Optimal Binary/Quaternary Adaptive Signature Design for Code-Division Multiplexing

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1 Globecom 0 - Wireless Communications Symposium Optimal Binary/Quaternary Adaptive Signature Design for Code-Division Multiplexing ili Wei, Member, IEEE and Wen Chen, Senior Member, IEEE Department of Electronic Engineering, Shanghai Jiao Tong University, China {liliwei, wenchen}@sjtu.edu.cn Abstract When data symbols modulate a signature waveform to move across a channel in the presence of disturbance, the adaptive real/complex signature that maximizes the signalto-interference-plus-noise ratio (SINR at the output of the maximum-sinr filter is the minimum-eigenvalue eigenvector of the disturbance autocovariance matrix. In digital communication systems the signature alphabet is finite and digital signature optimization is NP-hard. In this paper, we propose a new adaptive binary signature assignment for CDMA systems based on improved Fincke-Pohst (FP algorithm that achieve the optimal exhaustive search performance with low complexity. Then, we extend and propose the optimal adaptive signature assignment algorithm with quaternary signature sets. Simulation studies included herein offer performance comparisons with known adaptive signature designs and the theoretical upper bound of the complex/real eigenvector maximizer. Index Terms code-division multiplexing, spread-spectrum communications, signature sets, signal-to-interference-plus-noise ratio (SINR. I. INTRODUCTION In recent years, there has been renewed interest in optimal signature sets for the growing number of codedivision multiplexing applications. In the theoretical context of complex/real-valued signature sets, the early work of Welch [] on total-squared-correlation (TSC bounds was followed up by direct minimum-tsc design proposals []-[3] and iterative distributed optimization algorithms [4]-[6]. Minimum-meansquare-error (MMSE minimization is used for the design of signature sets for multiuser systems over multipath channels in [7]. Recently, new bounds on the TSC of binary signature sets were found [8] that led to minimum-tsc optimal binary signature set designs for almost all signature lengths and set sizes [8]-[0]. New bounds and optimal designs for minimum TSC quaternary signature sets are derived in []. Instead of previous static binary/quaternary signature design, we consider the NP-hard problem of finding the adaptive binary/quaternary signature in the code division multiplexing system with interference and multipath fading channels, that maximizes the SINR at the output of the maximum-sinr filter. Direct binary quantization of the minimum-eigenvalue eigenvector is proposed in [3]. The rank--optimal proposal that constructs binary signature based on two smallesteigenvalue eigenvectors is described in [4]. The slowest This work is supported by the National 973 Project #0CB3606, by NSF China # and #663059, by the National 973 Project #009CB descent method (SDM based adaptive binary signature assignment is presented in [5]. In this paper, we propose a new adaptive binary signature assignments based on improved Fincke-Pohst (FP algorithm. Instead of exhaustive searching, FP algorithm [6] enumerates all vectors lie in a suitable ellipsoid, with complexity of polynomial order in the searching dimension. FP algorithm has been modified and first applied to communication problems of lattice code decoder in [7], then used for MIMO and space-time codes decoder in [8], generally known as sphere decoding algorithm. In this work, we improve and apply FP algorithm in our adaptive finite-alphabet signature design problem. In addition, we extend our proposed adaptive binary signature assignment to adaptive quaternary signature assignment. We prove that, in general, the adaptive quaternary signature assignment with length can be equivalent to an adaptive binary signature assignment with length. II. SYSTEM MODE We consider