Spreading Code Assignment in an Ad hoc DS-CDMA Wireless Network

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1 Spreading Code Assignment in an Ad hoc DS-CDMA Wireless Networ Ynnan W, Qian Zhang, Wenw Zh, S.Y. ng Dept. of Electrical Engineering, Princeton University, Princeton, NJ Microsoft Research, China, 49 Zhichn Rd. Beiing 00080, P.R.China Abstract-In this paper, we eamine the problem of optimal spreading code assignment in an ad hoc DS-CDMA wireless networ. he problem is formlated in two ways, with the optimization criteria being minimizing the congestion measre, and minimizing the total consmed power, respectively. Either formlation is converted to an optimization of the weighted total sqared cross-correlation (WSC, after some mathematical simplification. We then devise algorithms to reach some local minima of the optimization. Simlation reslts show significant redction in the networ congestion measre and power consmption, compared with random code allocation. I. INRODUCION In a CDMA wireless networ, different sers are assigned different signatre waveforms to modlate their respective information bitstreams. In DS-CDMA, signatre waveforms are chip waveforms modlated by spreading code vectors. he signatre waveforms play an important role in controlling the interference among the sers, and conseqently the BER after mlti-ser detection []. he spreading code assignment problem arises as a resorce allocation strategy. Sppose there are sers and the dimension of the spreading code is N. One can view the problem as to allocate the locations of the sers on a N-dim sphere srface to optimize some system performance criteria. In particlar, a reasonable reqirement might be to balance the received qality of different sers to avoid resorce monopoly. he allocation strategy depends on the commnications model sed. Eisting models can be classified into two categories: the celllar model or the ad hoc model, shown in igre (A and (B respectively. he strctral differences lie in whether the receivers are co-located or distribted. he model in igre (A is a single cell abstraction of today s celllar commnications, where ot-of-cell interferences are treated as bacgrond noise. or the plin case as shown in igre (A, the Welch-Bond-Eqality (WBE seqences are fond to be the optimal seqence [-5], maimizing sm capacity, ser capacity and etc. rthermore, with optimal WBE seqences, the MMSE detector becomes single ser matched filter. By recognizing that the code set minimizing the total sqared cross-correlation (SC is the optimal WBE code set, in [6], the athors proposed iterative and distribted interference avoidance algorithms where sers pdate their spreading codes seqentially to gradally redce the SC. he ad hoc setting in igre (B is a more generic model than the celllar setting. It is st a collection of several commnication pairs. or one thing, it models the celllar networ by taing ot-of-cell interferences into consideration. or the other, it captres the ad hoc (or infrastrctre-less paradigm where a central base station is absent. hs the code assignment can be regarded as a more general form of freqency rese optimization in celllar planning. S BASE SAION S (A S 3 S R R (B S R 3 igre (A CEUAR MODE (B AD HOC MODE or the code assignment in the ad hoc model, prior arts [7][8] stdy the orthogonal code allocation (or eqivalently DMA timeslot assignment with a graph coloring approach. Assming a common transmission range, a directed graph is formed with each node denoting a commnication pair and each arc meaning a transmitter is within the range of the receiver. A distribted heristic graph coloring algorithm is then sed to assign the orthogonal channels. he nmber of colors is the dimension of the code reqired. However, there are several drawbacs of the scheme. irst, the nmber of channels reqired can t be nown beforehand, while in practice the dimension of the code is sally given. Second, it gives a physical layer simplification that two sers can not coeist if one falls within the other s range. hird, the scheme does not consider the possible heterogeneos power capabilities and QoS reqirements of the sers. In this paper, we investigate the optimal spreading code assignment problem in the ad hoc setting. he paper is organized as follows. In section II, we give a mathematical description of the problem. wo formlations, optimizing the congestion measre and optimizing the power consmption, are discssed in section III and IV, respectively. Since both formlations are simplified to a weighted total sqared crosscorrelation (WSC optimization, section V deals with solving the WSC optimization. Simlation reslts are presented in section V. inally, section VI concldes the paper. II. PROBEM ORMUAION AND ASSUMPIONS S 3 his wor was performed when Mr. W was a smmer intern at Microsoft Research, China /0/$7.00 (C 00 IEEE

