Y.S.Han Introduction to Binary Linear Block Codes 5 Probability distribution function for ri The signal energy per channel bit E has been normalized to 1. 0.5 0.45 Pr(ril1)Pr(ril0) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -4 -2 0 2 4 Ti School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 5 Probability distribution function for rj The signal energy per channel bit E has been normalized to 1. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -4 -2 0 2 4 rj P r(rj |1) P r(rj |0) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -4 -2 0 2 4 rj P r(rj |1) P r(rj |0) School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes 6 Binary Symmetric Channel 1.BSC is characterized by a probability p of bit error such that the probability p of a transmitted bit 0 being received as a 1 is the same as that of a transmitted 1 being received as a 0. 2.BSC may be treated as a simplified version of other symmetric channels.In the case of AWGN channel,we may assign p as ● p= Pr(rill)dri = Pr(ril0)dri 1 (rj+vE)2 ViNGe No drj Q((2E/No) School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 6 Binary Symmetric Channel 1. BSC is characterized by a probability p of bit error such that the probability p of a transmitted bit 0 being received as a 1 is the same as that of a transmitted 1 being received as a 0. 2. BSC may be treated as a simplified version of other symmetric channels. In the case of AWGN channel, we may assign p as p = ∫ ∞ 0 P r(rj |1)drj = ∫ 0 −∞ P r(rj |0)drj = ∫ ∞ 0 1 √ πN0 e − (rj + √ E) 2 N0 drj = Q ( (2E/N0) 1 2 ) School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes where 1-P 0 ·0 p 1. 。1 1-P Binary symmetric channel School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 7 where Q(x) = ∫ ∞ x 1 √ 2π e − y 2 2 dy ✟✟ ✟✟✟ ✟✯ ❍ ✲ q ❍❍❍❍❍❥ ✲ q q q 1 0 1 0 1 − p 1 − p p p Binary symmetric channel School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes 8 Binary Linear Block Code (BLBC) 1.An (n,k)binary linear block code is a k-dimensional subspace of the n-dimensional vector space Vn={(co,c1,...,cn-1)Ivcj cjGF(2)};n is called the length of the code,k the dimension. 2.Example:a (6,3)code C={000000,100110,010101,001011, 110011,101101,011110,111000} School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 8 Binary Linear Block Code (BLBC) 1. An (n, k) binary linear block code is a k-dimensional subspace of the n-dimensional vector space V n = {(c0, c1, . . . , cn−1)|∀cj cj ∈ GF(2)}; n is called the length of the code, k the dimension. 2. Example: a (6, 3) code C = {000000, 100110, 010101, 001011, 110011, 101101, 011110, 111000} School of Electrical Engineering & Intelligentization, Dongguan University of Technology
Y.S.Han Introduction to Binary Linear Block Codes 9 Generator Matrix 1.An (n,k)BLBC can be specified by any set of k linear independent codewords co,c1,...,ck-1.If we arrange the k codewords into a k x n matrix G,G is called a generator matrix for C. 2.Letu=(uou1,·,uk-1),where u∈GF(2). c=(co,c1,.,cn-1)=uG. 3.The generator matrix G'of a systematic code has the form of [IkA],where Ik is the k x k identity matrix. 4.G can be obtained by permuting the columns of G and by doing some row operations on G.We say that the code generated by G'is an equivalent code of the generated by G. School of Electrical Engineering Intelligentization,Dongguan University of Technology
Y. S. Han Introduction to Binary Linear Block Codes 9 Generator Matrix 1. An (n, k) BLBC can be specified by any set of k linear independent codewords c0,c1,…,ck−1. If we arrange the k codewords into a k × n matrix G, G is called a generator matrix for C. 2. Let u = (u0, u1, . . . , uk−1), where uj ∈ GF(2). c = (c0, c1, . . . , cn−1) = uG. 3. The generator matrix G′ of a systematic code has the form of [IkA], where Ik is the k × k identity matrix. 4. G′ can be obtained by permuting the columns of G and by doing some row operations on G. We say that the code generated by G′ is an equivalent code of the generated by G. School of Electrical Engineering & Intelligentization, Dongguan University of Technology