A Joint Design of Complex Signatures Sets for Multicellular Synchronous CDMA Network

A design using total squared correlation (TSC) measure for the overloaded complex spreading sequence sets on the uplink of multicellular synchronous CDMA systems is introduced. For equal power users, the design provides orthogonal Welch Bound Equality (WBE) complex spreading sequences with constant chip magnitude. We developed an algorithm, which allows obtaining a structured Gram matrix having a particular pattern for cross correlations of the multicellular signature sequence sets without alphabet constraints. By using the proposed algorithm spreading sequences with maximum equicorrelated amplitude are also obtained. The mathematical formalism is supported with selected numerical examples in the multicellular S-CDMA context.


I. INTRODUCTION
HE problem of designing complex spreading sequences (a.k.a codewords or signature waveforms) for single cell synchronous code division multiple access (S-CDMA) systems is a traditional one (see [9], [12] , [13], [14], and references therein).Recently, much attention has been paid to the problem of constructing and optimizing the signature sequences for synchronous (direct sequence) DS-CDMA systems [15], and [16] in order to understand the impact of such sequences on the wireless system performances.
The current CDMA technologies for second (2G) or third generation (3G) communication systems (all based on DS-CDMA, such that IS-95 A/B, cdma 2000, UTRA, W-CDMA, TD-CDMA) are suited only for low-speed continuous transmission applications such as voice, but not a good choice for high burst traffic, as in the case for future wireless communication systems (4G) [3], and [7].Therefore a new wave of research is required for novel CDMA complex spreading design techniques.
We focus on constructing complex valued signature sets for multicellular synchronous CDMA under the total squared Manuscript received July 12, 2006 correlation (TSC) criterion, which is a measure of multiple access interferences (MAI) [25] and [26].While the extension of some optimal results of from one cell to multiple cells is straightforward, the design of spreading sequences under TSC criterion is a more challenging task due to the amount of interference constraints that are considerably stricter than in the single-cell case 1 .
In a traditional deployment, base stations are located in the center of a cell.The coverage of a base station extends in both directions along the linear array.The decoding of any mobile user is undertaken by the nearest base station with co-channel interference simply treated as noise.All mobile users, which share the same two nearest base stations, will be grouped into the same cell.However, in the case of multicellular systems, we assume the cooperation of multiple base stations that share the same extended signal space and requiring that each base station to have its own power constraint.
We developed an algorithm, which allows obtaining a structured Gram matrix having a particular pattern for cross correlations of the multicellular signature sequence sets without alphabet constraints.It is an extension of the one presented in [1] in single cell case and it overcomes the limitations of the previous algorithms presented in [6], [15]- [17].It is based on Gram matrix approach using TSC ( [4], [5]) as a design criterion and Givens rotations [21].By using this algorithm it is possible to obtain complex spreading sequences with maximum equicorrelated amplitude and constant chip magnitude in the absence of multipath.It is different from the algorithms presented in [8] and [24] which are based on the inverse eigenvalue problem [22], and [15].The extension of the proposed algorithm for multipath channels [23] and unequal power users for each cell is an open problem.
The merit of this algorithm is two fold.When it is used for real numbers, then real spreading sequences are obtained, as 2G or 3G multicellular CDMA systems require them [3].When it is used for complex numbers, then complex spreading sequences for uplink overloaded multicellular S-CDMA systems are obtained.For the particular case of the overloaded 1 Some performance measure of interest, such that signal-to-interference ratio (SINR) or spectral efficiency converges to some deterministic values, which are functions of the empirical eigenvalue distribution of random matrices associated to signature spreading design.It should be noted here that the optimum spectral efficiency (without spreading) could be achieved with orthogonal spreading sequences when TSC is zero and the system load is unity.It is also known that even when the system load is higher then the unity, there exist spreading codes that incur no loss in capacity relative to multiaccess with no spreading.See also the footnote 2. S-CDMA when all the users in a cell have the same power, this algorithm is used to obtain complex spreading sequence sets with constant chip magnitude [1], [2], and [6].
For the future wireless communication systems beyond the third generation (B3G) and 4G, where the dominant load will be high speed burst type traffic, the complex spreading sequences with constant chip magnitude for fast start-up equalization and channel estimation ( [10], [11]) are definitely needed.This is also desirable for multiple antenna wireless communications [3], and equal gain unitary space-time modulation [7].
The organization of this paper is as follows.In Section II the characteristics of the uplink overloaded multicellular S-CDMA systems model under the TSC criterion are described.The proposed design for constant chip magnitude is developed in Section III.The algorithm is developed in Section IV.Selected numerical results are given in Section V. Conclusions and future works are drawn in Section VI.

