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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 8 Error Control Coding"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.1 page 384"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "generator matrix\n",
+ "[[ 1. 1. 1. 1. 1.]]\n",
+ "\n",
+ "parity-check matrix\n",
+ "[[ 1. 0. 0. 0. 1.]\n",
+ " [ 0. 1. 0. 0. 1.]\n",
+ " [ 0. 0. 1. 0. 1.]\n",
+ " [ 0. 0. 0. 1. 1.]]\n",
+ "\n",
+ "code word for binary one input\n",
+ "[[ 1. 1. 1. 1. 1.]]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from __future__ import division\n",
+ "from numpy import ones,zeros,identity,transpose,hstack,mat\n",
+ "\n",
+ "n =5# #block of identical 'n' bits\n",
+ "k =1# #one bit\n",
+ "m = 1## bit value = 1\n",
+ "I = identity(n-k) #Identity matrix\n",
+ "P = ones(n-k)##coefficient matrix\n",
+ "I=mat(I)\n",
+ "P=mat(P)\n",
+ "H = hstack([I,transpose(P)])##parity-check matrix\n",
+ "G = hstack([P, mat([1])])##generator matrix \n",
+ "x = m*G# #code word\n",
+ "print 'generator matrix\\n',G\n",
+ "print '\\nparity-check matrix\\n',H\n",
+ "print '\\ncode word for binary one input\\n',x"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.2 page 386"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "identity matrix Ik\n",
+ "[[ 1. 0. 0. 0.]\n",
+ " [ 0. 1. 0. 0.]\n",
+ " [ 0. 0. 1. 0.]\n",
+ " [ 0. 0. 0. 1.]]\n",
+ "\n",
+ "coefficient matrix P\n",
+ "[[1 1 0]\n",
+ " [0 1 1]\n",
+ " [1 1 1]\n",
+ " [1 0 1]]\n",
+ "generator matrix G\n",
+ "[[ 1. 1. 0. 1. 0. 0. 0.]\n",
+ " [ 0. 1. 1. 0. 1. 0. 0.]\n",
+ " [ 1. 1. 1. 0. 0. 1. 0.]\n",
+ " [ 1. 0. 1. 0. 0. 0. 1.]]\n",
+ "parity chechk matrix H\n",
+ "[[ 1. 0. 0. 1. 0. 1. 1.]\n",
+ " [ 0. 1. 0. 1. 1. 1. 0.]\n",
+ " [ 0. 0. 1. 0. 1. 1. 1.]]\n",
+ "Code words of (7,4) Hamming code\n",
+ "[[ 0. 0. 0. 0. 0. 0. 0.]\n",
+ " [ 1. 0. 1. 0. 0. 0. 1.]\n",
+ " [ 1. 1. 1. 0. 0. 1. 0.]\n",
+ " [ 0. 1. 0. 0. 0. 1. 1.]\n",
+ " [ 0. 1. 1. 0. 1. 0. 0.]\n",
+ " [ 1. 1. 0. 0. 1. 0. 1.]\n",
+ " [ 1. 0. 0. 0. 1. 1. 0.]\n",
+ " [ 0. 0. 1. 0. 1. 1. 1.]\n",
+ " [ 1. 1. 0. 1. 0. 0. 0.]\n",
+ " [ 0. 1. 1. 1. 0. 0. 1.]\n",
+ " [ 0. 0. 1. 1. 0. 1. 0.]\n",
+ " [ 1. 0. 0. 1. 0. 1. 1.]\n",
+ " [ 1. 0. 1. 1. 1. 0. 0.]\n",
+ " [ 0. 0. 0. 1. 1. 0. 1.]\n",
+ " [ 0. 1. 0. 1. 1. 1. 0.]\n",
+ " [ 1. 1. 1. 1. 1. 1. 1.]]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from __future__ import division\n",
+ "from numpy import ones,zeros,identity,multiply,mat,concatenate,hstack,transpose\n",
+ "\n",
+ "\n",
+ "k = 4# #message bits length\n",
+ "n = 7# #block length\n",
+ "m = n-k##Number of parity bits\n",
+ "I = identity(k) #identity matrix\n",
+ "I=mat(I)\n",
+ "print 'identity matrix Ik\\n',I\n",
+ "P =[[1,1,0],[0,1,1],[1,1,1],[1,0,1]]##coefficient matrix\n",
+ "P=mat(P)\n",
+ "print '\\ncoefficient matrix P\\n',P\n",
+ "G = hstack([P,I]) #generator matrix\n",
+ "print 'generator matrix G\\n',G\n",
+ "\n",
+ "H = hstack([identity(k-1),transpose(P)])##parity check matrix\n",
+ "print 'parity chechk matrix H\\n',H\n",
+ "\n",
+ "#message bits\n",
+ "m = [[0,0,0,0],[0,0,0,1],[0,0,1,0],[0,0,1,1],[0,1,0,0],[0,1,0,1],[0,1,1,0],[0,1,1,1],[1,0,0,0],[1,0,0,1],[1,0,1,0],[1,0,1,1],[1,1,0,0],[1,1,0,1],[1,1,1,0],[1,1,1,1]]\n",
