Further Optimization of the Decoding Method for Shortened Binary Cyclic Fire Code

Transcription

1 Further Optimization of the Decoding Method for Shortened Binary Cycic Fire Code Ch. Nanda Kishore Heosoft (India) Private Limited , Road No-12 Banjara His, Hyderabad, INDIA Phone: e-mai: ABSTRACT Channe coding adds redundant bits to the origina information bits in order to detect and correct possibe errors occurred during transmission. In GSM (Goba System for Mobie Communications), we use both bock coding and convoutiona coding for error contro. One of the channe coding schemes is CS-1 [1], which empoys shortened cycic Fire code for burst error correction. In this paper, we propose an efficient way of decoding shortened binary cycic Fire code, which is an improved version of our earier method [4]. This method is especiay suitabe for DSP impementation. The compexity of the proposed decoding scheme is much ower than that of conventiona methods [2,6] and ower than the method given in [4]. The proposed scheme is a straightforward method in the sense that it requires no knowedge of inverse cycic codes, which is a requirement for the patented method [3]. The proposed decoding scheme is appicabe to any binary shortened cycic code as we. 1. INTRODUCTION In the designated coding scheme CS-1 for GSM [1], the input to the bock encoder consists of a burst of 184 information bits {d (0), d (1),...,d (183)} and the encoder adds forty extra parity bits. These bits are added according to the binary shortened cycic Fire code. The decoding of the binary shortened cycic Fire code using conventiona methods is very compex and in the software impementation, it invoves three miion iterations for syndrome computation. An efficient agorithm for decoding binary shortened cycic Fire code was patented recenty [3]. In an earier paper [4], we presented an efficient method of decoding binary shortened cycic Fire code. Here, we describe further improvements in computationa compexity for the decoding of binary shortened cycic fire code. The method proposed here is of much ower compexity compared to that of conventiona methods [2,6], and is ess compex than the method given in [4]. This method is suitabe for DSP impementation. As the present and the next generation software defined radios (SDR) use DSP technoogy to increase system robustness and functionaity, this method is a ess compex soution for SDR. Note, however, that our scheme is straightforward method and entirey different from that of [3], as it requires no knowedge of inverse cycic codes. First we introduce the background materia briefy and present our scheme. Then we concentrate on impementation aspects with respect to CS-1 [1] and provide test procedure and resuts. 2. BACKGROUND 2.1 Cycic Codes An (n, k) inear code over a fied F is said to be a cycic code if, for every codeword C = (c 0, c 1,..., c n-1 ), the right cycic shift of C, viz. C 1 = (c n-1,c 0,c 1,..., c n-2 ), is aso a codeword. To deveop agebraic properties of cycic codes, we treat components of a code vector C, as coefficients of a poynomia. C(x) = c 0 + c 1 * x c n-1 * x n-1 (1) The encoding and decoding operations of a cycic code can be accompished by using inear shift registers with feed back connections. 2.2 Cycic Fire Code Fire codes are a type of cycic codes constructed systematicay for correcting burst errors. Let g 1 (x) be an irreducibe poynomia of degree m over GF(2). Let ρ be the smaest integer such that g 1 (x) divides x ρ +1. Let b be positive integer (b m) and 2b-1 is not divisibe by ρ. Foowing poynomia generates a b-burst errorcorrecting FIRE code with the number of parity check digits equa to m+2b-1. g(x) = (x 2b-1 + 1) * g 1 (x) (2) The ength n of this code is the east common mutipe of 2b-1 and ρ i.e., n = LCM (2b-1,ρ) (3)

2 2.3 Shortened Cycic Code Whie seecting a code, if a code of suitabe natura ength or suitabe number of information digits cannot be found, it is desirabe to shorten a code to meet the requirements. There are various methods to shorten a cycic code. One of such methods generay used is: given (n,k) cycic code C, consider the set of code vectors for which eading east significant (LSB) information digits are zero. There are 2 k- such code vectors and they form a inear sub code of C. If the zero information digits are deeted from each of these code vectors, we obtain a set of 2 k- vectors of ength n-. These 2 k- shortened vectors form an (n-,k- ) inear code. This code is caed shortened cycic code. The shortened cycic code is not cycic in the strict sense and properties of cycic codes cannot be appied directy. The minimum number of errors that a shortened cycic code can correct is same as that of the origina cycic code. The encoding of shortened cycic codes can be accompished by shift registers circuit with feed back connections, as usuay empoyed by the origina cycic encoding. This is so because the deeted eading