ECED 4504 Digital Transmission Theory

ECED 4504 Digital Transmission Theory Course Overview, Basic Characteristics of Block Codes

Topics today • Block codes • repetition codes • parity codes • Hamming codes • cyclic codes • Forward error correction (FEC) system error rate in AWGN • Encoding and decoding • Codes characterization • code rate • Hamming distance • error detection ability • error correction ability

A code taxonomy

Error-control coding: basics of Forward Error Correction (FEC) channel coding • Coding is used for error detection and/or error correction • Coding is a compromise between reliability, efficiency, equipment complexity • In coding extra bits are added for data security • Coding can be realized by two approaches • ARQ (automatic repeat request) • stop-and-wait • go-back-N • selective repeat • FEC (forward error coding) • block coding • convolutional coding • ARQ includes also FEC • Implementations, hardware structures Topic today

k-bits n-bits (n,k) block coder What is channel coding? • Coding is mapping of binary source (usually) output sequences of length k into binary channel input sequences n (>k) • A block code is denoted by (n,k) • Binary coding produces 2k codewords of length n. Extra bits in codewords are used for error detection/correction • In this course we concentrate on two coding types: (1) block, and (2) convolutional codes realized by binary numbers: • Block codes: mapping of information source into channel inputs done independently: Encoder output depends only on the current block of input sequence • Convolutional codes: each source bit influences n(L+1) channel input bits. n(L+1) is the constraint length and L is the memory depth. These codes are denoted by (n,k,L).

Representing codes by vectors • Code strength is measured by Hamming distance that tells how different code words are: • Codes are more powerful when their minimum Hamming distance dmin (over all codes in the code family) is large • Hamming distance d(X,Y) is the number of bits that are different between code words • (n,k) codes can be mapped into n-dimensional grid: valid code word

Hamming distance: The decision sphere interpretation • Consider two block code (n,k) words c1and c2 at the Hamming distance in the n-dimensional code space: • It can be seen that we can detect l=dmin-1 errors in the code words. This is because the only way to not to detect the error is that the error transforms the code into another code word. This requires change in d code bits. • Also, we can see that we can correct t=(dmin-1)/2 errors. If more errors occur, the received word may fall into the decoding sphere of another code word.

Example: repetition coding • In repetition coding bits are repeated several times • Can be used for error correction or detection • For (n,k) block codes that is a bound achieved by repetition codes. Code rate is anyhow very small • Consider for instance (3,1) repetition code, yielding the code rate • Assume binomial error distribution: • Encoded word is formed by the simple coding rule: • Code is decoded by majority voting, e.g. for instance: • Error in decoding is introduced if all the bits are inverted or two bits are inverted (by noise or interference), e.g. majority of bits is in-error

Repetition coding, cont. • In a three bit code word • one error can be corrected always, because majority voting can detect and correct one code word bit error always • two errors can be detected always, because all code words must be all zeros or all ones (but now the encoded bit can not be recovered) • Example:

Error rate for a simple repetitive code error rate pe Note that by increasing word lengthmore and more resistance to channelintroduced errors is obtained. code length n n

Parity-check coding • Repetition coding can greatly improve transmission reliability because • However, due to repetition transmission rate is reduced. Here the code rate was 1/3 (that is the ration of the bits to be coded to the encoded bits) • In parity-check coding a check bit is formed that indicates number of “1” in the word to be encoded. • Even number of “1” means that the the encoded word has even parity • Example: coding 2-bit words by even parity is realized by • Question: How many errors can be detected/corrected by parity-check coding?

Parity-check error probability • Note that the error is not detected if even number of errors have happened • Assume n-1 bit word parity coding, e.g. (n,n-1) code. Probability to have error in a code word: • single error can be detected (parity changed) • probability for two-bit error is Pwe=P(2,n) whereand note that for having more than two errors is highly unlikely and thus we approximate total error probability by

n-1 bit-word error probability • Without error correction we transmit n-1-bit word that will have a decoding error with the probabilitywhere simplification follows from the negligence of higher order terms, as for instance

Comparing parity-check coding and repetitive coding • Hence we note that parity checking is very efficient method of error detection: Example: • At the same time the information rate was reduced only by 9/10 • If the (3,1) repetitive coding would be used (repeating every bit three times) the code rate would drop to 1/3 and the error rate would beTherefore parity-check coding is very popular coding method of channel coding. (Note that explained error probability requires . successful retransmission) no encoding, n-1 bit word parity bit applied

Examples of block codes: a summary • (n,1) Repetition codes. High coding gain, but low rate • (n,k) Hamming codes. Minimum distance always 3. Thus can detect 2 errors and correct one error. n=2m-1, k = n - m • Maximum length codes. For every integer there exists a maximum length code (n,k) with n = 2k - 1,dmin = 2k-1 • Golay codes. The Golay code is a binary code with n = 23, k = 12, dmin = 7. This code can be extended by adding an extra parity bit to yield a (24,12) code with dmin = 8.Other combinations of n and k have not been found. • BCH-codes. For every integer there exist a code with n = 2m-1, and where t is the error correction capability • (n,k) Reed-Solomon (RS) codes. Works with ksymbols that consists of m bits that are encoded to yield code words of nsymbols. For these codes and • Nowadays BCH and RS are very popular due to large dmin, large number of codes, and easy generation