+ "\n",
+ "C = m*G#\n",
+ "C = (C%2)#\n",
+ "print 'Code words of (7,4) Hamming code\\n',C\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.3 page 389"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "remainder in polynomial form: \n",
+ " 2\n",
+ "1 x + 1 x\n",
+ "Parity bits are: [ 1. 1. 0.]\n",
+ "G:\n",
+ "[1, 1, 0, 1, 0, 0, 0]\n",
+ "[0, 1, 1, 0, 1, 0, 0]\n",
+ "[0, 0, 1, 1, 0, 1, 0]\n",
+ "[0, 0, 0, 1, 1, 0, 1]\n",
+ "\n",
+ "Generator Matrix G =\n",
+ "[1, 1, 0, 1, 0, 0, 0]\n",
+ "[0, 1, 1, 0, 1, 0, 0]\n",
+ "[1, 1, 1, 0, 0, 1, 0]\n",
+ "[1, 0, 1, 0, 0, 0, 1]\n",
+ "\n",
+ "Partiy Check matrix H =\n",
+ "\n",
+ "[1, 0, 0, 1, 0, 1, 1]\n",
+ "[0, 1, 0, 1, 1, 1, 0]\n",
+ "[0, 0, 1, 0, 1, 1, 1]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import poly1d, polydiv\n",
+ "#message sequence = [1,0,0,1]\n",
+ "g = poly1d([1,0,1,1]) #generator polynomial\n",
+ "m = poly1d([1,0,0,0])*poly1d([1,0,0,1]) #message sequence\n",
+ "q = polydiv(m,g)[0]\n",
+ "r = polydiv(m,g)[1]\n",
+ "p = r.coeffs\n",
+ "print 'remainder in polynomial form: \\n',r\n",
+ "print 'Parity bits are:',p\n",
+ "\n",
+ "def rev_coeffs(x):\n",
+ " X=[]\n",
+ " for i in reversed(x):\n",
+ " X.append(i)\n",
+ " return X\n",
+ "\n",
+ "\n",
+ "G = [rev_coeffs(g.coeffs),rev_coeffs((g*poly1d([1,0])).coeffs),rev_coeffs((g*poly1d([1,0,0])).coeffs),rev_coeffs((g*poly1d([1,0,0,0])).coeffs)]\n",
+ "M=len(G[-1])\n",
+ "for gg in G:\n",
+ " while len(gg)<M:\n",
+ " gg.append(0)\n",
+ "print \"G:\" \n",
+ "for gg in G:\n",
+ " print gg\n",
+ "\n",
+ "def fun1(a,x,y):\n",
+ " import numpy as np\n",
+ " z=[]\n",
+ " for xx,yy in np.nditer([a[x-1],a[y-1]]):\n",
+ " z.append(xx+yy)\n",
+ " a[x-1]=z\n",
+ " return a \n",
+ "\n",
+ "def modulo(a,i):\n",
+ " bb=[]\n",
+ " for aa in a[i-1]:\n",
+ " bb.append(aa%2)\n",
+ " a[i-1]=bb \n",
+ " return a\n",
+ "\n",
+ "G=fun1(G,3,1)#G(3,:) = G(3,:)+G(1,:);\n",
+ "G=modulo(G,3)#G(3,:) = modulo(G(3,:),2);\n",
+ "G=fun1(G,4,1)\n",
+ "G=fun1(G,4,2)#G(4,:) = G(1,:)+G(2,:)+G(4,:);\n",
+ "G=modulo(G,4)#G(4,:) = modulo(G(4,:),2);\n",
+ "print '\\nGenerator Matrix G ='\n",
+ "for ggg in G:\n",
+ " print ggg\n",
+ "\n",
+ "\n",
+ "#h = 1+D^-1+D^-2+D^-4;\n",
+ "#H_D = [D^4*h;D^5*h;D^6*h];\n",
+ "H_D=[poly1d([1,1,1,0,1]),poly1d([1,1,1,0,1,0]),poly1d([1,1,1,0,1,0,0])] \n",
+ "\n",
+ "\n",
+ "#H_num =numer(H_D);\n",
+ "#H = coeff(H_num);\n",
+ "H=[rev_coeffs(aa.coeffs) for aa in H_D]\n",
+ "\n",
+ "M=len(H[-1])\n",
+ "for hh in H:\n",
+ " while len(hh)<M:\n",
+ " hh.append(0)\n",
+ "H=fun1(H,1,3)\n",
+ "H= modulo(H,1) \n",
+ "print '\\nPartiy Check matrix H =\\n'\n",
+ "for hh in H:\n",
+ " print hh"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.4 page 395"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "remainder in polynomial form: \n",
+ " 2\n",
+ "1 x + 1 x\n",
+ "Parity bits are: [ 1. 1. 0.]\n",
+ "Table 8.3 Contents of the Shift Register in the Encoder of fig8.7 for Message Sequence(1001)\n",
+ "__________________________________________________________________________________________\n",
