3F4 Error Control Coding Dr. I. J. Wassell

Introduction Error Control Coding (ECC) Extra bits are added to the data at the transmitter (redundancy) to permit error detection or correction at the receiver Done to prevent the output of erroneous bits despite noise and other imperfections in the channel The positions of the error control coding and decoding are shown in the transmission model

Transmission Model X(w) Hc(w) Transmitter N(w) + Y(w) Receiver Digital Source Encoder Error Control Coding Line Modulator (Transmit Filter, etc) Channel Noise Sink Decoder Decoding Demod (Receive + Transmitter Receiver X(w) Hc(w) N(w) Y(w)

Error Models Binary Symmetric Memoryless Channel Assumes transmitted symbols are binary Errors affect ‘0’s and ‘1’s with equal probability (i.e., symmetric) Errors occur randomly and are independent from bit to bit (memoryless) 1-p p is the probability of bit error or the Bit Error Rate (BER) of the channel p IN OUT p 1 1 1-p

Error Models Many other types Burst errors, i.e., contiguous bursts of bit errors output from DFE (error propagation) common in radio channels Insertion, deletion and transposition errors We will consider mainly random errors

Error Control Techniques Error detection in a block of data Can then request a retransmission, known as automatic repeat request (ARQ) for sensitive data Appropriate for Low delay channels Channels with a return path Not appropriate for delay sensitive data, e.g., real time speech and data

Error Control Techniques Forward Error Correction (FEC) Coding designed so that errors can be corrected at the receiver Appropriate for delay sensitive and one-way transmission (e.g., broadcast TV) of data Two main types, namely block codes and convolutional codes. We will only look at block codes

Block Codes We will consider only binary data Data is grouped into blocks of length k bits (dataword) Each dataword is coded into blocks of length n bits (codeword), where in general n>k This is known as an (n,k) block code

Block Codes A vector notation is used for the datawords and codewords, Dataword d = (d1 d2….dk) Codeword c = (c1 c2……..cn) The redundancy introduced by the code is quantified by the code rate, Code rate = k/n i.e., the higher the redundancy, the lower the code rate

Block Code - Example Dataword length k = 4 Codeword length n = 7 This is a (7,4) block code with code rate = 4/7 For example, d = (1101), c = (1101001)

Codeword + possible errors (n bits) Error Control Process Source code data chopped into blocks Codeword (n bits) 101101 1000 Channel coder 1000 Dataword (k bits) Codeword + possible errors (n bits) Dataword (k bits) Channel decoder Channel Error flags

Error Control Process Decoder gives corrected data May also give error flags to Indicate reliability of decoded data Helps with schemes employing multiple layers of error correction

Parity Codes Example of a simple block code – Single Parity Check Code In this case, n = k+1, i.e., the codeword is the dataword with one additional bit For ‘even’ parity the additional bit is, For ‘odd’ parity the additional bit is 1-q That is, the additional bit ensures that there are an ‘even’ or ‘odd’ number of ‘1’s in the codeword

Parity Codes – Example 1 Even parity (i) d=(10110) so, c=(101101)

Parity Codes – Example 2 Coding table for (4,3) even parity code Dataword Codeword

Parity Codes To decode Code can detect single errors Calculate sum of received bits in block (mod 2) If sum is 0 (1) for even (odd) parity then the dataword is the first k bits of the received codeword Otherwise error Code can detect single errors But cannot correct error since the error could be in any bit For example, if the received dataword is (100000) the transmitted dataword could have been (000000) or (110000) with the error being in the first or second place respectively Note error could also lie in other positions including the parity bit

Parity Codes Known as a single error detecting code (SED). Only useful if probability of getting 2 errors is small since parity will become correct again Used in serial communications Low overhead but not very powerful Decoder can be implemented efficiently using a tree of XOR gates

Hamming Distance Error control capability is determined by the Hamming distance The Hamming distance between two codewords is equal to the number of differences between them, e.g., 10011011 11010010 have a Hamming distance = 3 Alternatively, can compute by adding codewords (mod 2) =01001001 (now count up the ones)

Hamming Distance The Hamming distance of a code is equal to the minimum Hamming distance between two codewords If Hamming distance is: 1 – no error control capability; i.e., a single error in a received codeword yields another valid codeword XXXXXXX X is a valid codeword Note that this representation is diagrammatic only. In reality each codeword is surrounded by n codewords. That is, one for every bit that could be changed

