channel coding

 Coding for Reliable Communication
Purpose :
Channel Coding
To overcome the problem of noise in the system, we might consider adding redundancy during the encoding process to overcome possible errors. Now describe a specific code and evaluate their performance for additive white Gaussian noise channel, linear block code and convolutional code.
To improve the error performance in symbol transmission over noise channel as shown in figure 1.
? In binary PCM ,each sample of the signal is represented by code word ,say, k bit .
? In the receiver recognized each code word in order to reconstruct the sample.
? Error may occurred in transmit ion as a result of noise
? Improvement the reliability of communication in the presence of noise is to increase the signal to noise ratio. An alternative is to add extra bit , at the expense of an increase in band width , to detect and correct the errors



Coding schemes
•    Forward error correction (FEC) where controlled redundant information is added to the symbol stream. The redundant information can be utilized for error detection and error correction . No return link required.
•    Automatic repeat-re quest(ARQ) where the symbol frame is retransmitted if symbol errors are detected in the reception. Only error detection necessary . A return link is required . introduce latency to the transmission.
•    Hybrid ARQ combines ARQ and FEC. Return link  necessary lover latency than pure ARQ.  





Coding Method
i.    Block coding where information symbols are segmented into frames to which redundant symbols are added to form code words, such Hamming code.
ii.    Convoluational coding where the redundancy is generated by feeding the original symbol stream through a shift register and the outputs corresponding to the used coding polynomial are multiplexed in time.
iii.    Concatened code with outer and inner coding . e.g. first block coding and the block coded symbol stream is then convolutionally coded.
iv.    Code interleaving where the subsequent coded symbols are spread (interleaving) over several frame . deinterleaved in the receiver will than spread error burst in the transmitted stream and make them appear more randomly before channel decoding .thus preserving the code gain which otherwise suffer from error burst.




Linear Block Coed  
A block code consist of a set of fixed length vectors called code word. The length of the code word is the number of elements in the vector and is denoted by n . the element of a code word are selected from an alphabet of q elements . when the alphabet consist of two elements 0,and 1 the code is binary code and the elements of a code word are selected from an alphabet having q elements (q>2), the code is nonbinary . its interesting to note that when q is a power of 2, i.e. q=2b where b is a positive integer each q ary elements has an equivalent binary representation consisting of  b  bits and thus a nonbinary code of a block length  N can be mapped into a binary code of a block length n =bN.




Hamming Codes
There are both binary and nonbinary  hamming codes discusses here binary Hamming codes . these comprise a class of codes with the property that