Generating block codes: Systematic block codes • In (n,k) block codes each sequence of k information bits is mapped into a sequence of n(>k) channel inputs in a fixed way regardless of previous information bits • The formed code family should be selected such that the code minimum distance is as large as possible -> high error correction or detection capability • A systematic block code: • the first k elements are the same as the message bits • the following q = n - k bits are the check bits • Therefore the encoded word isor as the partitioned representation

Block codes by matrix representation • Given the message vector M, the respective linear, systematic block code X can be obtained by the matrix multiplication by • The matrixG is the generator matrix with the general structure • whereIkis kxk identity matrix and P is a kxr binary submatrix ultimately determining the generated codes

Generating block codes • For u message vectors M (each consisting of k bits)the respective n-bit block codes Xare therefore determined by Generated check bits, from above, as for instance for k=4,

Forming the P matrix • The check vector B that is appended to the message in the encoded word is thus determined by the multiplication • The j:th element of B on the u:th row is therefore encoded by • For the Hamming code P matrix of k rows consists of all r-bit words with two or more "1":s arranged in any order! Hence P can be for instance

Generating a Hamming code: An example • For the Hamming codes n=2r-1, k = n - r, dmin=3 • Take the systematic (n,k) Hamming code with r=3 (the number of check bits) and n=23-1=7 and k=n - r=7-3=4. Therefore the generator matrix is • Note that in Hamming code the three last columns make up the P submatrix including all the 3-bit words that have 2 or more “1”:s. • For a physical realization of the encoder we now assume that the message contains the bits

Realizing a (7,4) Hamming code encoder • For these four message bits we have a four element message register implementation • Note that here the check bits [b1,b2,b3] are obtained by substituting the elements of P into equation B=MP or

Example* *S. Lin, E. Costello: Error Control Coding: Fundamentals and Applications

Listing generated Hamming codes • Going through all the combinations of the input vector X yields all the possible output vectors • Note that for the Hamming codes the minimum distance or weight w = 3 (the number of “1” on each row)

Decoding block codes • A brute-force method for error correction of a block code includes comparison to all possible same length code structures and choosing the one with the minimum Hamming distance when compared to the received code. • In practice applied codes can be very long and the extensive comparison would require much time and memory. For instance, to get the code rate of 9/10 with a Hamming code it is required that • This equation fulfills if the code length is at least k=57, and now n = 63. • There are different block codes in this case! Decoding by direct comparison would be quite unpractical! • This approach of comparing Hamming distance of the received code to the possible codes, and selecting the shortest one is the maximum likelihood detection and will be discussed more with convolutional codes

Syndrome decoding for error detection • In syndrome decoding a parity checking matrixH is designed such that multiplication with a code word produces all-zero matrix: • Therefore error detection of the received signal Y can be based on syndrome:that is always zero when a (correct) code word is received. (Note that the syndrome does not reveal errors if channel noise has produced another code word!) • The parity checking matrix is determined by or • Having parity checking matrix design such that the rows of HT are all different and contain at least one "1" a distinct syndrome for each single error pattern can be obtained -> enables error correction!

Syndrome decoding for error correction • Syndrome decoding can be used for error correction by checking the one-bit error pattern for each syndrome: • Example: Consider a (7,4) Hamming code with a parity check matrix • The respective syndromes and error vectors (showing the position of errors by "1") are

Syndrome is independent of code words • This design enables that the syndrome depends entirely on error pattern but not on particular code. Consider for instance • Syndrome does not determine the error pattern uniquely because there exists only 2q different syndromes (syndrome length is q) but there exists 2k different codes (for each symbol that must be encoded). • After error correction decoding double errors can turn out even triple errors • Therefore syndrome decoding is efficient when channel errors are not too likely, e.g. probability for double errors must be small. • For difficult channels there are more elaborated schemes using for instance extended Hamming codes or maximum likelihood methods (as the Viterbi-decoding)

Table lookup syndrome decoder circuit • The error vector is used for error correction by the circuit shown bellow: Error subtraction

Error rate in a modulated and channel coded system • Assume: • errors are corrected (upper bound, not achieved always, as in syndrome decoding) • Additive White Gaussian Noise channel (AWGN, error statistics in received encoded words same for each bit) • channel error probability a is small (used to simplify relationship between word and bit errors)

Bit and symbol error rate • Transmission error rate a is a function of channel signal and noise power. We will note later that for the coherent BPSK1 the bit error rate probability iswhere Eb is the transmitted energy / bit and h is the channel noise power spectral density [W/Hz]. • Due to the coding, energy / transmitted symbol is decreased and hence for the system using a (n,k) code with the rate RCthe error rate is where • However, coding can improve symbol error rate after decoding (=code gain) <-no code gain effect here 1Binary Phase Shift Keying

Bit errors and word errors • It is not self evident which one plays more important role for symbol errors, the energy decrease / symbol in the channel, or coding gain, thus for certain channel noise levels coding might be harmful. • Coding can correct up to errors. Therefore decoding error rate is upper bounded bywhere the simplification follows because higher terms of the summation are less significant in high SNR channels when • Note that this means that in average each unsuccessful (in-error) coded word contains in average t+1 erroneous bits

Bit errors and word errors, cont. • If there would be no ability to correct encoded words their error probability would be n-times the bit error probability or • However, the ability to correct t+1 errors decreases the word error rate towhere (the average value of the binomial distribution) and hence the encoded system error probability is

Error rate comparison Example, RC=11/15 • The error rate expression waswhere for BPSK • For the respective uncoded system (polar MF detection) error rate was

ECED 4504 Digital Transmission Theory