2 Consider an ad hoc wireless networ consisting of commnication pairs. et P be the transmission power of the -th transmitter, and be the channel path loss from the -th transmitter to the -th receiver. In free space, the power loss is modeled as inversely proportional to the distance sqared. We denote the normalized spreading code of the -th pair as s. he bacgrond white assian noise level at the -th receiver isσ. We assme single ser matched filter detector is sed at each receiver. hen the signal to interference pls noise ratio (SINR at the -th receiver is: P SINR = P s + σ. (. It is easily seen from (. that the qalities for different commnication pairs can be controlled throgh designing the spreading codes and throgh allocating different transmission powers. Before discssing the optimal spreading code assignment, we briefly review some reslts in the optimal power control [9] and introdce the notations to be sed in the seqel. We consider a general scenario where sers have different signal SINR reqirements. or a given set of spreading codes, the optimal power control problem is to satisfy the SINR reqirements with minimm total power: min P, s. t. SINR. (. With the following notations 0 s M s σ p (,, P P, σ 0 s O M M 0 s (. can be re-written as: ( I p, (.3 where means the component-wise. A power allocation can be fond to meet the set of SINR reqirements if and only if <, where ma λ, =,..., is the maimm modls of the eigenvales of. Since is nonnegative and irredcible, by the Perron-robenis theorem [0], has a real positive eigenvale eqal to. And if <, there is a component-wise minimm soltion: p opt = ( I. (.4 he spreading code assignment provides another degree of freedom. In particlar, and popt are fnctions of,, s. It is noted in [] that can be treated as a congestion measre of the wireless networ as it characterizes the margin to accommodate the traffic increases in the networ. hs, an optimization formlation can be given: min. (.5 s =, =,, Since the battery life of mobile devices is limited, it is of practical interest to conserve the energy as mch as possible. Hence, an alternative formlation is to minimize the total consmed power by choosing an appropriate code set, as shown in eqation (.6. min P = ( I (.6 s =, =,, or both (.5 and (.6, if N, the optimal soltion is readily obtained to be the orthogonal allocation as it decoples the channel into independent single-ser channels. Hence we are interested in the case of >N. Since it is difficlt to directly wor with either optimization, we see to se some bonding operations and approimations to simplify the problem in the following two sections. and III. OPIMIZIN HE CONESION MEASURE et D diag{,, }, * DD. * has the following epression: = s, if *. (3. Since it is a similarity transform, both and * have the same set of eigenvales. By sbstitting and with min{, }, we obtain a symmetric matri # : min{, } = s, if #. (3. * # Since 0 where means the element-wise, we have = * # by a property of non-negative matrices [0]. With an application of the Rayleigh qotient for real symmetric matri:, we have A λ ma he -norm of a vector v is defined as: v v /0/$7.00 (C 00 IEEE

3 . (3.3 # # = = Overall, we get a lower bond for which is epressed as a weighted total sqared cross-correlation (WSC. We denote this as WSC and choose to minimize it instead of : min{, } min s,. (3.4 s =, =,, IV. OPIMIZIN HE POWER CONSUMPION As for (.6, we can approimate (I- - with its first order aylor epansion: P = ( I ( I + = +. (4. Sch an approimation is good if is near zero, i.e., when the networ is not very congested. With the approimation, we arrive at a minimization of another WSC epression which we denote as WSC: min s, =, σ s =,, (4. V. SOVIN WSC OPIMIZAION In section III and IV, both optimization criteria have been converted to a minimization of the weighted total sqare cross-correlation (WSC. It is still difficlt to find the eact optimal soltion for the general WSC problem min a, s =, s. (5. =,, We restrict or attention first to the case of orthogonal allocation. Since there are only N orthonormal bases in a N-dim space, the problem is eqivalent to optimally partition the sers into N clsters. Associated with each pair of the sers is a cost a + a, if the two sers and are within the same clster. he total cost is to be minimized. Unfortnately, finding the eact optimal partition has an eponential compleity. hs we develop a heristic algorithm as follows: Algorithm. Sort the sers according to some measre of importance, e.g. + a which is the total costs to all other sers.. Each of the first N sers pics one distinct orthonormal basis. 