These outputs can be written in a more convenient form: with y(i) = Hc m (i) n(i) y(i) = (y(i); ; y K (i)) T ; c m (i) = (c m (i); ; c m K(i)) T and n

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1 Binary Codes for synchronous DS-CDMA Stefan Bruck, Ulrich Sorger Institute for Network- and Signal Theory Darmstadt University of Technology Merckstr. 25, 6428 Darmstadt, Germany Tel.: , Fax: sbruecknesi.tu-darmstadt.de Abstract In this paper synchronous DS-CDMA with binary linear block coding is investigated for AWGN. First the asymptotic error probability for coded transmission is derived for the optimal decoder. Then it is shown how the asymptotic order of the error probability depends on the codes of the users and how the codes have to be designed to be asymptotically good. It is further demonstrated that separated multiuser detection and decoding has an inherent loss in asymptotic order. Finally a detection and decoding algorithm is presented which is able to reduce this disadvantage. Introduction In the area of spread spectrum communications research eorts on Code Division Multiple Access (CDMA) systems have steadily increased in recent years. Originally used in military systems these Spread Spectrum systems are also investigated with regard to dierent commercial applications, e.g. cellular mobile radio systems or wireless indoor communication []. All the users share the same frequency band, so they are not orthogonal, in general. To achieve optimal performance, all the users have to be detected jointly. This Maximum Likelihood (Sequence) detector was found by Verdu [2]. It was further shown that the optimal detector is able to cope with different received energies []. So, the near-far problem is not inherent in CDMA. Rather, it is just the inability of the conventional detector to exploit the interuser interference. A lot of work was done on suboptimal multiuser detectors in the last years. Up to now almost always uncoded systems were considered. Recently, the Maximum Likelihood detector for a system with convolutional codes was presented [4]. It was shown there that the optimal algorithm consists of a trellis with a number of states which is the product of the number of users and the constraint length of the codes. Detection (user separation) and decoding must be done jointly. But for practical applications, the complexity is too large. So, suboptimal detectors have to be constructed. The easiest method is to separate multiuser detection and decoding. The user separation is done bitwise or symbolwise, the decoding is done for K users independently. Asymptotically, an uncoded DS-CDMA system is equivalent to a single user system with reduced energy []. The reduction factor is called asymptotic eciency. In section 2 we derive the asymptotic order of the error probability for a synchronous DS-CDMA system with BPSK modulation and binary block coding for the optimal joint detector and decoder and AWGN. It turns out, that a coded system is also equivalent to a single user system with reduced energy. The energy reduction depends on the code properties and leads to the denition of asymptotic distance. In section upper and lower bounds for the asymptotic distance are derived. It is shown how the codes of the users can be constructed jointly to reduce the energy loss. In section 4 binary linear codes based on these criteria are constructed. It is further shown in section 5, that separated multiuser detection and decoding leads necessarily to a bad asymptotic error probability. A suboptimal low complexity algorithm which does not suffer from this disadvantage is proposed. Simulations are used to illustrate the benets of the methods presented. 2 Joint Maximum Likelihood Detection and Decoding We assume BPSK modulation and perfect bit synchronization. Each user k is assigned a signature waveform s k (t); t 2 [; T ] with energy! k and a binary linear block code C k with length n and Hamming weight d H. We assume for convenience, that the code parameters are the same for all codes. However, the K codes may be dierent. The mapping from binary to real symbols is done in the usual way:! ;!? Modulated codewords are denoted as c m k for user k and the corresponding modulated code as Ck m. If all the users transmit over a white gaussian multiple access channel, the received signal can be written as KX r(t) = c m k (i)s k (t? it ) n(t) k= i= where c m k (i) is the ith modulated code symbol for user k and n(t) is additive white gaussian noise with variance =2. If all codewords of all the users are equally likely a sucient statistic for a Maximum Likelihood decision are the outputs of a bank of matched lters: y k (i) = Z T r(t it )s k (t)dt