+ "Shift Input Register Contents\n",
+ "__________________________________________________________________________________________\n",
+ "1 1 1 1 0\n",
+ "2 0 0 1 1\n",
+ "3 0 1 1 1\n",
+ "4 1 0 1 1\n",
+ "____________________________________________________________________________________________\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import poly1d,polydiv\n",
+ "#message sequence = [1,0,0,1]\n",
+ "g = poly1d([1,0,1,1]) #generator polynomial\n",
+ "m = poly1d([1,0,0,0])*poly1d([1,0,0,1])# #message sequence\n",
+ "q= polydiv(m,g)[0]\n",
+ "r= polydiv(m,g)[1]\n",
+ "p = r.coeffs\n",
+ "print 'remainder in polynomial form: \\n',r\n",
+ "print 'Parity bits are:',p\n",
+ "print 'Table 8.3 Contents of the Shift Register in the Encoder of fig8.7 for Message Sequence(1001)'\n",
+ "print '__________________________________________________________________________________________'\n",
+ "print 'Shift Input Register Contents'\n",
+ "print '__________________________________________________________________________________________'\n",
+ "print '1 1 1 1 0'\n",
+ "print '2 0 0 1 1'\n",
+ "print '3 0 1 1 1'\n",
+ "print '4 1 0 1 1'\n",
+ "print '____________________________________________________________________________________________'"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.5 page 396"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "remainder in polynomial form : \n",
+ " 2\n",
+ "2 x + 2 x\n",
+ "Syndrome bits for error free codeword are: [0.0, 0.0, 0.0]\n",
+ "remainder in polynomial form for errored codeword : \n",
+ " 2\n",
+ "2 x + 3 x + 1\n",
+ "Syndrome bits for errored codeword are: [0.0, 1.0, 1.0]\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import poly1d,polydiv\n",
+ "\n",
+ "#message sequence = [0,1,1,1,0,0,1]\n",
+ "\n",
+ "g = poly1d([1,0,1,1]) # #generator polynomial\n",
+ "C1 = poly1d([1,0,0,1,1,1,0]) #error free codeword\n",
+ "C2 = poly1d([1,0,0,0,1,1,0]) #middle bit is error\n",
+ "#[r1,q1] = pdiv(C1,g)#\n",
+ "\n",
+ "q1 = polydiv(C1,g)[0]\n",
+ "r1 = polydiv(C1,g)[1]\n",
+ "\n",
+ "S1 = (r1).coeffs\n",
+ "S1 = [xx%2 for xx in S1]\n",
+ "print 'remainder in polynomial form : \\n',r1\n",
+ "print 'Syndrome bits for error free codeword are:',S1\n",
+ "q2 = polydiv(C2,g)[0]\n",
+ "r2 = polydiv(C2,g)[1]\n",
+ "S2 = (r2).coeffs\n",
+ "S2 = [xx%2 for xx in S2]\n",
+ "print 'remainder in polynomial form for errored codeword : \\n',r2\n",
+ "print 'Syndrome bits for errored codeword are:',S2"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.6 page 399"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "n = 3\n",
+ "n-k = 2\n",
+ "Code rate:r = k/n = 0.333\n",
+ "It can correct any error upto = 2\n"
+ ]
+ }
+ ],
+ "source": [
+ "from __future__ import division\n",
+ "#Single-error-correcting RS code with a 2-bit byte\n",
+ "\n",
+ "m =2# #m-bit symbol\n",
+ "k = 1**2# #number of message bits\n",
+ "t =1# #single bit error correction\n",
+ "n = 2**m-1# #code word length in 2-bit byte\n",
+ "p = n-k# #parity bits length in 2-bit byte\n",
+ "r = k/n# #code rate\n",
+ "print 'n =',n\n",
+ "print 'n-k =',p\n",
+ "print 'Code rate:r = k/n = %.3f'%r\n",
