Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.586 49 — Repeat–Accumulate CodesG T =33H =33Figure 49.5. The generator matrixand parity-check matrix of asystematic low-densitygenerator-matrix code. The codehas rate 1/ 3.G T =33A, p =33Figure 49.6. The generator matrixand generalized parity-checkmatrix of a non-systematiclow-density generator-matrixcode. The code has rate 1/ 2.ExamplesRepetition code. The generator matrix, parity-check matrix, and generalizedparity-check matrix of a simple rate- 1/ 3 repetition code are shown in figure 49.4.Systematic low-density generator-matrix code. In an (N, K) systematic lowdensitygenerator-matrix code, there are no state variables. A transmittedcodeword t of length N is given byt = G T s, (49.6)where[IKG T =P], (49.7)with I K denoting the K ×K identity matrix, and P being a very sparse M ×Kmatrix, where M = N − K. The parity-check matrix of this code isH = [P|I M ]. (49.8)In the case of a rate- 1/ 3 code, this parity-check matrix might be representedas shown in figure 49.5.Non-systematic low-density generator-matrix code. In an (N, K) non-systematiclow-density generator-matrix code, a transmitted codeword t of length N isgiven byt = G T s, (49.9)where G T is a very sparse N × K matrix. The generalized parity-check matrixof this code isA = [ G T |I N], (49.10)and the corresponding generalized parity-check equation is[ ] sAx = 0, where x = . (49.11)tWhereas the parity-check matrix of this simple code is typically a complex,dense matrix, the generalized parity-check matrix retains the underlyingsimplicity of the code.In the case of a rate- 1/ 2 code, this generalized parity-check matrix mightbe represented as shown in figure 49.6.Low-density parity-check codes and linear MN codes. The parity-check matrixof a rate-1/3 low-density parity-check code is shown in figure 49.7a.(a)3 3(b)Figure 49.7. The generalizedparity-check matrices of (a) arate- 1/ 3 Gallager code with M/2columns of weight 2; (b) a rate- 1/ 2linear MN code.