2 These outputs can be written in a more convenient form: with y(i) = Hc m (i) n(i) y(i) = (y(i); ; y K (i)) T ; c m (i) = (c m (i); ; c m K(i)) T and n(i) is a K-dimensional vector with zero mean and covariance matrix 2 H. H is a crosscorrelation matrix with components Z T H ij = s i (t)s j (t)dt The joint Maximum Likelihood detector and decoder selects the most likely modulated codeword matrix ^C m opt = (^c m opt() ^c m opt(n)): ^C m opt = arg max p(y(); ; y(n)j^c m (); ;^c m (n)) = arg max (2y(i) T^c m (i)? ^c m (i) T H^c m (i)) i= {z } =: L(^C m ) Each term in the sum corresponds to the ML metric in the uncoded case [5]. Now we want to consider the probabilty that an error is made by the ML approach. It is assumed that the codeword matrix C m was sent by the K users and ^C m = C m?2e was selected to be the most likely one. An error between two codewords c m i und ^c m i is dened as e i = 2 (cm i? ^c m i ) 2 f?; ; g n The probability that an error event E, where E is a matrix consisting of the errors of all users, might happen is Lemma : P (L(^C m ) > L(C m )) = Q vu u t 2 e(i) T He(i) i= e(i) denotes the ith column of the matrix E. The proof is given in [6]. A The term e(i) T He(i) is the necessary noise power to cause the error pattern e(i) for the K users in the ith symbol. If only one user were active then e(i) T He(i) reduces to P e 2 (i) =: w H, where w H is the Hamming weight of the error for user. Then Lemma is equal to the probability that w H errors occur for single user transmission. Let without loss of generality the probability of error for user be of interest. This probability can be easily bounded by applying the union bound and Lemma. Lemma 2: X P (^c m 6= c m ) Q E;e 6= vu u t 2 e(i) T He(i) i= A where e is the rst row of the matrix E and the sum goes over all possible error matrices. For low signal to noise ratios this upper bound for the probability of error is loose, but for large signal to noise ratios the bound is rather tight, because the union bound was applied. In this region that term in the sum with smallest argument determines the behaviour of the probability of error. So far the upper bound is valid for xed energies of all the users. In mobile applications the received energies are not constant, but they vary, because the power control might not be perfect. The capability of the receiver to cope with the worst energy constellation of the interfering users is known as near-far resistance []. Therefore we dene: Denition: Let! be the energy per bit of user and let w = (!2; ;! K ) be the energies of the interfering users. Then d A (CjC2; ; C K ) = min min E;e 6= w! e(i) T He(i) i= is called the asymptotic distance of the code C for the set of codes (C; ; C K ). If minimization is done only over E for xed energies, the term is called coded asymptotic eciency. This denition for coded transmission is equivalent to the denition of near-far resistance for uncoded transmission []. A similar denition for coded CDMA including the code rate was given in [4]. Note, that the coded asymptotic eciency is not upper bounded by as the asymptotic eciency in the uncoded case. We call this term a distance in the worst case, because later on it will turn out that the value of d A (CjC2; ; C K ) depends on the properties of the codes C k. With this denition it follows that the probability of decoding error for user is asymptotically of the same order as Q r 2Rd A E b with d A = d A (CjC2; ; C K ). For coded single user transmission the error probability is asymptotically of the same order as Q r 2Rd H E b R is the code rate and E b is the energy per information bit, i.e.! = R E b. So, for a multiple access situation with AWGN the Hamming distance d H has to be replaced by the asymptotic distance d A (CjC2; ; C K ). Bounds for the Asymptotic Distance Now some bounds for the asymptotic distance are derived. We assume that all the signature waveforms are linearly independent, then the crosscorrelation matrix H is positive denite, i.e. the matrix has only positive eigenvalues. The correlations between the K? interfering users are described by a (K? ) (K? ) dimensional submatrix. So, it is convenient to decompose the normalized!!

