0 . 2. Thus, we can expect one bit out of five to be erased. Find the generator matrix G of a linear (5 , 4) binary code C with the property that if one bit of the five transmitted is erased, the four data bits can be recovered successfully from the four bits received (without error). Hint: G needs to have the property that if any column of G is deleted, the remaining 4 4 matrix is invertible. But you dont need to write down G and then write out formal proofs of the invertibility of 5 different 4 4 submatrices of G ! Simply describe how the decoder will recover the data bits if the first bit of the codeword is erased, if the second bit of the codeword is erased, ..., the fifth bit of the codeword is erased. 2. [Decoding a Reed-Solomon on an erasure channel] Consider a (7 , 4) Reed-Solomon code being used over a 7-ary erasure channel with input alphabet { , 1 , 2 , 3 , 4 , 5 , 6 } constituting GF(7), the field of 7 elements, and output alphabet { , 1 , 2 , 3 , 4 , 5 , 6 , ? } . The 4 data symbols to be transmitted are elements of GF(7) and as denoted as d ,d 1 ,d 2 ,d 3 . Suggestion: Before starting on this problem, it would probably help to make a table of inverses in the field, that is, for each x , 1 x 6, find a number y , 1 y 6 such that xy...
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PS06 - University of Illinois Spring 2011 ECE 361: Problem...

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