Hamming Distance If Hamming distance is: 2 – can detect single errors (SED); i.e., a single error will yield an invalid codeword XOXOXO X is a valid codeword O in not a valid codeword See that 2 errors will yield a valid (but incorrect) codeword

Hamming Distance If Hamming distance is: 3 – can correct single errors (SEC) or can detect double errors (DED) XOOXOOX X is a valid codeword O in not a valid codeword See that 3 errors will yield a valid but incorrect codeword

Hamming Distance - Example Hamming distance 3 code, i.e., SEC/DED Or can perform single error correction (SEC) 10011011 X 11011011 O 11010011 O 11010010 X This code corrected this way X is a valid codeword O is an invalid codeword

Hamming Distance The maximum number of detectable errors is That is the maximum number of correctable errors is given by, where dmin is the minimum Hamming distance between 2 codewords and means the smallest integer

Linear Block Codes As seen from the second Parity Code example, it is possible to use a table to hold all the codewords for a code and to look-up the appropriate codeword based on the supplied dataword Alternatively, it is possible to create codewords by addition of other codewords. This has the advantage that there is now no longer the need to held every possible codeword in the table.

Linear Block Codes If there are k data bits, all that is required is to hold k linearly independent codewords, i.e., a set of k codewords none of which can be produced by linear combinations of 2 or more codewords in the set. The easiest way to find k linearly independent codewords is to choose those which have ‘1’ in just one of the first k positions and ‘0’ in the other k-1 of the first k positions.

Linear Block Codes For example for a (7,4) code, only four codewords are required, e.g., So, to obtain the codeword for dataword 1011, the first, third and fourth codewords in the list are added together, giving 1011010 This process will now be described in more detail

Linear Block Codes An (n,k) block code has code vectors d=(d1 d2….dk) and c=(c1 c2……..cn) The block coding process can be written as c=dG where G is the Generator Matrix

Linear Block Codes Thus, ai must be linearly independent, i.e., Since codewords are given by summations of the ai vectors, then to avoid 2 datawords having the same codeword the ai vectors must be linearly independent

Linear Block Codes Sum (mod 2) of any 2 codewords is also a codeword, i.e., Since for datawords d1 and d2 we have; So,

Linear Block Codes 0 is always a codeword, i.e., Since all zeros is a dataword then,

Error Correcting Power of LBC The Hamming distance of a linear block code (LBC) is simply the minimum Hamming weight (number of 1’s or equivalently the distance from the all 0 codeword) of the non-zero codewords Note d(c1,c2) = w(c1+ c2) as shown previously For an LBC, c1+ c2=c3 So min (d(c1,c2)) = min (w(c1+ c2)) = min (w(c3)) Therefore to find min Hamming distance just need to search among the 2k codewords to find the min Hamming weight – far simpler than doing a pair wise check for all possible codewords.

Linear Block Codes – example 1 For example a (4,2) code, suppose; a1 = [1011] a2 = [0101] For d = [1 1], then;

Linear Block Codes – example 2 A (6,5) code with Is an even single parity code

Systematic Codes For a systematic block code the dataword appears unaltered in the codeword – usually at the start The generator matrix has the structure, k R R = n - k P is often referred to as parity bits

Systematic Codes I is k*k identity matrix. Ensures dataword appears as beginning of codeword P is k*R matrix.

Decoding Linear Codes One possibility is a ROM look-up table In this case received codeword is used as an address Example – Even single parity check code; Address Data 000000 0 000001 1 000010 1 000011 0 ……… . Data output is the error flag, i.e., 0 – codeword ok, If no error, dataword is first k bits of codeword For an error correcting code the ROM can also store datawords

Decoding Linear Codes Another possibility is algebraic decoding, i.e., the error flag is computed from the received codeword (as in the case of simple parity codes) How can this method be extended to more complex error detection and correction codes?