3.Each of the following -N sers selects the best clster to oin in. he criterion is to minimize the increase of the WSC with the addition of the ser in consideration. A related wor [6] considered the SC minimization problem in a celllar wireless networ. he sers are at first assigned with random codes. hen a distribted algorithm is condcted at the sers seqentially and iteration by iteration, where in each step a ser pdates its spreading code only to redce the SC. We note that the techniqe can be etended to the WSC minimization problem (5.. he ey observation is that the single ser adaptation version of (5. (only a particlar s is the variable min, a s = s can be solved. Etracting the terms related with s in (5., we have (5. min + a s = s ( ( a + a s s. (5.3 s = he above minimization can be solved by a agrange mltiplier approach. he soltion is st the eigenvector corresponding to the minimal eigenvale of the matri + a s. he adapted iterative algorithm can arrive at a local minimm of the problem since the WSC can only decrease at each step. he main drawbac is its compleity and the signaling overhead incrred to inform the receivers abot the code pdate. With the techniqe of single ser adaptation, we as the qestion whether performance gains can be achieved in step 3 of algorithm by looing at not only the N bases, bt all possible s. Withot loss of generality, we assme the N orthonomal bases are the standard bases { v v =, v = 0, i}. hen i ii = diag{ i + a s s = v + a,, s = vn + a. (5.4 } Conseqently, if p to now all sers are assigned with the orthonormal bases, the optimal single ser adaptation wold still sggest an orthonormal basis in the channel where the total cost is minimized /0/$7.00 (C 00 IEEE

4 VI. SIMUAION RESUS We generate the ad hoc commnication settings as follows: transmitters are randomly drawn from a rectanglar region [ 0,] [0,], and each receiver for a transmitter at (, y is randomly drawn from a rectanglar region [ 0.5, + 0.5] [ y 0.5, y + 0.5]. his model is chosen to reflect that in an ad hoc networ, a node can only directly commnicate with a neighbor. An eample setting is shown in igre where each arrow represents a commnication pair with the tail being the transmitter and the head being the receiver. igre also presents the orthogonal allocation reslt with algorithm by plotting each pair with the color (also shown with the nmber corresponding to its orthonormal basis. We can see from igre that the basic idea in the ad hoc spreading code assignment is to assign nearby pairs different codes. color nmber (spreading gain dimension N, we rn it first and then (A~(A3 are rn with the N given by (A. able -able 4 give the simlation reslts for ~4, respectively. or each traffic setting, we test over 0 geographic settings. he first three rows are the performance criteria: the congestion measre, the observed probability of having a valid power allocation, and the total consmed power ( I which is the average for the cases when a valid allocation eists. < ( I ABE RESUS OR, =, E(N=6.75, N [ 4, ] A A WSC optimization WSC optimization WSC WSC.0.03 ABE RESUS OR, =, E(N=6.75, N [ 4, ] igre USER OCAIONS AND HE COORIN WIH AORIHM We simlate for cases for the traffics in the networ: ( ight niform traffic where =, N~7 and =4dB for every pair. ( ight non-niform traffic where =, N~7, and half of the sers have = 3dB and the other half 6dB. (3 Heavy niform traffic where =3, N~3, =4dB for every pair. (4 Heavy non-niform traffic where =3, N~3, and half of the sers have = 3dB and the other half 6dB. he reslts are collected for the following algorithms: (A raph coloring (prior arts [7][8] (A Random assignment (each code is niformly drawn from the srface of a nit N-sphere. (A3 Iterative algorithm for WSC minimization (A4 Orthogonal assignment (algorithm. Since both algorithm (A and (A3 depend on the realization of the random spreading codes, we average the reslts over 00 trials. And since (A cannot control the resltant < ( I A A WSC optimization WSC optimization WSC WSC < ( I ABE 3 RESUS OR 3, =3, E(N=.75, N [,5 ] A A WSC optimization WSC optimization WSC WSC /0/$7.00 (C 00 IEEE