east significant zero information digits do not affect the parity-check bits computation. However, the decoding process may become invoved whie computing the syndrome. The process woud invove shifting the syndrome cycicay after the entire received vector has been shifted into the shift register. For the bock code used in CS-1 [1], the number of shift operations wi be around three miion to get the proper syndrome. These extra shifts of the syndrome register cause undesirabe decoding deay and redundant computations. One method of avoiding these extra shifts is given in [4]. In the next section, we present a better method when impementing the syndrome computation mechanism on a DSP. 3. PROPOSED DECODING SCHEME The decoding process of binary shortened cycic Fire code invoves 1) Syndrome computation and 2) Error trapping and correction. In the derivation of decoding method we refer n as the tota number of bits in the received shortened code and k as the corresponding number of information bits in the received code. The order of the origina cycic code from which the shortened cycic code is derived is N-1, where N = n +. The coefficients g 0, g n-k the first and the ast coefficients of generator poynomia g(x), are assumed to be 1, as these are aways equa to one for any cycic code. 3.1 Syndrome computation The syndrome computation of the shortened cycic codes can be achieved using inear feedback shift registers (LFSR). But, after the entire received poynomia has been shifted into the LFSR, the syndrome register must be shifted right cycicay times to get proper syndrome, where is number of eading zeros of cycic code. For arge as in the case of bock decoding of GSM Packet Data Traffic Channe bock type-1 [1], these extra shifts of the syndrome register cause undesirabe deay in decoding and woud need redundant computations. This can be avoided by modifying the decoding process by the method given in [4]. However, we need simutaneous mutipication and division operations to get the syndrome. Though, simutaneous mutipication and division can be impemented very easiy using the circuit given in [2], whie impementing the code on a DSP this takes more cyces than poynomia division operation. Therefore a different method is considered here to arrive at ess compex soution than the method given in [4] for this situation. Let r(x) be the received poynomia of a shortened cycic code, and R(x) = x * r(x) be the unshortened version of this code. We can write R(x) = a 1 (x)*g(x) + S(x) (4) where S(x) is the syndrome obtained when R(x) is divided by g(x) and a 1 (x) is quotient. Let p(x) be the remainder resuting from dividing x by g(x) and a 2 (x) is quotient, then p(x) = x + a 2 (x)*g(x) (5) The actua syndrome of cycic code can aso be computed by using syndrome of shortened cycic code as described beow. Let s(x) be the syndrome of a received shortened cycic code. Then, r(x) = a 3 (x)*g(x) + s(x) (6) where a 3 (x) is the quotient. However note, here r(x) is not cycic and s(x) is not proper syndrome. The proper syndrome of the cycic code S(x) is obtained by x * r(x) = a 4 (x) * g(x) + S(x) (7) Substituting Equation (6) in (7) to get x * [a 3 (x)*g(x) + s(x)] = a 4 (x)*g(x) + S(x) (8) Simpifying Equation (8), we obtain x * s(x) = a 5 (x)*g(x) + S(x) (9) where a 5 (x) = x * a 3 (x) + a 4 (x). Note from (9) that S(x) is the remainder obtained by dividing x * s(x) by g(x). Mutipying both sides of Equation (5) with s(x) and using (9), we get p(x) * s(x) = a 6 (x) * g(x) + S(x) (10) where a 6 (x) = a 2 (x) * s(x) + a 5 (x) Equation (10) suggests that we can obtain the actua syndrome of cycic code S(x) by mutipying the syndrome of shortened cycic code with p(x) and dividing the product with g(x). The syndrome of the shortened code is computed by dividing the received poynomia with generator poynomia g(x). Since g(x) has ony 6 non zero coefficients for GSM CS-1 [1], this division can be impemented with very ess compexity. The syndrome of the

3 shortened code so computed, is mutipied with p(x) and dividing the product with g(x) gives the actua syndrome of the cycic Fire code. Here simutaneous mutipication and division is performed on the syndrome poynomia of shortened code, which is of order n-k-1. So this method of syndrome computation is ceary a ess compex soution than the method given in [4], for which syndrome is computed by p(x)*r(x) = a(x)*g(x) + S(x) (11) Once syndrome computation is over, the process of error trapping and correction for shortened cycic code described in [4] takes pace to compete the decoding process. This process is briefy described in the next sub section. 