Parity Check Matrix A linear block code is a linear subspace Ssub of all length n vectors (Space S) Consider the subset Snull of all length n vectors in space S that are orthogonal to all length n vectors in Ssub It can be shown that the dimensionality of Snull is n-k, where n is the dimensionality of S and k is the dimensionality of Ssub It can also be shown that Snull is a valid subspace of S and consequently Ssub is also the null space of Snull

Parity Check Matrix Snull can be represented by its basis vectors. In this case the generator basis vectors (or ‘generator matrix’ H) denote the generator matrix for Snull - of dimension n-k = R This matrix is called the parity check matrix of the code defined by G, where G is obviously the generator matrix for Ssub- of dimension k Note that the number of vectors in the basis defines the dimension of the subspace

Parity Check Matrix So the dimension of H is n-k (= R) and all vectors in the null space are orthogonal to all the vectors of the code Since the rows of H, namely the vectors bi are members of the null space they are orthogonal to any code vector So a vector y is a codeword only if yHT=0 Note that a linear block code can be specified by either G or H

Parity Check Matrix So H is used to check if a codeword is valid, R = n - k The rows of H, namely, bi, are chosen to be orthogonal to rows of G, namely ai Consequently the dot product of any valid codeword with any bi is zero

Parity Check Matrix This is so since, and so, This means that a codeword is valid (but not necessarily correct) only if cHT = 0. To ensure this it is required that the rows of H are independent and are orthogonal to the rows of G That is the bi span the remaining R (= n - k) dimensions of the codespace

Parity Check Matrix For example consider a (3,2) code. In this case G has 2 rows, a1 and a2 Consequently all valid codewords sit in the subspace (in this case a plane) spanned by a1 and a2 In this example the H matrix has only one row, namely b1. This vector is orthogonal to the plane containing the rows of the G matrix, i.e., a1 and a2 Any received codeword which is not in the plane containing a1 and a2 (i.e., an invalid codeword) will thus have a component in the direction of b1 yielding a non- zero dot product between itself and b1

Parity Check Matrix Similarly, any received codeword which is in the plane containing a1 and a2 (i.e., a valid codeword) will not have a component in the direction of b1 yielding a zero dot product between itself and b1 c1 c2 c3 a1 a2 b1

Error Syndrome For error correcting codes we need a method to compute the required correction To do this we use the Error Syndrome, s of a received codeword, cr s = crHT If cr is corrupted by the addition of an error vector, e, then cr = c + e and s = (c + e) HT = cHT + eHT s = 0 + eHT Syndrome depends only on the error

Error Syndrome That is, we can add the same error pattern to different codewords and get the same syndrome. There are 2(n - k) syndromes but 2n error patterns For example for a (3,2) code there are 2 syndromes and 8 error patterns Clearly no error correction possible in this case Another example. A (7,4) code has 8 syndromes and 128 error patterns. With 8 syndromes we can provide a different value to indicate single errors in any of the 7 bit positions as well as the zero value to indicate no errors Now need to determine which error pattern caused the syndrome

Error Syndrome For systematic linear block codes, H is constructed as follows, G = [ I | P] and so H = [-PT | I] where I is the k*k identity for G and the R*R identity for H Example, (7,4) code, dmin= 3

Error Syndrome - Example For a correct received codeword cr = [1101001] In this case,

Error Syndrome - Example For the same codeword, this time with an error in the first bit position, i.e., cr = [1101000] In this case a syndrome 001 indicates an error in bit 1 of the codeword

Comments about H The minimum distance of the code is equal to the minimum number of columns (non-zero) of H which sum to zero We can express Where do, d1, dn-1 are the column vectors of H Clearly crHT is a linear combination of the columns of H

Comments about H For a codeword with weight w (i.e., w ones), then crHT is a linear combination of w columns of H. Thus we have a one-to-one mapping between weight w codewords and linear combinations of w columns of H Thus the min value of w is that which results in crHT=0, i.e., codeword cr will have a weight w (w ones) and so dmin = w

Comments about H For the example code, a codeword with min weight (dmin = 3) is given by the first row of G, i.e., [1000011] Now form linear combination of first and last 2 cols in H, i.e., [011]+[010]+[001] = 0 So need min of 3 columns (= dmin) to get a zero value of cHT in this example

Standard Array From the standard array we can find the most likely transmitted codeword given a particular received codeword without having to have a look-up table at the decoder containing all possible codewords in the standard array Not surprisingly it makes use of syndromes

Standard Array The Standard Array is constructed as follows, c1 (all zero) e1 e2 e3 … eN c2+e1 c2+e2 c2+e3 …… c2+eN c2 cM+e1 cM+e2 cM+e3 cM+eN cM s0 s1 s2 s3 sN All patterns in row have same syndrome Different rows have distinct syndromes The array has 2k columns (i.e., equal to the number of valid codewords) and 2R rows (i.e., the number of syndromes)