+ "print 'It can correct any error upto =',(2*t)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.7 page 401"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Result:\n",
+ "[1.0, 1.0] \n",
+ "\n",
+ "[1.0, 0.0] \n",
+ "\n",
+ "[1.0, 1.0] \n",
+ "\n",
+ "[1.0, 1.0] \n",
+ "\n",
+ "[0.0, 1.0] \n",
+ "\n",
+ "[0.0, 1.0] \n",
+ "\n",
+ "[1.0, 1.0] \n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import convolve,ones\n",
+ "g1 = [1,1,1] # The input Top Adder Sequence\n",
+ "g2 = [1,0,1] #The input Bottom Adder Sequence\n",
+ "m =[1,1,0,0,1] # The message sequence\n",
+ "x1 = [round(xx) for xx in convolve(g1,m)]\n",
+ "x2 = [round(xx) for xx in convolve(g2,m)]\n",
+ "x1 = [xx%2 for xx in x1]\n",
+ "x2 = [xx%2 for xx in x2]\n",
+ "N = len(x1)\n",
+ "x=[]\n",
+ "for i in range(0,len(x1)):\n",
+ " x.append([x1[N-i-1],x2[N-i-1]])\n",
+ "print 'Result:' \n",
+ "for xx in x:\n",
+ " print xx,'\\n'"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.8 page 404"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "top output sequence\n",
+ "1 \t1 \t1 \t1 \t0 \t0 \t1 \t\n",
+ "bottom output sequence\n",
+ "1 \t0 \t1 \t1 \t1 \t1 \t1 \t"
+ ]
+ }
+ ],
+ "source": [
+ "from numpy import poly1d\n",
+ "g1D=poly1d([1,1,1]) #generator polynomial 1\n",
+ "g2D=poly1d([1,0,1]) #generator polynomial 2\n",
+ "mD=poly1d([1,1,0,0,1]) #message sequence polynomial representation\n",
+ "x1D=(g1D*mD) #top output polynomial\n",
+ "x2D=(g2D*mD) #bottom output polynomial\n",
+ "x1=x1D.coeffs\n",
+ "x2=x2D.coeffs\n",
+ "x1=x1.tolist()\n",
+ "X1=[]\n",
+ "for i in reversed(x1):\n",
+ " X1.append(i)\n",
+ "X2=[]\n",
+ "for i in reversed(x2):\n",
+ " X2.append(i)\n",
+ "print 'top output sequence'\n",
+ "for xx in X1:\n",
+ " print xx%2,'\\t',\n",
+ "print '\\nbottom output sequence' \n",
+ "for xx in X2:\n",
+ " print xx%2,'\\t',"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example8.11 page 409"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "branch metric for correct reception : 0.8822\n",
+ "branch metric for any one correct recption: -3.7027\n",
+ "branch metric for no correct reception : -8.2877\n"
+ ]
+ }
+ ],
+ "source": [
+ "from __future__ import division\n",
+ "from math import log\n",
+ "\n",
+ "r = 1/2# #code rate\n",
+ "n =2# #number of bits\n",
+ "pe = 0.04# #transition probility \n",
+ "p = 1-pe## probability of correct reception\n",
+ "gama_1 = 2*log(p,2)+2*(1-r)# #branch metric for correct reception\n",
+ "gama_2 = log(pe*p,2)+1# #branch metric for any one correct recption\n",
+ "gama_3 = 2*log(pe,2)+1# #branch metric for no correct reception\n",
+ "print 'branch metric for correct reception : %.4f'%gama_1\n",
+ "print 'branch metric for any one correct recption: %.4f'%gama_2\n",
+ "print 'branch metric for no correct reception : %.4f'%gama_3"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 2",
+ "language": "python",
+ "name": "python2"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 2
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython2",
+ "version": "2.7.9"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
generated by cgit (git 2.34.1) at 2024-12-20 04:45:57 +0000