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Block Error Correction Codes and Convolution Codes S-72.333 Postgraduate Course in Radio Communications - Wireless Local Area Network (WLAN) 1 Block Error Correction Codes and Convolution Codes Mei Yen, Cheong can be transmitted across the channel at rate less than the Abstract—In wireless communications, error control is an channel capacity with arbitrarily low error rate. Since the important feature for compensating transmission impairments publication of this theory more than 40 years ago, control such as interference and multipath fading which cause high bit coding theorists have been working towards discovering these error rates in the received data. Forward Error Correction codes. Among the error control codes found since then are (FEC) is one of the data link layer protocols for error control. some block codes such as BCH and Reed-Solomon codes and This paper gives an overeview to error control coding employed in FEC, particularly block codes and convolutional codes. convolutional codes which will be discussed in this paper. Finally, some consideration of code selection will be discussed. The paper is organized as follows. In Section II the concept of FEC will be introduced. Section III discusses first block codes I. INTRODUCTION in general and then some particular cyclic codes, namely BCH codes and Reed-Solomon codes. Convolutional code is he trend towards portable personal computers or presented in section IV and finally section V discusses some T workstations are rising quickly in the recent years. considerations of code selection and some methods to WLAN networks are becoming increasingly popular. Users of enhanced error control schemes. the network are demanding for ever higher quality of service (QoS) and multi-variety services (not only data, but also packet voice, video, etc.). Unlike wireline systems, wireless II. FORWARD ERROR CORRECTION and mobile environment is more prone to transmission impairments due to multipath propagation and movement of Forward error correction schemes add redundant bits to the the mobile stations. original transmitted frame in order to obtain known structures or patterns in the final transmitted frame. Figure 1 depicts the Particularly, in WLAN systems which employ FEC operation where the encoder maps the k-bit transmitted omnidirectional antennae in order to obtain good coverage, data into an n-bit codeword and decoding and error correction to obtain the original data. the transmitted signal power is not concentrated to the intended user. On top of that, the transmitted signal is more scattered and multipath effect becomes more pronouns. Interference is also an issue in WLAN environment due to its unlicensed frequency spectrum. With the aim of providing high QoS and reliable transmission in this hostile wireless environment with restricted received signal level, development of appropriate physical layer, data link layer and network layer protocols is important. In this paper, we are interested particularly in error control protocols in data link layer. There are largely two categories of error Figure 1: Operation of FEC based on block coding control mechanisms, namely FEC and Automatic Repeat Request (ARQ). The focus of the paper will be on error With these known patterns, the decoder at the receiver will be control coding employed in FEC. able to detect and possibly correct the erroneous bits in the received frames. To be more precise, there are four possible Shannon’s theory on channel coding states that, given a outcomes at the decoder output:- channel there exist error control codes such that information S-72.333 Postgraduate Course in Radio Communications - Wireless Local Area Network (WLAN) 2 i. Received frame is identical to the transmitted éù ddmin = minêú()ccij , . frame. The FEC decoder produces the original ij¹ ëû transmitted data block as output. ii. The received frame is different from the A code is capable of correcting up to t bits of errors if the transmitted frame but has a certain error pattern code satisfy dmin ³ 2t +1 . Putting this in another way, the know to the decoder. The FEC decoder will be maximum number of guaranteed correctable errors per able to detect and correct these erroneous bits codeword is and produces the original transmitted block as output. êd - 1ú iii. The received frame is different from the t = ê min ú, ê 2 ú transmitted frame. The error pattern of the ë û received frame can be detected but cannot be corrected by the FEC decoder. The FEC decoder where ëx û stands for the largest integer not exceeding x. For reports an uncorrectable error. error detection, the code that satisfies td=-min 