5 < ( I ABE 4 RESUS OR 4, =3, E(N=.75, N [,5 ] A A WSC optimization WSC optimization WSC WSC Note that at present, the ad hoc networ is configred as a random access networ. We do not claim that schedled access, as the optimal allocation we are prsing here, is feasible immediately. In fact, many important problems remain nanswered, sch as the mobility spport, dealing with dynamic traffic arrivals and departres in the networ. rom able to able 4, we can see that random allocation is mch worse than the others. One reason is that in an ad hoc networ, the problem is highly geographic dependent, while complete random allocations do not tae advantage of the spatial separation. In light traffic, the graph-coloring algorithm A has 59% higher than that of A3, and 9% higher power consmption. In heavy traffic, A is mch worse than A3. And since it is naware of the different QoS reqirements, its performance in the non-niform traffic cases is worse that that of the niform traffic cases. When comparing A3 and A4, we see that as far as the optimization obective WSC is concerned, the two algorithms have almost the same performance. Yet A4 has a mch lower compleity. However, an interesting phenomenon is that A4 has on average 45% higher than that of A3, while the power consmptions are abot the same. One possible case might be the strctre restriction of orthogonal allocation. And it might be in part de to that optimizing the lower bonds is not as relevant as optimizing the approimation. o evalate the lower bonding operation and the approimation, we eamine the vs. WSC progression and the power consmption vs. WSC progression, dring the gradal redction of the WSC in A3. wo eample crves are plotted as igre 3 (A and (B. Empirically, we observe that the althogh the (total power can not be assred to monotonically decrease as WSC(WSC decreases, the trends of the two are the same. VII. CONCUSIONS AND UURE WOR In this paper, we stdied the spreading code assignment in an ad hoc wireless networ. he problem is converted into two forms of WSC minimization. And heristic algorithms to reach some local minima of the WSC minimizations were designed. he proposed orthogonal allocation algorithm has low compleity and can achieve promising performance. hrogh simlations, we demonstrate the significant performance gains of strctred code design, compared with the sally adopted random spreading code assignment. (A igre 3 (B (A VS. WSC PRORESSION (3 (B OA POWER VS. WSC PRORESSION (3 REERENCES [] S. Verd, Mltiser Detection, Cambridge University Press, 998. [] M. Rpf and J.. Massey, Optimal seqence mltisets for synchronos code division mltiple-access channels, IEEE rans. on Information theory, vol. 40, pp. 6-66, Jly 994. [3] J.. Massey and. Mittelholzer, Welch s bond and seqence sets for code-division mltiple-access systems, in Seqences II: Methods in Commnication, Secrity and Compter Sciences, R. Capocelli, A. De Santis, and U. Vaccaro, Eds, pp , 993. Springer. [4] P. Viswanath, V. Anantharam, D. N. C. se, Optimal seqences, power control, and ser capacity of synchronos CDMA systems with linear MMSE mltiser receivers, IEEE rans. on Information heory, vol.45, pp , Sept [5] P. Viswanath and V. Anantharam, Optimal seqences and sm capacity of synchronos CDMA systems, IEEE rans. on Information theory, vol. 45, pp , Sept [6] S. Uls and R. D. Yates, Iterative constrction of optimm signatre seqence sets in synchronos CDMA systems, IEEE rans. on Information heory, 47(5: pp , Jly 00. [7]. H, Distribted code assignments for CDMA pacet radio networs, IEEE/ACM rans. on Networing, vol., pp , Dec [8] S. Ramanathan, A nified framewor and algorithm for channel assignment in wireless networs, Wireless Networs, pp. 8-94, 5(999. [9] N. Bambos, oward power-sensitive networ architectres in wireless commnications: concepts, isses, and design aspects, IEEE Personal Commnications, pp.50-59, Jne 998. [0] E. Seneta, Non-negative Matrices and Marov Chains, Springer Verlag, nd ed. 98. [] S. V. Hanly, Congestion measres in DS-CDMA networs, IEEE rans. on commnications, vol. 47, pp , Mar /0/$7.00 (C 00 IEEE