a multiuser CDMA-type environment with processing gain/signature length, where K signals/users transmit simultaneously in frequency and time. Each user transmits over N resolvable multipath fading channels. Assuming synchronization with the signal of user k, upon carrier demodulation, chip matched-filtering and sampling at the chip rate over a presumed multipath extended data bit period of + N chips, we obtain the received vector r(m E k b k (mh k s k +z k +i k +n, m 0,,..., ( where b k (m {±}, m 0,,,... is the mth data bit; E k represents transmitted energy per bit period; s k is the signature assigned to user k, for binary alphabet s k {±} and for quaternary alphabet s k {±, ±j} where j. H k C (+N is the user k channel matrix of the form h k, H k h k, h k, h k,n h k,n 0 0 h k,n h k, h k,n with entries h k,n, n,..., N, complex Gaussian random variables to model fading phenomena; z k C +N rep- ( 447

2 resents comprehensively multiple-access-interference (MAI to user k by the other K users, i.e. z k K i i k Ei b i (mh i s i ; i k C +N denotes multipath induced inter-symbol-interference (ISI to user k by its own signal; and n is a zero-mean additive Gaussian noise vector with autocorrelation matrix σ I +N. Information bit detection of user k is achieved via linear minimum-mean-square-error (MMSE filtering (or, equivalently, max-sinr filtering as follows ˆbk sgn ( Re { w H MMSE,kr } (3 where w MMSE,k cr H k s k C +N, R E{r r H }, c > 0, { } H is the Hermitian operator, Re{ } denotes the real part of a complex number, and E{ } represents statistical expectation. The output SINR of the filter w MMSE,k is given by { w ( H } E MMSE,k Ek b k H k s k SINR MMSE,k (s k { w E H MMSE,k (z k + i k + n } E k s H k H H k R k H ks k (4 where R k E {(z k + i k + n (z k + i k + n H} is the autocorrelation matrix of combined channel disturbance. For mathematical convenience we disregard the ISI component in R k and approximate R k by R k E {(z k + n (z k + n H}. For notational simplicity we define the matrix Q k H H k R k H k. (5 Then, the output SINR in (4 can be rewritten as SINR MMSE,k (s k E k s H k Q k s k. (6 Our objective is to find the signature s k that optimizes (maximizes SINR MMSE,k of (6, in binary alphabet s k {±} and quaternary alphabet s k {±, ±j} respectively. For the case of binary alphabet s k {±}, let Q kr denote the real part of the complex matrix Q k, i.e. Q kr Re{Q k }. The binary signature s k {±} that maximizes SINR MMSE,k of (6 is equivalent to s (b arg max s k {±} s T k Q k s k arg max s k {±} s T k Q kr s k. (7 The superscript (b indicates that s (b is binary; { }T is the transpose operator. The quaternary signature s k {±, ±j} that maximizes SINR MMSE,k of (6 is given by s (q arg max s H s k {±,±j} k Q k s k, (8 the superscript (q indicates that s (q is quaternary. III. OPTIMA BINARY SIGNATURE ASSIGNMENT Regarding the binary optimization in (7, we first do the follow transformation s (b arg max s {±} s T Q kr s arg min s {±} s T (αi Q kr s, (9 where α is a parameter greater than the maximum eigenvalue of the matrix Q kr and let W αi Q kr. (0 By definition, matrix W is Hermitian positive definite. The Cholesky s factorization of matrix W yields W B T B, where B is an upper triangular matrix. Then equation (9 will lead to s (b arg min s {±} s T Ws arg min s {±} B s F, ( where F denotes the Frobenius norm. The original Finche-Pohst (FP algorithm [6] searches through the discrete points s in the -dimensional Euclidean space which make the corresponding vectors z Bs inside a sphere of given radius C centered at the origin point, i.e. Bs F z F C. This guarantees that only the points that make the corresponding vectors z within the square distance C from the origin point are considered in the metric minimization. Compared with the original FP algorithm in [6], we have two main modifications: (i The original FP algorithm are searching for integer points, i.e. s Z, while our searching alphabet is antipodal