3.2 Error trapping and Correction Let the received poynomia of a shortened cycic code r(x) be represented as r(x) = c 0 + c 1 * x +..., + c n-1 * x n-1 (12) and unshortened version of this code R(x) = represented as x * r(x) be R(x) = C 0 + C 1 * x +..., + C N-1 * x N-1 (13) and et R 1 (x) be the right cycic shift of R(x). Then the reationship between R(x) and R 1 (x) is given by R 1 (x) = x * R(x) + C N-1 * (x N +1) (14) Let the actua syndrome of shortened cycic code be represented as S(x) = s 0 + s 1 * x + s 2 * x , + s n-k-1 * x n-k-1 (15) Let S 1 (x) be the syndrome of R 1 (x) and g(x) = 1 + g 1 * x +..., + g n-k-1 * x n-k-1 + x n-k be the generator poynomia then R(x) = a(x) * g(x) + S(x) (16) and S 1 (x) = R 1 (x) mod g(x) (17) where mod is the moduo operator. Substituting Equation (14) in (17) we get, S 1 (x) = [x * R(x) + C N-1 * (x N +1)] mod g(x) (18) For cycic codes we can write (x N +1) = g(x) * h(x) where h(x) is the parity-check poynomia. Then Equation (18) can be written as, S 1 (x) = x * R(x) mod g(x) (19) Substituting Equation (16) in (19) we get, S 1 (x) = x * S(x) mod g(x) (20) The Equation (20) suggests that the syndrome S 1 (x) of R 1 (x) is the remainder obtained by dividing x * S(x) with g(x) where S(x) is the syndrome of R(x). Now x * S(x) mod g(x) can be computed by right shifting the syndrome register once cycicay with S(x) as its initia contents using the circuit of Figure 1, shown in [4]. The proof for this is provided in [4]. This property is used in the error trapping and correction mechanism. Let r(x) be the received poynomia of a shortened cycic code. Further, et R(x) = x * r(x) is the received poynomias of unshortened version of this code. Let g(x) be the generator poynomia and E(x) be the error poynomia of the unshortened cycic code, then the actua syndrome of cycic code is, S(x) = R(x) mod g(x) = E(x) mod g(x) (21) Now suppose errors are confined to b high order positions x n-b, x n-b+1,..., x n-1 of r(x). These bits are identica to x N-b, x N-b+1,..., x N-1 bits of R(x). Then if R(x) is cycicay shifted right b times, errors are confined to ow order positions x 0, x 1,x 2,..., x b-1 of R b (x). The corresponding error pattern is then E b (x) =e N-b + e N-b+1 *x + e N-b+2 *x ,+ e N-1 *x b-1 (22) As degree of E b (x) is ess than that of g(x), from (21) S b (x) = E b (x) mod g(x) = E b (x) (23) Therefore errors in the burst can be corrected by adding b ow order syndrome register bits with x N-b, x N-b+1,..., x N-2, x N-1 bits of R(x). These bits are same as x n-b, x n-b+1,..., x n-2, x n-1 bits of r(x). So the burst error in the received poynomia can be corrected by adding b ow order syndrome register bits with these bits of r(x) over moduo 2. Suppose errors are not confined to b high order positions of r(x) or R(x), but they are confined to b consecutive positions x i, x i+1, x i+2,..., x i+b-1 of r(x). These bits are same as x +i, x +i+1, x +i+2,..., x +i+b-1 bits of R(x). After (n-i) right cycic shifts of R(x), errors are confined to b consecutive ow order positions of R n-i (x). And from Equation (21), S n-i (x) corresponds to E n-i (x). Therefore errors can be corrected by adding b ow order syndrome register bits to x +i, x +i+1, x +i+2,..., x +i+b-1 of R(x) which are same as x i, x i+1, x i+2,..., x i+b-1 bits of r(x). So by adding b eft side bits of LFSR to these bits of r(x), singe burst error can be corrected. Thus, the process of Error trapping and correction consists of cycicay shifting the syndrome register for n times and each time checking the ast n-k-b stages of syndrome register for zero bits. Where n is the number of encoded bits of shortened cycic code. If the ast n-k-b bits of syndrome register are zero for i th shift (1 i n), the process of shifting is stopped and burst error is corrected by adding ow order b bits of syndrome register to appropriate bits of r(x). This process takes n cock cyces. One may note here that in [3], error trapping is based on checking LSB (Least Significant Bit) of syndrome register, shifting the

4 syndrome towards LSB, and adding the pre computed and stored version of x n-1 mod g(x) to it if the LSB is 1. Our method of error trapping is a straightforward method and an extension to the error trapping of origina cycic codes to efficienty decode shortened cycic codes. The decoding procedure described above can be appied to burst error correction in the bock decoding for CS-1, which empoys binary shortened cycic Fire code. This is given in the next section. 