II. SYSTEM MODEL AND TOTAL SQUARED CORRELATION
Modeling the 4G CDMA network, including all the parameters such as signal propagation, fading, free space loss, etc., is a challenging task.In this paper, we focused on TSC metric, which is more tractable on the uplink than the sum capacity [16], and it was successfully applied in [25] for the next CDMA technologies in the presence of multipath.In addition, we assume that each base station has its own "policy" on power control assignment.However, for wireless cellular type systems we consider a more restrictive condition: the sum of transmitted power for each cell is constant one.In other words, the sum of elements on the main diagonal of the Gram matrix associated to each cell-spreading signature design is constant.This assumption will allow us to use a common signal space for all the cells signature design, while the powers of different signals in the multicellular system are vastly different.
We consider first the synchronous uplink vector multiple access channels (VMAC) of single-cell S-CDMA system with independent active users and the processing gain ( [15], [16]).In the presence of noise vector K N z , the received signal in one symbol interval is given by The model of uplink multiple cell synchronous S-CDMA systems used in this paper is a generalization of (1) and it is given by ( 2) (the same model was used in [25]).Our work is different from CDMA systems with random spreading, where both the number of users and the processing gain go to infinity, while their ratio goes to same finite constant.We consider the quasi-synchronous case where chip synchro ed.
nism g system received vector can be expressed as it is assum The lon where 1 ( ,..., ) ( , ,..., ) The composite spreading signature matrix S associated to multicellular system is o

sum of two matrices by considering
A and B , i.e., 0 If we consider a composite spreading matrix of the form in fact, we assume the case of equal spreading gain for each cell.This brings us back to single cell case in dimensional signal space considered in [6], [15] and [16].When model ( 2) is particularized to noncolaborative scenario it allows analyzing the case where in each cell the users have different spreading gain as in the future wireless CDMA networks.
eigenvalue, its determinant is zero, and Choleski ecomposition with pivoting can not be applied in order to e matrix d obtain th i S ) may be expressed as matrix mplex spreading sequences across cells are

S S S S S S S S (4)
where * is the complex conjugate transpose operation.The dimensionality of the extended signal space of the where the user co G G to be designed is 1 Given the composite spreading signature S , by definition, the TSC metric is the Frobenius norm of the Gram m trix (4) a ding signatures ulticellular of its eigenvalues associated to all sprea system and it is expressed in term TSC is inimized in [4], and [5] where for m single cell case, i.e. a generic cell , the corresponding spreading signature matrix i i S satisfies the following conditions a ) en ] , and [25] is the ula for multicellular systems, which still is an open problem.III.CONSTANT CHIP MAGNITUDE DESIGN l for unitary power users (6a) and unequal power users (6b), respectively.When the users have unitary power the condition (6a) corresponds to Welch Bound Equality sequences (WBE) [18], [19] and the condition (6b) corresponds to so called generalized WBE sequ ces [24 ( N i I identity matrix of order i N ) [21].TSC metric defined in ( 5) is a measure of the total amount of interferences in the multicellular S-CDMA system and it includes as a particular case the TSC criterion considered in [4] and [5] for a single cell.In addition, it is an alternative to sum capacity form It is instructive to focus on the single cell first.We will consider the cel i for convenience, after that the generalization to M cells follows.In the case of equally correlated signature sequences the cross-correlation ximum ac i The ma hievable crosscorrelation amplitude with K equally correlated signature sequences satisfies (8).For the set is called a simplex [18] and the corresponding Gram ngle cell overloaded CD For the si where MA system given in (1), and . ) ) , ij g j i j ρ = ± ≠ [18].
,…, 1 ( , , c s t given in (10) using the attern imposed in (11), the following system of equations in unknown ) The equation ( 12a) is a quadratic one whose discriminator depe on the values of i K and i N being associated to the first i N users.The equation (12b) corresponds to the next  The Jacobi method [21] of calculating the eigenvalues of a symmetric matrix use Givens (Jacobi) rotations.The idea behind Jacobi's method is to reduce systematically the off diagonal entries of a matrix.Under TSC design we are going in the opposite direction.The proposed design for constant chip magnitude overcomes the limitation of numerical search method based on generalized Lloyd algorithm developed in [6] du e to highly structured matrix ((7) in [6]) which does not satisfy the bandwidth constraint for each cell as it is discussed next.
id complex spreading sequences for erloaded multicellular CDMA on e following theorem [1] that allows constructing a structured ting matrices in (4)

