# coding4 - Coding for Communications Lecture 4 Linear block...

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1 Coding for Communications Lecture 4 Linear block codes • The information bit stream is chopped into blocks of k bits. • Each block is encoded to a larger block of n bits. • The coded bits are modulated and sent over the channel. • The reverse procedure is done at the receiver. Data block Channel encoder Codeword k bits n bits rate Code bits Redundant n k R n-k c = n n H c c c = ) , , , ( 2 1 K c k k H m m m = ) , , , ( 2 1 K m

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2 A linear binary block encoder generates the output bits as a linear binary combination of the input bits n k k n n r k k r r r k k r r k k k k g m g m c g m g m c g m g m c g m g m c g m g m c , 1 , 1 1 , 1 , 1 1 , 1 , 1 1 , 1 , 2 1 2 1 , 1 , 1 1 1 ... ... ... ... ... ... ... + + = + + = + + = + + = + + = + + ) 2 ( , GF g j i Linear block encoder ) , , ( 1 n c c K = c ) , , ( 1 k m m K = m It can be written in matrix form as n x k full rank matrix = n k k n g g g g , 1 , , 1 1 , 1 ... ... ... ... ... G mG c = n k k n n r k k r r r k k r r k k k k g m g m c g m g m c g m g m c g m g m c g m g m c , , 1 1 1 , 1 , 1 1 1 , , 1 1 2 , 2 , 1 1 2 1 , 1 , 1 1 1 ... ... ... ... ... ... ... + + = + + = + + = + + = + + = + + + We are using indices starting from 1
3 Also in this case Given the equation mG c = c is a linear combination or the rows of G . If G is nonsingular (full rank), the rows of G are a basis for the code C Hamming weight and minimum distance ± The Hamming weight of the vector U , denoted by w( U ), is the number of non-zero elements in U . ± The Hamming distance between two vectors U and V , is the number of elements in which they differ. ± The minimum distance of a block code is ) ( ) ( V U V U, = w d ) ( min ) , ( min min i i j i j i w d d U U U = =

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4 Error detection and correction capability ± Error detection capability is given by ± Error correcting-capability t of a code is defined as the maximum number of guaranteed correctable errors per codeword, that is = 2 1 min d t 1 min = d e Linear block codes –cont’d ± A matrix G is constructed by taking as its rows the vectors of the basis, . n V k V C Bases of C mapping } , , , { 2 1 k b b b K = = n k k k n n k b b b b b b b b b b b , 2 , 1 , , 2 22 1 , 2 1 12 1 , 1 1 L M O M L L M G
5 Linear block codes – cont’d ± Encoding in (n,k) block code ± This is a linear combination of the linearly independent rows of G. mG c = k n k n m m m c c c m m m c c c b b b b b b k 2 1 + + + = = 2 2 2 1 1 2 1 2 1 2 ) , , , ( ) , , , ( ) , , , ( K K M K K Linear block codes – cont’d ± Example: Block code (6,3) = = 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 3 2 1 b b b G 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 Message vector Codeword

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6 Linear systematic block code ± Systematic block code (n,k) ± For a systematic code, the first (or last) k elements in the codeword are information bits.
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coding4 - Coding for Communications Lecture 4 Linear block...

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