binary, i.e. s {±}. Hence, the bounds to calculate each entry are modified, or further tightened to make the algorithm work faster; (ii According to the binary signature vector obtained by applying the direct sign operator [3] on the real maximum-eigenvalue eigenvector of Q kr, denoted as Quantized Binary or s (b quant, we can have a very proper square distance setting as C s (b T (b quant W s quant, ( such that the searching sphere radius is big enough to have at least one signature point fall inside, while in the meantime small enough to have only a few signature points within. et b ij, i,,,, j,,, denote the entries of the upper triangular matrix B; let s i, i,,, denote the entries of searching signature s. According to (, the signature points that make the corresponding vectors z Bs inside the given radius C can be expressed as 448

3 The entries s, s,, s are chosen as follows: for a s T Ws B s F b ii s i + b ij s j chosen s, we can choose a candidate for s satisfying its i bounds requirements as in (9 for k. If a candidate for s does not exist, we go back to choose other s. Then g ii search for s that meets the bounds requirement for this i new s. We follow the same procedure to choose s, and k so on. When a set of s, s,, s is chosen, one signature g ii + g ii candidate vector s [s, s,, s ] T is obtained. We record ik i all the candidate signature vectors such that the entries satisfy C (3 their bounds requirements, and choose the one that gives the where g ii b ii and g ij b ij /b ii for i,,,, j i + smallest s T Ws metric.,,. Obviously the second term of (3 is non-negative, Note that this searching procedure will return all candidates hence, to satisfy (3, it is equivalent to consider for every that satisfy s T Ws C and gives the one with minimum k,,,, value. There is at least one candidate vector s (b quant satisfying all the bounds requirements, since that is how we set C in (. On the other hand, the optimal exhaustive binary search g ii C. (4 result s (b exhaustive will also fall inside the search bounds, since ik Then, we can start work backwards to find the bounds for signature entries s, s,, s one by one. To evaluate the element s k of the signature vector s, referring to (4 we will have g ii C, (5 that leads to s k gkk If we denote ik ( C ( ik+ g ii s i + g ijs j jk+ g kjs j gkk k ( C ( ik+ g ii s i + g ijs j jk+ g kjs j (6. jk+ C g kj s j, (7 ik+ g ii s i + g ij s j + g k+,k+ ( k+ + s k+, (8 and take consideration of s k {±}, the bounds for s k can be expressed as B k s k UB k, (9 where UB k min B k max ( (,,,. (0 Note that for given radius C and the matrix W, the bounds for s k only depends on the previous evaluated s k+, s k+,, s. s (b T (b exhaustive Ws exhaustive T s(b (b quant W s quant C. ( Hence, we are guaranteed to find the optimal exhaustive binary search result by the proposed improved FP algorithm. Algorithm FP Based Binary Signature Design Algorithm For the binary signature optimization of s (b arg max s {±} s T Q kr s: Step : et q k, be the real maximum-eigenvalue eigenvector of Q kr with eigenvalue λ k,. Apply the direct sign operator on q k, and obtain s (b quant sgn(q k,. Then construct matrix W as W αi Q kr, where α is a parameter set greater than the maximum eigenvalue of the matrix Q kr, i.e. α > λ k,. Set the square distance as C s (b quant T (b Ws quant. Step : Operate Cholesky s factorization of matrix W yields W B T B, where B is an upper triangular matrix. et b ij, i,,,, j,,, denote the entries of matrix B. Construct a new upper triangular matrix G where g ii b ii, g ij b ij /b ii, for i,,,, j i +,,. Step 3: Search the candidate vector s with entries s,, s according to the following procedure. Input: Matrix W, Matrix G and the radius C obtained by Step and Step. Output: The vector s min {±} satisfies s T Ws C and gives the minimum s T Ws metric. (i Start from 0, C C, metric C, s min s (b quant and k. (ii Set the upper bound UB k and the lower bound B k as follows UB k min B k max ( (,,,, 449