3 crosscorrelation matrix R in r T R = r R K? R = W?=2 HW?=2 with W = diag(!; ;! K ). With this decomposition the rst bound can be formulated: Theorem : The asymptotic distance is upper and lower bounded by d H (? r T R? K? r) d A(CjC2; ; C K ) d H Corollary : If C = = C K, then d A (CjC2; ; C K ) = d H (? r T R? K?r) For K = 2 it is then For a proof see [6]. d A (CjC2) = d H (? 2 ) This corollary says, that the worst asymptotic behaviour of the joint Maximum Likelihood detector and decoder is achieved, if all the K codes are the same. Now code properties for a good asymptotic distance are investigated. Let I = f2; ; Kg be the indices of the interfering users and I s a subset of I. We dene U;I s = fc : c 2 C ^ c 2 C j 8 j 2 I s g S;I s (k) = fc : c = c c 2 8 c 2 U;I s ^ c 2 2 C k g 8k =2 I s. is the mod 2 addition in GF(2). Further the matrix D Is = diag(d2; ; d K ) is dened with d k = dh (S;I s (k)) k =2 I s k 2 I s Is is the smallest (positive) eigenvalue of the matrix R if the rst and all rows and columns with indices in I s are cancelled. With these denitions the following theorem is proved in [6]: Theorem 2: For the asymptotic distance holds: with d A (CjC2; ; C K ) min I s dh (U;I s )(? r T A? I s r) A Is = R K? Theorem 2 simplies for K = 2: Is d H (U;I s ) D I s Corollary 2: For the asymptotic distance with K = 2 users holds with d A = d A (CjC2): d A min d 2 d H H? ; d H (U2)(? 2 ) d H d H (S(2)) These lower bounds state which parameters of the codes are important for a large asymptotic distance and therefore, for an at least asymptotically good error rate. If the Hamming distances of the intersection codes are larger than the Hamming distance of the original code than the asymptotic distance will be larger than the lower bound. Also large Hamming distances of the sum codes reduce the energy loss due to interuser interference further. 4 Code Constructions In the last section it has been shown what the important parameters for a large asymptotic distance are. Now we construct some codes for K = 2 and K = based on these distances. All the codes are BCH codes, spectral techniques are used to describe them [7]. Example: The rst code is a (7,4,) Hamming Code. We choose the zeros in the spectral domain as given in the table below. The intersection code must contain the zeros of both codes. The sum code has a zero at those positions where both codes have a zero: A A A2 A A4 A5 A6 C C2 U2 S(2) From this table it follows, that the intersection code U2 consists of the all zero and the all one word and has distance d H (U2) = 7. The sum code ist the complete GF(2) 7 and has just distance d H (S(2)) =. If both d A equal code bound lower bound Figure : lower bound and equal code bound for asymptotic distance of (7,4,) Hamming code codes are the same, then d A (CjC2) = d H (? 2 ) by Corollary. Figure shows the gain of the constructed codes by using the lower bound of Corollary 2 compared to the case, where both user share the same code. Example: The second example for K = 2 is a (5,9,) BCH code. For the two user codes we choose the zeros as shown in the table: A A A2 A A4 A5 A6 C C2 U2 S(2) A7 A8 A9 A A A2 A A4 C C2 U2 S(2)

4 The intersection code is a (5,5,7) BCH code with d H (U2) = 7, the sum code is a (5,,2) BCH code with d H (S(2)) = 2. Here the coding gain is still larger (g. d A equal code bound lower bound Figure 2: lower bound and equal code bound for asymptotic distance of (5,9,) BCH code 2), because the distance of the sum code is 2 compared to in the former example. Example: For more than two users, there should at least be one pair of users, who have dierent codes. Let us choose K = with the following zeros in the spectral domain: A A A2 A A4 A5 A6 C C2 C U2 U U2 S2() S(2) S(2) S() The matrices D Is are D ; = D = ; D 2 = ; D 2 = The intersection codes and the sum codes are either the original Hamming code or the (7,,7) repetition code. The examples show, that on the one side dierent zeros in the spectral domain are needed for large Hamming distances of the intersection codes. On the other side there should be joint zeros, so that the sum codes have a large Hamming distance. 