Evaluation Of An Error Control Codec Information Technology Essay

military rating Of An Error Control legislationc In cultivateation Technology EssayThe assignments object is to physical body and evaluate an wrongful conduct control encryptc. This aims to prove in practice the play cypher theory.In the first part there is a design of an en work outr and its simulation. From the en enrolr simulation we domiciliate figure how the code joints be generated and when a codeword is valid.The decipherer purpose is to rec everyplace the codeword from the legitimate word. To accomplish this, syndrome theory was utilise. A design and a simulation of the decipherer is shown in answer 2.Final, a codec is designed with an extension of XOR admittances to introduce errors. The main reason of this is to commiserate why overplay code can get hold 2 errors and reverse only one.Introduction to ham linear overgorge codesNoise ca utilizes errors ( data distortion) during transmission. So, a received center has arcsecond errors. A repercussion o f noise is the variation of one or more here and nows of a codeword. Alteration of a bit means inversion of its situation because signals have double star form. Some examples of abuzz communication demarcations are a) an analogue telephone line which, oer which cardinal modems communicate digital training and b) a disk drive. in that respect are ii solutions that can achieve perfect communication over an imperfect noisy communication occupation, physical and system solution. Physical modifications g wrangleing the cost of the communication stemma.Information theory and cryptanalytics theory digest an alternative approach we accept the minded(p) noisy channel as it is and add communication systems to it, so that we can detect and set the errors introduced by the channel. As shown in figure 1, we add an encoder before the channel and a decoder after it. The encoder encodes the source message s into a genetic message t, adding redundancy to the original message in some way. The channel adds noise to the transmitted message, yielding a received message r. The decoder uses the known redundancy introduced by the encoding system to infer twain the original signal and the added noise. jut 1 Error catch up withing codes for the double star biradial channel 1The only cost for system solution is a computational requirement at the encoder and decoder.Error detective work and error even outionIn order to make error correction manageable, the bit errors must be detected. When an error has been detected, the correction can be obtained by a) receiver asks for reiterate transmission of the incorrect codeword until a correct one has been received semiautomatic Repeat Request (ARQ) b) victimization the structure of the error correcting code to correct the error in the lead Error field of study (FEC). Forward Error Correction is been use for this assignment.Error DetectionAutomatic Repeat RequestForward Error CorrectionBlock CodeBlock CodeConvolutiona l Code propose 2 The main methods to introduce error correction codingLinear block codesLinear block codes are a pattern of mirror symmetry delay codes that can be characterized by the (n, k) nonation. The encoder transforms a block of k message digits into a longer block of n codeword digits constructed from a given alphabet of elements. When the alphabet consists of two elements (0 and 1), the code is a binary code comprising binary digits (bits).4 The on qualifying assignment of linear block codes is curtail to binary codes.The output of an information source is a sequence of binary digits 0 or 1 (since we discuss about binary codes). In block coding, this binary information sequence is segmented into message blocks of located length. Each block can represent any of 2k distinct messages. The channel encoder transforms each k-bit data block into a larger block of n bits, called code bits. The (n-k) bits, which the channel encoder adds to each data block, are called redunda nt or coincidence bits. Redundant or space-reflection symmetry bits carry no information. such(prenominal) a code is referred to as an (n, k) code. 5 The encoding result is the codeword.Any writer matrix of an (n, k) code can be reduced by row operations and column permutations to the systematic form. 6 We call a systematic code the code where the first k digits (information or message bits) of the code word are exactly the same as the message bits block and the go away n-k digits are the parity bits as it shown below.Message Information bitsRedundant or parity bitskn-kn digit codewordFigure 3 (n, k systematic block code)Encoding and Decoding of Linear Block CodesThe informant matrix is a matrix of basis vectors. The generator matrix G for an (n, k) block code can be used to generate the trance n-digit codeword from any given k-digit data sequence. The H and corresponding G matrices for the current assignment block code (6, 3) are shown below H is the parity block matrix G is the generator matrixThe first threesome columns are the data bits and the rest three columns are the parity bits. magisterial code manner of speaking are sometimes written so that the message bits occupy the leave hand helping of the codeword and the parity bits occupy the salutary hand portion. This reordering has no effect on the error detecting or error correction properties of the code.4 Study of G shows that on the left of the dotted partition there is a 33 unit slice matrix and on the right of the partition there is a parity check section. This part of G is the transpose of the left hand portion of H. As this code has a whizz error correcting capability thusly dmin, and the lading of the codeword must be 3. As the identity matrix has a genius one in each row then the parity check section must contain at least two ones. In addition to this constraint, rows cannot be identical. 7 The parity check bits are selected so they are independent of each other.The Hamming outdo between two code words is defined as the chassis of bits in which they differ. The weight of a binary codeword is defined as the number of ones which it contains (the number of the nonzero elements-bits).The codeword is given by the multiplication of data bits and the generator matrix. The operations of modulo-2 multiplication (AND) and modulo-2 addition (EXOR) are used for the binary field. EXOR addition AND multiplicationThe parity check equations are shown belowIf the components of the output transmission satisfy these equationsthen the received codeword is valid.These equations can be written in a matrix formwhere c is the codeword.The syndromeLet c be a code vector which was transmitted over a noisy channel. At the receiver we might obtain a corrupted vector r. Decoder must acquire c from r. The decoder computes,S=Hrwhere S is called the syndrome and r is the received vector (arranged as a column vector) then if,then r is not a code word. The syndrome is the result of a parity check performed on r to determine whether r is a valid fraction of the codeword set. If r is a member the syndrome S has a value 0. If r contains detectable errors, the syndrome has some nonzero value. The decoder will take actions to locate the errors and correct them in the case of FEC.No column of H can be all zeros, or else an error in the corresponding codeword position would not affect the syndrome and would be undetectable. in all columns of H must be unique. If two columns of H were identical, errors in these two corresponding codeword positions would be indistinguishable. 