4 and s k B k. (iii Set s k s k +. If s k 0, set s k. For s k UB k, go to (v; else go to (iv. (iv If k, terminate and output s min ; else set k k + and go to (iii. (v For k, go to (vi; else set k k, and k jk+ where c g kj s j, + g k+,k+ ( k+ + s k+ then go to (ii. (vi We get a candidate vector s that satisfies all the bounds requirements. If s T Ws metric, then update s min s and metric s T Ws. Go to (iii. Step 4: Once we get the optimal s min from Step 3 that returns the minimum s T Ws metric, the optimal adaptive binary signature that maximizes the SINR at the output of MMSE filter is s (b s min. Note that the solution obtained through this algorithm is guaranteed to be optimal, which means the same as exhaustive searching. IV. OPTIMA QUATERNARY SIGNATURE ASSIGNMENT For the quaternary signature optimization of (8, a heuristic approach will be direct quantization signature vector obtained by applying the sign operator on real part and imaginary part of the complex maximum-eigenvalue eigenvector of Q k. However, this is a suboptimal approach and the performance is inferior as shown in simulation section. For a quaternary signature s {±, ±j}, a transform is made as s ( jc, ( such that c { j, + j, j, + j}. Note that if the real part and imaginary part of vector c are denoted as c R Re{c} and c I Im{c}, the transform will lead to c R {±} and c I {±}, two binary antipodal sequences. By definition, Q k in (8 is Hermitian positive definite. So we can operate on the matrix of Q k Cholesky decomposition Q k U H U, (3 where U is an upper triangular matrix. Then ( H ( s H Q k s ( jc Q k ( jc Uc F. (4 Define y Uc and let y R Re{y} and y I Im{y}, U R Re{U} and U I Im{U}. Then, it is easy to obtain the following equation [ ] [ ] [ ] yr UR U I cr. (5 y I U I U R Hence, combining equations (4 and (5 will lead to c I s H Q k s [ ] [ ] UR U I cr U I U R c I F [ ] T [ ] T [ ] cr UR U I UR U I c I U I U R U I U R }{{}}{{} c T Q kr [ cr c I [ ] cr, (6 c I }{{} c ] {±} is a binary signature with length. Note that c c R + jc I is complex signature with length. Now, as shown in (6, the quaternary signature optimization with length in (8 is transformed into the following binary signature optimization problem with length c (b opt arg max c T QkR c. (7 c {±} After we get the optimal [ binary sequence ] c (b opt of length, split c (b opt into c (b c (b opt c (b, where c (b and c(b I,opt I,opt are binary sequences in length, i.e. c (b {±} and c (b I,opt {±}. Then, the optimal quaternary signature can be constructed as s (q opt ( j ( c (b + jc(b I,opt. (8 We summarize the quaternary optimization problem of (8 and propose the following algorithm. Algorithm FP Based Quaternary Signature Design Algorithm For the quaternary signature optimization of s (q arg max s {±,±j} s H Q k s: Step : We operate on the complex matrix of Q k Cholesky decomposition Q k U H U. et U R Re{U} and U I Im{U}. Construct real matrix Q kr as follows Q kr [ ] T [ ] UR U I UR U I. U I U R U I U R Step : Solve the following binary signature optimization problem with signature length based on FP Based Binary Signature Design Algorithm c (b Step 3: Split c (b opt opt arg max c T QkR c. c {±} [ ] c (b c (b I,opt, where c (b and c(b I,opt are binary sequences in length. Then, the optimal quaternary signature can be constructed as s (q opt ( ( j c (b + jc(b I,opt. According to the same analysis as in the previous section, FP Based Quaternary Signature Design Algorithm is also 4430

