6.27 Error-correcting codes: channel decoding

Because the idea of channel coding has merit (so long as the code is efficient), let's develop a systematic procedure forperforming channel decoding. One way of checking for errors is to try recreating the error correction bits from the dataportion of the received block $\widehat{c}$ . Using matrix notation, we make this calculation by multiplying the received block $\widehat{c}$ by the matrix $H$ known as the parity check matrix . It is formed from the generator matrix $G$ by taking the bottom, error-correction portion of $G$ and attaching to it an identity matrix. For our (7,4) code ,

When the received bits $\widehat{c}$ do not form a codeword, $H\widehat{c}$ does not equal zero, indicating the presence of one or more errors induced by the digital channel. Because the presenceof an error can be mathematically written as $\widehat{c}=c\mathop{\mathrm{xor}}e$ , with $e$ a vector of binary values having a 1 in those positions where a bit erroroccurred.

Consequently, $H\widehat{c}=Hc\mathop{\mathrm{xor}}e=He$ . Because the result of the product is a length- $(N-K)$ vector of binary values, we can have $2^{(N-K)}-1$ non-zero values that correspond to non-zero error patterns $e$ . To perform our channel decoding,

• compute (conceptually at least) $H\widehat{c}$ ;
• if this result is zero, no detectable or correctable error occurred;
• if non-zero, consult a table of length- $(N-K)$ binary vectors to associate them with the minimal error pattern that could have resulted in the non-zero result; then
• add the error vector thus obtained to the received vector $\widehat{c}$ to correct the error (because $c\mathop{\mathrm{xor}}e\mathop{\mathrm{xor}}e=c$ ).
• Select the data bits from the corrected word to produce the received bit sequence $\widehat{b}(n)$ .