1 can detect iv. The received frame contains a typically rare up to t bits of errors in a block. pattern that the decoder cannot detect any errors. The FEC decoder produces a k-bit block In general, the encoding operation of linear block codes can that is different from the original transmitted be represented with multiplying the generator matrix with the block as output. data block to be transmitted as These scenarios are depicted in Figure 2. c = mG , where m = ém mm... ù and ë 01 k û ég ùég ggL ù ê 111121úên ú êg úêg ggL ú ê 221222úên ú G ==ê úê ú. êMúêMM M ú ê úê ú ëêgk ûëúêgkk12ggL kn ûú G is a matrix consisting of k rows and n column. Since an ()n,k linear block code has 2k valid codewords, it is a Figure 2: FEC decoding scenarios subspace of dimension k. Consequently, the row of G must be linearly independent, i.e. they must be the basis for the ()n,k code. The basis vectors for the code are not unique, and hence III. BLOCK CODES G is not unique. Block code schemes involves dividing the input data bit A linear block code is said to be a systematic code if the first stream into block of k-bit streams and then mapping each k- k bits of the code words are the data bits and the following bit block into n-bit block called codeword, where nk> in n - k bits are the parity check bits. Any generator matrix G the encoding process. ()nk- check bits are added to each k- can be reduced by row operations to the systematic form as follow. bit block. The ratio ()nkk-/ is called the redundancy of the code and the ratio kn/ is called the code rate. é100L 0ppL pù ê 11 12 1nk- ú ê pp pú ê010L 021 22L 2nk- ú Before we go further, some important terms shall be defined G = ê ú to ensure clarity in the discussion to follow. êMMMO MMMO Mú ê pp pú ëê000L 1kk12L knk- ûú i) Hamming distance d c , c between two binary ()1 2 = []IPk |. code words c1 and c2 is the number of bits the two code words disagree. At the receiver, the decoding operation of a linear block code ii) For an error correcting code that consists of involves a parity check matrix H that is used to detect and codewords éùcc, ,..., c, the minimum distance possibly correct bit errors. Associate with any ()nk, linear ëû12 s code, there is a dual code of dimension nk-. The generator is defined as matrix for the dual code is the parity check matrix H for the S-72.333 Postgraduate Course in Radio Communications - Wireless Local Area Network (WLAN) 3 linear block code. The rows of H spans a subspace of instance, we represent a four digit binary message, []1010 dimension nk- , which is the null space of G. For as 1 + x 2 . The encoding process involves a polynomial systematic codes, H can be deduce from G as generator g()x and the data block to be transmitted. HPI=- '| . []n- k For a given (n, k) cyclic code, the generator polynomial has certain properties related to n and k. The generator n To prove the claim that the generator matrix for the dual code polynomial g()x must be a factors of x + 1 and of degree is the parity check matrix, we denote the received data block n - k . This procedure is best explained with an example. as Example 1: Encoding process yce=+ =+mG e, We consider a (7, 4) cyclic code for encoding a data block m =[]1010 . To find the generator polynomial g()x for an where e is the error vector. ()()nk, = 7, 4 code, we first factorize the polynomial 7 x + 1 and find the factor with degree nk-=.3 Since H is the null space of G, the matrices are orthogonal and GH' = 0 . Then 32 g()xxx= ()++ 1 yH''=+() mG e H To encode the data block m = []1010 , we multiply its =+mGH' eH ' 2 corresponding polynomial mx()=+ x 1 with g()x . = eH'. c ()xmxx= ()()g SeH= ' is known as the syndrome of the error pattern. Note that the syndrome is purely a function of the error pattern and = (xxxx332+++)( 1) not the transmitted code word. Hence H must be the dual =+++xxxx654. space of G. The parity check results in S0= when no error detected, which is equivalent to e0= . When errors are detected, S is Thus the data block is encoded to the codeword nonzero and indicates the positions of the error bits. Hence c =.[]1110010 the error bits can be corrected. In the cases of the number of errors exceeding the error correcting capacity of the code but Another approach for encoding is without using polynomial within the detection capability, the syndrome indicates function is to construct the generator matrix G. This uncorrectable errors. There exist some rare error patterns that approach allows us to encode systematic codes where the the syndrome reports no error. first k bits of the code are the transmitted data follow by n-k bits of parity check bits.
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