5 guaranteed to find the optimal exhaustive searching result with complexity order of polynomial in the signature length. We note that by using the same quaternary binary equivalence procedure, we can also extend our previous proposed SDM Based Binary Signature Design Algorithm in [5] to solve the quaternary signature optimization of (8. We denote it as SDM Based Quaternary Signature Design Algorithm which performance comparisons will follow in the simulation studies section. V. SIMUATION STUDIES We first compare performance of adaptive binary signature assignment of the following benchmarks: (i Real max- EV : The real maximum-eigenvalue eigenvector of Q kr Re{Q k }; (ii Exhaustive Binary : The binary signature assigned by exhaustive search; (iii Quantized Binary : The binary signature obtained by applying the sign operator on the real maximum-eigenvalue eigenvector of Q kr [3]; (iv Rank Binary : The adaptive rank- binary signature design [4]; (v SDM Based Binary Algorithm : The adaptive binary signature assignment in [5] based on the top P real maximum-eigenvalue eigenvectors; (v FP Based Binary Algorithm : The optimal binary signature assignment proposed in this paper. We consider a code-division multiplexing multipath fading system model with 6 and N 3. The signal power of the user of interest is set to E 0dB, while E, E 3,, E K are uniformly spaced between 8dB and db. The interfering spreading signatures are randomly generated. For comparison purposes, we evaluate the SINR loss, the difference between SINR of the optimal real signature (Real max-ev and other adaptive binary signature assignment algorithms. The results that we present are averages over 000 randomly generated interferences and channel realizations. In Fig., we plot the SINR loss as a function of the number of interferences. We observe that SDM based binary algorithm and FP based binary algorithm offer superior performance than direct quantized binary and rank assignments. Furthermore, FP based binary algorithm actually achieves exactly the optimal exhaustive binary search assignment as we expected. Then we investigate the multiuser binary signature assignment in a sequential user-after-user manner based on various adaptive binary signature assignments. In such an approach, each user s spreading signature is updated one after the other. Several multiuser adaptation cycles are carried out until numerical convergence is observed. In Fig., for a total of K 8 users, we plot the SINR loss of one user of interest as a function of multiuser adaptation cycle. Still, SDM based binary algorithm and FP based binary algorithm offer superior performance than the direct quantized binary and rank assignments. FP based binary algorithm achieves exactly the optimal exhaustive binary search assignment. We repeat our studies for adaptive quaternary signature assignment algorithms: (i Complex max-ev : The complex maximum-eigenvalue eigenvector of Q k ; (ii Exhaustive Quaternary : The quaternary signature assigned by exhaustive SINR oss(db Quantized Binary Rank Binary SDM Based Binary Algorithm FP Based Binary Algorithm Exhaustive Binary Number of Interferences Fig.. SINR oss of various adaptive binary signature assignments versus number of interferences (6. SINR oss (db Quantized Binary Rank Binary SDM Based Binary Algorithm FP Based Binary Algorithm Exhaustive Binary Multiuser Adaptation Cycle Fig.. SINR oss of various adaptive binary signature assignments versus multiuser adaptation cycle (6, K8. search; (iii Quantized Quaternary : The quaternary signature obtained by applying sign operator on real part and imaginary part of the complex maximum-eigenvalue eigenvector of Q k, mentioned in the beginning of section IV; (iv SDM Based Quaternary Algorithm : The SDM based quaternary signature design based on the quaternary-binary equivalence procedure and the application of SDM based binary signature assignment in [5]; (v FP Based Quaternary Algorithm : The quaternary signature design algorithm proposed in this paper as FP Based Quaternary Signature Design Algorithm. The SINR loss for 443