Standard Array The standard array is formed by initially choosing ei to be, All 1 bit error patterns All 2 bit error patterns …… Ensure that each error pattern not already in the array has a new syndrome. Stop when all syndromes are used

Standard Array Imagine that the received codeword (cr) is c2 + e3 (shown in bold in the standard array) The most likely codeword is the one at the head of the column containing c2 + e3 The corresponding error pattern is the one at the beginning of the row containing c2 + e3 So in theory we could implement a look-up table (in a ROM) which could map all codewords in the array to the most likely codeword (i.e., the one at the head of the column containing the received codeword) This could be quite a large table so a more simple way is to use syndromes

Standard Array This block diagram shows the proposed implementation Compute syndrome Look-up table + cr s e c

Standard Array For the same received codeword c2 + e3, note that the unique syndrome is s3 This syndrome identifies e3 as the corresponding error pattern So if we calculate the syndrome as described previously, i.e., s = crHT All we need to do now is to have a relatively small table which associates s with their respective error patterns. In the example s3 will yield e3 Finally we subtract (or equivalently add in modulo 2 arithmetic) e3 from the received codeword (c2 + e3) to yield the most likely codeword, c2

Hamming Codes We will consider a special class of SEC codes (i.e., Hamming distance = 3) where, Number of parity bits R = n – k and n = 2R – 1 Syndrome has R bits 0 value implies zero errors 2R – 1 other syndrome values, i.e., one for each bit that might need to be corrected This is achieved if each column of H is a different binary word – remember s = eHT

Hamming Codes Systematic form of (7,4) Hamming code is, The original form is non-systematic, Compared with the systematic code, the column orders of both G and H are swapped so that the columns of H are a binary count

Hamming Codes The column order is now 7, 6, 1, 5, 2, 3, 4, i.e., col. 1 in the non-systematic H is col. 7 in the systematic H.

Hamming Codes - Example For a non-systematic (7,4) code d = 1011 c = 1110000 + 0101010 + 1101001 = 0110011 e = 0010000 cr= 0100011 s = crHT = eHT = 011 Note the error syndrome is the binary address of the bit to be corrected

Hamming Codes Double errors will always result in wrong bit being corrected, since A double error is the sum of 2 single errors The resulting syndrome will be the sum of the corresponding 2 single error syndromes This syndrome will correspond with a third single bit error Consequently the ‘corrected’ codeword will now contain 3 bit errors, i.e., the original double bit error plus the incorrectly corrected bit!

Bit Error Rates after Decoding For a given channel bit error rate (BER), what is the BER after correction (assuming a memoryless channel, i.e., no burst errors)? To do this we will compute the probability of receiving 0, 1, 2, 3, …. errors And then compute their effect

Bit Error Rates after Decoding Example – A (7,4) Hamming code with a channel BER of 1%, i.e., p = 0.01 P(0 errors received) = (1 – p)7 = 0.9321 P(1 error received) = 7p(1 – p)6 = 0.0659 P(3 or more errors) = 1 – P(0) – P(1) – P(2) = 0.000034

Bit Error Rates after Decoding Single errors are corrected, so, 0.9321+ 0.0659 = 0.998 codewords are correctly detected Double errors cause 3 bit errors in a 7 bit codeword, i.e., (3/7)*4 bit errors per 4 bit dataword, that is 3/7 bit errors per bit. Therefore the double error contribution is 0.002*3/7 = 0.000856

Bit Error Rates after Decoding The contribution of triple or more errors will be less than 0.000034 (since the worst that can happen is that every databit becomes corrupted) So the BER after decoding is approximately 0.000856 + 0.000034 = 0.0009 = 0.09% This is an improvement over the channel BER by a factor of about 11

Perfect Codes If a codeword has n bits and we wish to correct up to t errors, how many parity bits (R) are needed? Clearly we need sufficient error syndromes (2R of them) to identify all error patterns up to t errors Need 1 syndrome to represent 0 errors Need n syndromes to represent all 1 bit errors Need n(n-1)/2 to syndromes to represent all 2 bit errors Need nCe = n!/(n-e)!e! syndromes to represent all e bit errors

Perfect Codes So, If equality then code is Perfect Only known perfect codes are SEC Hamming codes and TEC Golay (23,12) code (dmin=7). Using previous equation yields

Summary In this section we have Used block codes to add redundancy to messages to control the effects of transmission errors Encoded and decoded messages using Hamming codes Determined overall bit error rates as a function of the error control strategy