IV. THE PROPOSED ALGORITHM
If we know the optimal eigenvalues of Gram matrix specified by ( 5), then based on (4), we need to construct a structured Gram matrix of the form required by ( 4), and at the same time, we need to preserve the bandwidth 6 of spreading sequences of each cell.The algorithm proposed in [15], (reproduced completely with its original proof in [17]) and those used in [16] based on the so called T-transform [20], may fail to generate val overloaded multicellular S-CDMA systems under TSC measure defined in [5].
The main drawbacks of the algorithms presented in [15]- [17] (when they are used for construction of the complex spreading sequences for ov systems) is the requirement of ordering the eigenvalues of composite Gram matrix [8].
Our algorithm generalizes the algorithms given in [1]; it is amenable to online implementation due to the finite number of steps required for convergence.Focusing on TSC, the proposed algorithm might be used for decentralized transmitter adaptation in multicellular systems and it does not require updates of sequences at each step.Hence it is suitable for a distributed implementation.The algorithm is based th orthogonal matrix necessary in genera .rotations. 6The bandwidth is specified by the dimensionality of initial signal space of the dimension .
utput: The Gram matrix such that

Update:
, For solve the in for 1 T = GU and system given (9 for obtaining j α and j β for an imposed j γ . 6.For (9) for complex spreading the orthogonal matrix used for ents on main diagonal of Gram matrix is 0 The Gram matrix obtained with the proposed Algorithm has one of the following patterns: The general form of ( , ) p p and ( , ) (14) Proof: The Algorithm starts with and 1 p = q K j = imposing for the element 11 / ( ) (1, / ,..., / , 0,..., 0 , ( ) / ) The Gram matrix obtained with the algorithm after the first iteration can be written as 1 11 1 x G (15) where is of order The Algorithm will continue with and q 2 p = K j = . We can apply this procedure at most times.Since the Frobenius norm is preserved by orthogonal transformations, we find that, after the first iteration, the TSC is given by , Thus, under the TSC design, the absolute value of the inner product for the vector is exactly given by (10).In the last iteration we obtain .Again, since TSC is preserved by using Givens rotations, we have TSC TSC TSC TSC y y G (17) We observe that, due to symmetry, the column vector y has all equal elements.Thus, the vector y can have the form ( ,..., ) for real crosscorrelation case and ( ,..., ) for complex crosscorrelation case, which concludes the lemma.Since the Gram matrix is normal ( for some unitary matrix U ) we will prove next, that for each pattern in (14), there exist a unique orthogonal matrix , which considerably simplifies the design.* = G GU V Lemma 4.2: The orthogonal matrix used to construct matrices given by ( 14) is unique.

V
Proof: The Gram matrix of order K i given by ( 14) is a normal matrix since it commutes with its Hermitian adjoint.By Theorem 2.5.8 [22] there is an orthogonal matrix such that Q 1 2 ( , ,..., ), 1 where each j G is either a real 1 by 1 matrix or is a real 2 by 2 matrix of the following form With the proposed Algorithm we can find an orthogonal matrix such that U 1 ( / ,..., / , 0,..., 0) The value of the determinant of given by ( 13) is 1 so is a unitary matrix.From (16) there is an orthogonal matrix such that V U Q 2 ( / ,..., / , 0,..., 0) It is easy to check that 1 G and are real commuting normal matrices and by Theorem 2.5.15 [22] there exists a unique real orthogonal matrix such that 2 G P 1 T P G P and 2 T P G P to be of the form (13) which concludes the proof.