4Hamming code can correct a single bit error and detect two bit errors assuming no correction is attempted.Answers to assignment questions labor movement 1Design the encoder for a (6,3) Hamming single error correcting codec using the interleaved P1P2D1P3D2D3 format. You can implement your parity generation using XOR portals. Simulate your lap to check for correct operation.Answer 1An encoder is a device used to change a signal (such as a bitstream) or data into a code. The code may serve any of a number of purposes such as compressing information for transmission or storage, encrypting or adding redundancies to the input code, or translating from one code to another. This is usually done by means of a programmed algorithm, especially if any part is digital, while closely analog encoding is done with analog travelry. 3 Encoder creates the codeword in a conspiracy of information and parity bits.Interleaving is a way to arrange data in a non-contiguous way in order to increase performance and revoke burst errors. In our case we use interleaved to protect the data bits from nonstop error.Figure 4 Encoder design for a (6,3) Hamming single error correcting codecSince the encoder is for a (6,3) Hamming single error correcting codec, that means there are 3 information bits and 3 parity bits. Thus, 8 code words are generated from the encoder 2k where k are information b its 23=8.The H and G matrix are shown below for a (6, 3) Hamming code exclusively possible code wordsMessage x G =Codeword lean000 x G0000000001 x G0010113010 x G0101013100 x G1001103011 x G0111104101 x G1011014110 x G1100114111 x G1110003Table 1 All possible code wordsThe minimum distance is dmin=3Figure 5 Encoder simulationChecking if c=(D1D2D3P1P2P3) is a codewordThe EXOR gate () is a logic gate that gives an output of 1 when only one of its inputs is 1.X1 (Input)X2 (Input)Y1 (Output)000011101110Table 2 Truth table of EXOR admission c is a valid codeword.Task 2Design the decoder for a (6,3) Hamming single error correcting coded using the interleaved P1P2D1P3D2D3 format. You can use a 3-to-8 line decoder for syndrome decoding and XOR gates for the controlled inversion. Simulate your circuit to check for correct operation.Answer 2A decoder is a device which does the reverse of an encoder, undoing the encoding so that the original information can be retrieved. The same method used to encode is usually just reversed in order to decode. 3Decoder tries to reveal the correct data word from the codeword. Thus means that here is the process where spotting and correction of codeword take place.Figure 6 Decoder design for a (6.3) Hamming single error correcting codecDecode the received codewordFigure 7 Decoder simulationr is the received word 111000 r is code wordTask 3Join your encoder to decoder and add an XOR gate with an input in each bit transmission line to admit you to introduce errors into the transmission. Simulate your circuit and check that it can cope with the sextuplet single errors as expected.Answer 3Figure 8 Codec desingFigure 9 Six single errorsAs it shown from the above figure the codec can cope with the six single errors. This is possible becauseMessage x G =EncoderCodewordWeight000 x G0000000001 x G0010113010 x G0101013100 x G1001103011 x G0111104101 x G1011014110 x G1100114111 x G1110003Table 3 All possible code words and their ha mming weightThe minimum distance of a linear block code is defined as the smallest Hamming distance between any pair of code words in the code.5The minimum distance is dmin=3.The error correcting capability t of a code is defined as the maximum number of guaranteed correctable errors per codeword.where t is the error correcting capabilityFor dmin=3 we can see that all t=1 bit error patterns are correctable.In general, a t-error correcting (n, k) linear code is capable of correcting a total of 2n-k error patterns.4Task 4By experimenting with your implemented codec, examine the effect, in terms of additional errors, of (i) all 15 double errors, (ii) all 20 triple errors, (iii) all 15 quaternary errors, (iv) all 6 quintuple errors, (v) the single sextuple error. Note. You only involve consider one of the 8 possible input data words. wherefore?Answer 4(i)Figure 10 15 double errors(ii)Figure 11 20 triple errors(iii)Figure 12 15 tetrad errors(iv)Figure 13 6 quintuple errors(v)Figu re 14 The single sextuple errorSince the error correcting capability is t1, our codec cant detect or correct more than 1 error. Thus, the above results.Task 5Calculate the post codec probability of a code being in error, A(n), for each of the five categories examined in Task 4. Then calculate the general number of errors per 6 bit word, Eav, given by the following toughie based on the binomial distributionas function of the channel bit error probability p. Plot the decoded error probability as function of p. Over what range of p do you conclude that this codec is utilitarian and why?Answer 5A(n)=1-(number of correct errors/number of total errors)A(n) is going to be always 1 except the case where the codec detects and corrects 1 single error then A(n)=1Using matlab for the plotp=00.011Eav=15*p.2.*(1-p).4+20*p.3.*(1-p).3+15*p.4.*(1-p).2+6*p.5.*(1-p).1+p.6.*(1-p).0pd=Eav/6plot(p,pd)xlabel(Bit error probability (p))ylabel(Decoder error probability Pd(p))grid onFigure 15 Plot of dec oder error probability (pd) as function of pConclusionsParity bits must be added for the error espial and correction.Hamming distance is the criterion for error detection and correction.Error detection can be done with addition of one parity bit but error correction needs more parity bits (Hamming code).Hamming code can detect 2 bit errors assuming no correction is attempted.Hamming code can correct only a single bit error.The ability to correct single bit errors comes at a cost which is less than sending the entire message twice. displace a message twice is not accomplish an error correction.