<< Chapter < Page | Chapter >> Page > |
So-called linear codes create error-correction bits by combining the data bits linearly. The phrase "linearcombination" means here single-bit binary arithmetic.
$0\mathop{\mathrm{xor}}0=0$ | $1\mathop{\mathrm{xor}}1=0$ | $0\mathop{\mathrm{xor}}1=1$ | $1\mathop{\mathrm{xor}}0=1$ |
$\xb7(0, 0)=0$ | $\xb7(1, 1)=1$ | $\xb7(0, 1)=0$ | $\xb7(1, 0)=0$ |
The length- $K$ (in this simple example $K=1$ ) block of data bits is represented by the vector $b$ , and the length- $N$ output block of the channel coder, known as a codeword , by $c$ . The generator matrix $G$ defines all block-oriented linear channel coders.
As we consider other block codes, the simple idea of the decoder taking a majority vote of the received bitswon't generalize easily. We need a broader view that takes intoaccount the distance between codewords. A length- $N$ codeword means that the receiver must decide among the $2^{N}$ possible datawords to select which of the $2^{K}$ codewords was actually transmitted. As shown in [link] , we can think of the datawords geometrically. We define the Hamming distance between binary datawords ${c}_{1}$ and ${c}_{2}$ , denoted by $d({c}_{1}, {c}_{2})$ to be the minimum number of bits that must be "flipped" to gofrom one word to the other. For example, the distance between codewords is 3 bits. In our table of binary arithmetic, wesee that adding a 1 corresponds to flipping a bit. Furthermore, subtraction and addition are equivalent. We can express theHamming distance as
Show that adding the error vector col[1,0,...,0]to a codeword flips the codeword's leading bit and leaves the rest unaffected.
In binary arithmetic (see [link] ), adding 0 to a binary value results in that binary value while adding 1results in the opposite binary value.
The probability of one bit being flipped anywhere in a codeword is $N{p}_{e}(1-{p}_{e})^{(N-1)}$ . The number of errors the channel introduces equals the number of ones in $e$ ; the probability of any particular error vector decreases with the number of errors.
Notification Switch
Would you like to follow the 'Fundamentals of electrical engineering i' conversation and receive update notifications?