5 Remarks on Suboptimal Receivers In this section we turn to suboptimal receivers. As already mentioned, a suboptimal receiver may consist of a multiuser detector and K independent decoders. A lot of research has been done in the last years on suboptimal multiuser detectors. For an overview on this research area see [8]. We will demonstrate that any separation of detection and decoding is not able to use the benets of the code constructions of the last section, even if the ML detector and ML decoders are used. Instead, some form of multistage decoding could be used. Only the two user case is considered, because this example is sucient to illustrate the eects. The Maximum Likelihood detector and Maximum Likelihood decoders are used. The detector provides hard values to the decoders, but it is also possible to use soft outputs of the detector. For large signal to noise ratios, the worst case bit error probability p after multiuser detection for user is bounded by [5] p constq s 2(? 2 )RE b A conste? (? 2 )RE b After decoding the word error probability p c for large signal to noise ratios is approximately: p c const p dh=2 const?(? 2 )d H RE b e 2 Here the asymptotic behaviour is determined by the factor (? 2 )d H =2. This is even below the equal code bound from section and does not depend on the codes used. Thus the word error rate is at least asymptotically independent on the codes. The equal code bound can be achieved with soft detector outputs [9], but not exceeded, even if the asymptotic distance is larger. The reason, why the asymptotic eciency of this suboptimal receiver is as small as possible is, that the multiuser detector does not take into account the code construction. Due to this construction, it is not possible that after decoding d H errors occur in both codewords at the same places. The errors must dier in at least one place or both codewords must contain more than d H errors. That means, a larger noise power is needed to lead to wrong codewords for both users compared to the equal code case. Thus, the decoded codewords can be used for successive cancellation. This leads to the following algorithm: use matched lter outputs by any suboptimal multiuser receiver for the n codesymbols independently to detect the two users decode the two users with independent decoders. The results are ^c m and ^c m 2, respectively.

5 subtract p!!2 ^c m from the n matched lter outputs for user 2 and p!!2 ^c m 2 from the n matched lter outputs for user. These are the corrected matched lter outputs. use the signs of the corrected matched lter outputs for a second decoding step by independent decoders. The results are ^c m2 and ^c m2 2, respectively. choose arg max (L(^c m,^c m2 2 ), L(^c m2,^c m 2 )) as nal result. The probability that both decoding results in the rst step are wrong determines the behaviour of the algorithm. This probability is reduced for the constructed codes, thus the algorithm is able to benet from the construction. The extension of this algorithm to K users is straightforward. Soft values can also be used in each step. 6 Simulation Results First we consider a two user system. Each user has a (7,4,) Hamming code. The correlation between the signature waveforms is 5/7 and the received energies are equal. The error probability is not as bad as possible, because the worst case interfering energy is less than that of the considered user [5]. Nevertheless the eective energy of user is reduced, because the crosscorrelation is large. Figure shows the bit error probability if both users share the same code and if the dierent Hamming codes of section 4 are used. It is seen that the joint code construction leads to a gain of approximately db for a bit error rate of?. Although the theory is only asymptotically valid, a gain is achieved for moderate signal to noise ratios... BER.. equal codes di. codes e Figure : joint ML detection and decoding Now we consider a suboptimal receiver with separated multiuser detection and decoding. Because we just want to demonstrate the inability of this approach to benet from the code construction we use the ML principle in both detection and decoding. The decoders use reliability information about the