Decoding strategies Error detection decoding If then, set , otherwise, set . Error correction decoding- Minimum Likelihood Decoding (MLD) If then, set , otherwise, find the codeword that satisfies (5.1.6) , set . If more than one word satisfies inequality, what means that more than one codeword...
Joint and conditional entropy of correlated sources If two sources X and Y (Fig. 2.1.2) are considered: ; , , then, conditional entropy and joint entropy can be defined. For two sources X and Y , conditional probabilities (2.1.5): and joint probabilities (2.1.6) can be introduced: . Relation bet...
Properties of entropy P1. equality, if and only if the messages are all equally probable (for uniform distribution of probabilities), . P2. equality, if and only if both sources generate messages statistically independent, ; . P3. P4. equality, ...
Syndrome based decoding If , i.e. if v is a codeword, then If , i.e. if v is not a codeword, then As can be seen, product: or, in a matrix form contains information relating to an error added to a codeword and is called the error syndrome . Then, error detection and error correction decoding st...
B inary Symmetric Channel (BSC) where p is bit error probability . Transition probability diagram: 0 01 1 1p p 1p p ...
SELECTED ERROR CONTROL CODES Four most important error control codes are discussed, one error detecting code: single-parity-check code , and three error detecting and correcting codes: product code , repetition code , Hamming code . ...
Classification of codes Uniquely decodable codes Information codes Error control codesPrefix condition codes Block codes Convolutional codesVariable length c. Fixed length c. Linear codes Optimum (Huffman) Error detecting c. Error correcting c. Single parity check c. Simple binary Gray BCD Hexadeci...
Error correction limit : the maximum number of errors that a block code can correct. To correct every single error, , as an error pattern added to a code word is uniquely correctable if the resulting redundant word v is closer to than any other word (5.1.6). If , then single errors added to these ...
Arbitrary Hamming code ( n , k ) fulfills (6.4.3). So, the next Hamming codes are codes (15,11), (31,26), etc. All they have error syndromes, columns of the parity-check matrix, being binary words that encode a column number, position of nonzero bit in z . Parity-check bits are bits: . Problem 6.4...
Binary Markov source of the first order For such source: M =2, L =1, and the source N =2 states are: . The source is defined (modeled) by the original (0 th order) source probability assignment: and conditional probabilities: . The source can be expressed graphically the nodes correspond to sta...
