problem2

# problem2 - UNIVERSITY OF CALIFORNIA SAN DIEGO Electrical...

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UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 259B - Winter Quarter 2011 Probabilistic Coding Problem Set #2 Due Thursday, January 27, 2011 1. This problem looks at encoders that generate the same convolutional code, and addresses the concept of catastrophic encoders and generator matrices. (a) The following rate 2/3 generator matrices (encoders) all generate the same code. Prove this explicitly for generator matrices G 3 ( D ) ,G 4 ( D ) , and G 5 (D) . G 1 ( D )= " 11 + D + D 2 1+ D D D + D 2 1 # G 2 ( D " 11+ D + D 2 D 01 + D 1 # G 3 ( D " 1 D 0 01+ D 1 # G 4 ( D " D 0 D D D + D 2 1 # G 5 ( D " 10 D 1+ D 1 1+ D # (b) Determine which of the encoders are catastrophic. (c) For the catastrophic encoders, give an example of an in±nite weight input sequence u ( D ) that produces a ±nite weight output sequence v ( D ).

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2. Consider a rate m/ ( m + 1) convolutional encoder with parity-check matrix of the form H ( D )=[ h m ( D ) ,...,h 1 ( D ) ,h 0 ( D )] where, for some 1 ˜ m<m (assuming memory ν 2) : h j ( D )=0 , ˜ m<j m, h j ( D )=0+ h ν - 1 j D ν - 1 + ...
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problem2 - UNIVERSITY OF CALIFORNIA SAN DIEGO Electrical...

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