detection results. As predicted in the former section, the loss is the same for equal and dierent codes (g. 4)... BER.. equal codes di. codes e Figure 4: separated ML multiuser detection and ML soft decoding In Figure 5 the error rates for the multistage algorithm are shown. Here, also soft values are used for each stage. It is seen, that the performance for dierent codes is better than that for equal codes. Thus, the proposed algorithm is able to benet from the code construction... BER.. equal codes di. codes e Figure 5: proposed multistage algorithm Finally, the three user example of section 4 is considered. The crosscorrelation matrix R is chosen to be R = 5=7 =7 5=7 5=7 =7 5=7 A The matrices D Is are given in section 4. The eigenvalues are ; = 2 7 ; 2 = ; = 2 does not exist. Applying Theorem 2 gives a lower bound of.597 for the asymptotic distance. Figure 6 shows the error performance for this example. The energies of the interfering users are chosen to be:!2 =

6 25=6! and! = =6!. Although users and share the same code, there is a gain of nearly db to the equal code case (g. 6). If all the codes are the same, the asymptotic distance is exactly.429. Now the distribution of the codes to the users is changed. If users and 2 shared the same code, the lower bound for the asymptotic distance would be.445. Here only a small gain can be achieved. The third possibility is, that users 2 and share the same code. Then the lower bound is.596. The case that users and have the same code is the best regarding the asymptotic distance. The simulation results in gure 6 and 7 show the same behaviour... BER.. equal codes C = C e Figure 6: joint ML detection and decoding for three users.. BER.. C = C 2 C 2 = C e Figure 7: joint ML detection and decoding for three users The lower bounds for the asymptotic distances and the coded asymptotic eciencies for the chosen energies are listed in the table below: codes lower bound asymptotic eciency C = C2 = C C = C C = C C2 = C Summary First the asymptotic order of the error probability of joint Maximum Likelihood detection and decoding was derived. It turned out that this order is dependent on the codes. Therefore, we introduced the denition of asymptotic distance. For this distance upper and lower bounds were given. In the two user case, the exact asymptotic distance can be calculated [6]. It was shown that the asymptotic order of the error probability is minimal if all users share the same code. For dierent codes the asymptotic distance can be larger than the lower bound if the codes are constructed jointly. That means, the error performance is at least asymptotically better than in the equal code case. Further separated multiuser detection and decoding was investigated. We explained, why this approach is not able to achieve the designed distance. We stated an algorithm based on multistage decoding, which achieves a better asymptotic order. Simulations illustrated the theoretical results. References [] A. J. Viterbi, CDMA. Addison-Wesley, 995. [2] S. Verdu, \Minimum probability of error for asynchronous gaussian multiple-access channels," IEEE Trans. Inform. Theory, vol. 2, pp. 85{96, January 986. [] S. Verdu, \Optimum multiuser asymptotic eciency," IEEE Trans. Commun., vol. 4, pp. 89{897, September 986. [4] T. Giallorenzi and S. Wilson, \Multiuser ML sequence estimator for convolutionally coded asynchronous DS- CDMA systems," IEEE Trans. Commun., vol. 44, pp. 997{8, August 996. [5] R. Lupas and S. Verdu, \Linear multiuser detectors for synchronous code-division multiple-access channels," IEEE Trans. Inform. Theory, vol. 5, pp. 2{ 6, January 989. [6] S. Bruck and U. Sorger, \Binary codes with inner spreading for gaussian multiple access channels," in Int. Seminar on Coding Theory, Thahkadzor, Armenia, Oct [7] R. E. Blahut, Theory and Practice of Error Control Codes. Addison-Wesley, 984. [8] A. Klein, Multi-user detection of CDMA signals - algorithms and their application to cellular mobile radio. PhD thesis, University of Kaiserslautern, 996. [9] S. Bruck, \Multiuser detection for coded transmission and AWGN," in Winter School on Coding and Information Theory, Moelle, Sweden, Dec. 996.