6 SINR oss(db Quantized Quaternary SDM Based Quaternary Algorithm FP Based Quaternary Algorithm Exhaustive Quaternary Number of Interferences Fig. 3. SINR oss of various adaptive quaternary signature assignments versus number of interferences (8. SINR oss (db Quantized Quaternary SDM Based Quaternary Algorithm FP Based Quaternary Algorithm Exhaustive Quaternary Multiuser Adaptation Cycle Fig. 4. SINR oss of various adaptive quaternary signature assignments versus multiuser adaptation cycle (8, K4. quaternary assignments are the difference between SINR of the optimal complex signature (Complex max-ev and other adaptive quaternary assignment algorithms. We plot the SINR loss as a function of the number of interferences in Fig. 3, and as a function of multiuser adaptation cycle in Fig. 4. We obtain the same results as previous adaptive binary simulations. We propose a new adaptive binary signature assignments based on improved FP algorithm, that returns the optimal exhaustive searching result with low complexity. In addition, we extend to adaptive quaternary signature assignments and prove that, the adaptive quaternary signature assignment with length can be equivalent to an adaptive binary signature assignment with length. Simulation studies show the comparisons with the optimal FP based binary/quaternary signature assignment and previous existing signature assignments. REFERENCES []. R. Welch, ower bounds on the maximum cross correlation of signals, IEEE Trans. Info. Theory, vol. IT-0, pp , May 974. [] J.. Massey and T. Mittelholzer, Welch s bound and sequence sets for code division multiple access systems, Sequences II: Methods in Communication Security, and Computer Science, vol. 47, pp , Springer-Verlag, New-York, 993. [3] M. Rupf and J.. Massey, Optimum sequence multisets for synchronous code-division multiple-access channels, IEEE Trans. Info. Theory, vol. 40, pp. 6-66, July 994. [4] S. Ulukus and R. D. Yates, Iterative construction of optimum signature sequence sets in synchronous CDMA systems, IEEE Trans. Info. Theory, vol. 47, pp , July 00. [5] C. Rose, S. Ulukus, and R. D. Yates, Wireless systems and interference avoidance, IEEE Trans. Wireless Commun., vol., pp , July 00. [6] P. Anigstein and V. Anantharam, Ensuring convergence of the MMSE iteration for interference avoidance to the global optimum, IEEE Trans. Info. Theory, vol. 49, pp , Apr [7] J. I. Concha and S. Ulukus, Optimization of CDMA signature sequences in multipath channels,, in Proc. IEEE Vehic. Tech. Conf., vol. 3, pp , Rhodes, Greece, May 00. [8] G. N. Karystinos and D. A. Pados, New bounds on the total squared correlation and optimum design of DS-CDMA binary signature sets, IEEE Trans. Commun., vol. 5, pp. 48-5, Jan [9] C. Ding, M. Golin, and T. Klϕve, Meeting the Welch and Karystinos- Pados bounds on DS-CDMA binary signature sets, Designs, Codes and Cryptography, vol. 30, pp , Aug [0] V. P. Ipatov, On the Karystinos-Pados bounds and optimal binary DS- CDMA signature ensembles, IEEE Commun. ett., vol. 8, pp. 8-83, Feb [] G. N. Karystinos and D. A. Pados, The maximum squared correlation, total asymptotic efficiency, and sum capacity of minimum total-squaredcorrelation binary signature sets, IEEE Trans. Info. Theory, vol. 5, pp , Jan [] M. i, S. N. Batalama, D. A. Pados and J. D. Matyjas, Minimum totalsquared-correlatioin quaternary signature sets: new bounds and optimal designs, IEEE Trans. Commun., vol. 57, no., pp , Dec [3] T. F. Wong and T. M. ok, Transmitter adaptation in multicode DS- CDMA systems, IEEE J. Select. Areas Commun., vol. 9, pp. 69-8, Jan. 00. [4] G. N. Karystinos and D. A. Pados, Rank--optimal adaptive design of binary spreading codes, IEEE Trans. Info. Theory, vol. 53, no. 9, pp , Sept [5]. Wei, S. N. Batalama, D. A. Pados and B. W. Suter, Adaptive binary signature design for code-division multiplexing, IEEE Trans. Wireless Commun., vol. 7, no. 7, pp , July 008. [6] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comput., vol. 44, pp , Apr [7] E. Viterbo and J. Boutros, A universal lattice code decoder for fading channels, IEEE Trans. Info. Theory, vol. 45, no. 5, pp , July 999. [8] O. Damen, A. Chkeif, and J. C. Belfiore, attice code decoder for space-time codes, IEEE Commun. etters, vol. 4, no. 5, pp. 6-63, May 000. VI. CONCUSIONS 443