4. BLOCK DECODER FOR CS Decoder design In this section, we concentrate on the impementation of binary shortened cycic Fire decoding circuit for coding scheme-1 of GSM packet data traffic channe [1]. The encoding rue for this coding scheme is given as foows. The bock of 184 information bits is protected by 40 extra bits used for error correction and detection. These bits are added to the 184 bits according to a shortened binary cycic code (FIRE code) using the generator poynomia: g(x) = (x ) * (x 17 + x 3 + 1) (24) The encoding of the cycic code is performed in a systematic form, which means that, in GF(2), the poynomia: v(x) = d(0)x d(1)x d(183)x 40 + p(0) x 39 +p(1)x ,+p(38)x + p(39) (25) Where {p(0),p(1),...,p(39)} are the parity bits, when divided by g(x) yieds a remainder equa to: s(x) = 1 + x + x x 39 (26) This code is capabe of correcting singe 12-bit burst error. The period ρ of (x 17 + x 3 + 1) is The order n of unshortened version of this code is The number of eading zero bits of this code is The bit configuration of the code word for this binary shortened cycic Fire code is as shown in Figure 2 of [4]. The bock of bits indicate eading east significant zero bits, and bocks of 184 and 40 indicate number of information and parity bits respectivey. At the encoding side the 40 parity check bits can be computed by pushing 184 information bits from right side in to inear shift registers with feed back connection with circuit shown in Figure 1 of [4]. After 184 information bits are cocked into the LFSR, we get the 40 parity bits in LFSR. This is so because the deeted eading zero information digits do not affect the paritycheck bits computation. However, for decoding this code word, if we want to use the same circuit used for encoding, after pushing 184 information bits and 40 parity bits into the syndrome register, we need to shift the syndrome register cycicay right times to get proper syndrome. This can be avoided by using any one of the two methods, either the method given in [4] or the method described in this paper for the syndrome computation. However, the method described here is efficient than the method given in [4] whie impementing on a DSP. The predetermined poynomia, p(x) is given in [4]. Then using the Equation (10), we can find the proper syndrome of the binary shortened cycic Fire code, S(x). The error trapping and correcting process invoves, shifting syndrome register (with S(x) as initia contents) cycicay right 224 times and each time checking for a zeros for the ast 28 stages of syndrome register. If ast 28 stages of syndrome register (LFSR) turn out to be a zeros for i th shift (1 i 224), then burst error is trapped into eft 12 stages of shift register. Which aso indicates that x 224-i, x 224-i+1,..., x 224-i+11 of r(x) might be corrupted bits. Then the process of shifting the syndrome register is stopped. And 12-burst error can be corrected by adding first 12 bits of syndrome register to x 224-i, x 224-i+1,..., x 224-i+11 bits of the received code over GF(2). For the occurrence of burst error of ength ess than 12, ony those number of bits need to be corrected. If a right 28 stages of syndrome register are never zero for 224 shifts, then we can say uncorrectabe burst errors occurred in the received code word. This process of the error trapping and correction takes 224 cock cyces. 4.2 Test procedure and resuts The decoder is tested according to the test procedure given in [4] and is found that the decoder described in this paper is abe to correct a possibe singe burst errors for burst ength ess than or equa to 12 in a cases. 5. CONCLUSIONS In this paper, an efficient method of decoding binary shortened cycic Fire code is deveoped. Its impementation with inear shift registers with feed back connections is discussed. Present decoding method impementation requires fewer processing operations than conventiona methods. The decoding method is capabe of correcting a possibe error patterns with singe burst error. The impementation detais for the bock decoder of CS-1 [1] are described. This method is a better option than that of [4], whie impementing the binary shortened cycic Fire decoder on a DSP. Our method of decoding is a straightforward method and entirey different from that of [3] in the sense that it requires no knowedge of inverse cycic codes, which is a requirement for the patented method. The present method of decoding is appicabe to any shortened cycic code as we. 6. ACKNOWLEDGMENT The author wishes to thank S. Rama Rao, Vice President and Genera Manager (India operations), of Heosoft for the support and encouragement extended during the course of work. The author aso wishes to thank Dr. Y. Yoganandam, Technica Director of Heosoft, and Dr. V. Umapathi Reddy, Chief Technica Officer of Heosoft for their vauabe suggestions and encouragement.

5 7. REFERENCES [1] ETSI EN v8.5.1 ( ) - Digita ceuar teecommunications system (Phase 2+); Channe coding (GSM version Reease 1999) [2] Shu Lin and Danie J. Costeo, Jr, Error Contro Coding: Fundamentas and Appications. Prentice-Ha, Inc. Pubishing Company, 1983 (Chapters 4,9) [3] Ramesh, et a, Shortened Fire code Error Trapping Decoding method and apparatus- United States Patent 5,936,978. Cycic Fire Code, Internationa Conference on Persona Wireess Communications (IEEE-ICPWC), December 2002, P [5] Vijay K. Bhargava, Forward Error Correction Schemes For Digita Communications, IEEE Communications Magazine, January [6] John G. Proakis, Digita Communications, McGRAW-HILL INTERNATIONAL EDITION, 2001 (Chapter.8) [4] Ch. Nanda Kishore, Efficient Decoding of Shortened Binary