V. NUMERICAL RESULTS
Experiment 5.1 Generating orthogonal WBE complex spreading sequences with constant chip magnitude.
Consider two cells with having 8 users with the same spreading gain gives the distribution of the optimal eigenvalues used for construction.Running the proposed algorithm with the above data we obtained the following matrices where the Gram matrices for the users are obtained as 1 1.0000 0.3333 0.3333 0.3333 0.3333 1.0000 0.3333 0.3333 0.3333 0.3333 1.0000 0.3333 0.3333 0.3333 0.3333 1.0000 1.0000 0.3333 0.3333 0.3333 0.3333 1.0000 0.3333 0.3333 0.3333 0.3333 1.0000 0.3333 0.3333 0.3333 0.3333 1.0000

S
One might think it is possible to design complex spreading sequences for individual cells and concatenate their correlation matrices as diagonal blocks in a composite blockdiagonal correlation matrix as long as the cells operate orthogonally to each other.This is true in a non-collaborative scenario when users communicate independently and are supervised by only one base station 8 .In this case the T -transform provides the same results as the algorithm proposed in this paper.However, in a collaborative scenario and in the same extended signal space, users need to have the same signature waveforms when they are "seen" by different base stations.This condition is satisfied if there exist a matrix U as it is provided by Theorem 4.1.Experiment 5.2 Generating orthogonal WBE complex spreading sequences with real maximum equal crosscorrelation amplitude: We also consider two cells with 12 users with the same spreading gain and .The following vectors and give the distribution of the diagonal of composite Gram matrix and the optimal eigenvalues used for construction.Running the proposed algorithm with the above data we obtained the following structured Gram matrix where the Gram matrices for the users are obtained again with patterns specified by Lemma 4.1: As in the previous experiment we can observe that the bandwidth of spreading sequences in each cell, specified by and , respectively, is preserved in the signal space of dimension 1 N 2 N 8 The notion of collaborative base stations is crucial to the idea of intercell interference mitigation.The extension of our results to overloaded "cognitive radios" is left for future research.Thank you to anonymous reviewer for this perspective.

K K
+ of the composite Gram matrix .

Experiment 5.3 Generating orthogonal complex spreading sequences with complex maximum equal crosscorrelation amplitude:
For this experiment we used the optimal eigenvalue vector (2, 2,0,0, 2, 2,0,0) = y and the corresponding Gram matrices for each cell with four users are obtained as

G
The above two matrices specified by Lemma 4.1 are in fact the same since we can check that .

VI. CONCLUSIONS AND FUTURE WORK
We developed a distributed algorithm for 4G wireless communications providing complex spreading sequences with constant chip magnitude and maximum cross correlation amplitude for multicellular S-CDMA.In this paper we have focused on symbol-synchronous CDMA systems in the presence of AWGN.The extension of our results to the asynchronous situation and considering colored noise is interesting and also an important open problem.Our current efforts are directed towards solving this important open question.

APPENDIX
Proof of the Theorem 4.1: a) For the particular case when 1 M = the Theorem 4.1 is the same with Theorem 4.1 [2].

.. N x x
≥ ≥ ≥ x .The vector is said to be majorized by the vector x y , denoted if [20] x y ≺ a) 1 1 , 1 ,2,..., i) Verify for 2 M = and 2 N = .We start with 1 M = .Condition ( 22) is equivalent to 1 1 x y ≤ , 1 2 1 2 x x y y + = + and becomes since respectively.Consider the initial elements on diagonal i.e. the vectors which can be written using (22.a) for the matrix of order as 2N + 2 ].In contrast with the underloaded S-CDMA systems ( i i K N ≤ ), the composite Gram matrix of (3) is positive semidefinite (there exist at least one null ] 2 , the Gram matrix of equally correlated ces has the following form: Now, we extend the constant chip design for M overloaded cells, 1 i M ≤ ≤ , focusing on orthogonal E sequences for each cell.Our design for constant chip m itu ased numbers with modulus one for h cell i , and c eac onsider the following i K unit vectors with

N
In order to on truc the Gram matrix

Theorem 4 . 1 : 2 (
Given the vectors 1 there exist an orthogonal matrix U such that the diagonal entries of the matrix orthogonalThe proof is given in the Appendix 7 .

b) Consider 1 M
> and we use induction on .Without loss of generality we can assume within a cell the eigenvalues (

2 N
, what remains is to prove that .If this is true under the induction assumption there exist an orthogonal matrix (check conditions (22.a) and (22.b) again.For condition (22.a) it is enough to show that By using (22.b) for the matrix of order we have ; Revised on July 15, 2007.This paper was presented in part at 2006 IEEE Radio and Wireless Symposium, RWS' 06, 17-19 Jan 2006, San Diego, and at 40 th Annual Conference on Information Sciences and Systems, CISS '06,22-24 March 2006, Princeton.