Is the additive white Gaussian noise with mean and variance and In our example, as, we can correct up-to 1 error. So the error correction capability of code with distance is. If each code word is visualized as a sphere of radius, then the largest value of which does not result in overlap between the sphere is,Īny code word that lies with in the sphere is decoded into the valid code word at the center of the sphere. To determine the error correction capability, let us visualize that we can have valid code words from possible values. So, the number of errors which can be detected is. If an error of weight occurs, it is possible to transform one code word to another valid code word and the error cannot be detected. For the coded output sequence listed in the table above, we can see that the minimum separation between a pair of code words is 3. Hamming distance computes the number of differing positions when comparing two code words. Table: Coded output sequence for all possible input sequence Minimum distance The matrix of valid coded sequence of dimension Sl No The operator denotes exclusive-OR (XOR) operator. This type of code matrix where the raw message bits are send as is is called systematic code.Īssuming that the message sequence is, then the coded output sequence is :
Since an identity matrix, the first coded bits are identical to source message bits and the remaining bits form the parity check matrix. Using the example provided in chapter eight (example 8.1-1) of Digital Communications by John Proakis , let the coding matrix be, The coding operation can be denoted in matrix algebra as follows: With a Hamming code, we have 4 information bits and we need to add 3 parity bits to form the 7 coded bits. In this post, let us focus on the soft decision decoding for the Hamming (7,4) code, and quantify the bounds in the performance gain. An earlier post we discussed hard decision decoding for a Hamming (7,4) code and simulated the the bit error rate.
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