Optimal and Suboptimal Finger Selection Algorithms for MMSE Rake Receivers in Impulse Radio Ultra-Wideband Systems 1

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1 Optima and Suboptima Finger Seection Agorithms for MMSE Rake Receivers in Impuse Radio Utra-Wideband Systems Sinan Gezici, Mung Chiang, H. Vincent Poor and Hisashi Kobayashi Department of Eectrica Engineering Princeton University, Princeton, NJ Abstract Convex reaxations of the optima finger seection agorithm are proposed for a minimum mean square error (MMSE) Rake receiver in an impuse radio utra-wideband system. First, the optima finger seection probem is formuated as an integer programming probem with a non-convex objective function. Then, the objective function is approximated by a convex function and the integer programming probem is soved by means of constraint reaxation techniques. The proposed agorithms are suboptima due to the approximate objective function and the constraint reaxation steps. However, they can be used in conjunction with the conventiona finger seection agorithm, which is suboptima on its own since it ignores the correation between mutipath components, to obtain performances reasonaby cose to that of the optima scheme that cannot be impemented in practice due to its compexity. The proposed agorithms everage convexity of the optimization probem formuations, which is the watershed between easy and difficut optimization probems. Index Terms Utra-wideband (UWB), impuse radio (IR), MMSE Rake receiver, convex optimization, integer programming. I. INTRODUCTION Since the US Federa Communications Commission (FCC) approved the imited use of utra-wideband (UWB) technoogy [], communications systems that empoy UWB signas have drawn considerabe attention. A UWB signa is defined to be one that possesses an absoute bandwidth arger than 500MHz or a reative bandwidth arger than 20% and can coexist with incumbent systems in the same frequency range due to its arge spreading factor and ow power spectra density. UWB technoogy hods great promise for a variety of appications such as short-range high-speed data transmission and precise ocation estimation. Commony, impuse radio (IR) systems, which transmit very short puses with a ow duty cyce, are empoyed to impement UWB systems ([2]-[6]). In an IR system, a train of puses is sent and information is usuay conveyed by the position or the poarity of the puses, which correspond to Puse Position Moduation (PPM) and Binary Phase Shift Keying (BPSK), respectivey. In order to prevent catastrophic coisions among different users and thus provide robustness against mutipeaccess interference, each information symbo is represented by This research is supported in part by the Nationa Science Foundation under grants ANI , CNS , and CCR , and in part by the New Jersey Center for Wireess Teecommunications. a sequence of puses; the positions of the puses within that sequence are determined by a pseudo-random time-hopping (TH) sequence specific to each user [2]. The number N f of puses representing one information symbo can aso be interpreted as puse combining gain. Commony, users in an IR-UWB system empoy Rake receivers to coect energy from different mutipath components. A Rake receiver combining a the paths of the incoming signa is caed an a-rake (ARake) receiver. Since a UWB signa has a very wide bandwidth, the number of resovabe mutipath components is usuay very arge. Hence, an ARake receiver is not impemented in practice due to its compexity. However, it serves as a benchmark for the performance of more practica Rake receivers. A feasibe impementation of mutipath diversity combining can be obtained by a seective- Rake (SRake) receiver, which combines the M best, out of L, mutipath components [7]. Those M best components are determined by a finger seection agorithm. For a maxima ratio combining (MRC) Rake receiver, the paths with highest signa-to-noise ratios (SNRs) are seected, which is an optima scheme in the absence of interfering users and inter-symbo interference (ISI). For a minimum mean square error (MMSE) Rake receiver, the conventiona finger seection agorithm is to choose the paths with highest signa-to-interferencepus-noise ratios (SINRs). This conventiona scheme is not necessariy optima since it ignores the correation of the noise terms at different mutipath components. In other words, choosing the paths with highest SINRs does not necessariy maximizes the overa SINR of the system. In this paper, we formuate the optima MMSE SRake as a nonconvex, integer-constrained optimization, where the aim is to choose the finger ocations of the receiver so as to maximize the overa SINR. Whie computing the optima finger seection is NP-hard, we present severa reaxation methods to turn the (approximate) probem into convex optimization probems that can be very efficienty soved by interior-point methods, which are poynomia time in the worst case, and very fast in practice. These optima finger seection reaxations produce significanty higher average SINR than the conventiona one that ignores the correations, and represent a numericay efficient way to strike a baance between SINR optimaity and computationa tractabiity. The remainder of the paper is organized as foows. Sec-

2 Fig.. An exampe time-hopping impuse radio signa with puse-based poarity randomization, where N f =6, N c =4, the time hopping sequence is {2,,2,3,,0} and the poarity codes are {+,+,-,+,-,+}. tion II describes the transmitted and received signa modes in a mutiuser frequency-seective environment. The finger seection probem is formuated and the optima agorithm is described in Section III, which is foowed by a brief description of the conventiona agorithm in Section IV. In Section V, two convex reaxations of the optima finger seection agorithm, based on an approximate SINR expression and integer constraint reaxation techniques, are proposed. The simuation resuts are presented in Section VI, and the concuding remarks are made in the ast section. II. SIGNAL MODEL We consider a synchronous, binary phase shift keyed TH- IR system with K users, in which the transmitted signa from user k is represented by: s (k) E k tx (t) = N f k= j= d (k) j b (k) j/n f p tx(t jt f c (k) j T c ), () where p tx (t) is the transmitted UWB puse, E k is the bit energy of user k, T f is the frame time, N f is the number of puses representing one information symbo, and b (k) j/n f {+, } is the binary information symbo transmitted by user k. In order to aow the channe to be shared by many users and avoid catastrophic coisions, a time-hopping (TH) sequence {c (k) j }, where c (k) j {0,,..., N c }, is assigned to each user. This TH sequence provides an additiona time shift of c (k) j T c seconds to the jth puse of the kth user where T c is the chip interva and is chosen to satisfy T c T f /N c in order to prevent the puses from overapping. We assume T f = N c T c without oss of generaity. The random poarity codes d (k) j are binary random variabes taking vaues ± with equa probabiity ([8]-[0]). Consider the discrete presentation of the channe, α (k) = [α (k) α (k) L ] for user k, where L is assumed to be the number of mutipath components for each user, and T c is the mutipath resoution. Then, the received signa can be expressed as K E k L r(t) = α (k) d (k) j b (k) N j/n f f j= = p rx (t jt f c (k) j T c ( )T c )+σ n n(t), (2) where p rx (t) is the received unit-energy UWB puse, which is usuay modeed as the derivative of p tx (t) due to the effects Fig. 2. The receiver structure. There are M mutipath components, which are combined by the MMSE combiner. of the antenna, and n(t) is zero mean white Gaussian noise with unit spectra density. We assume that the time-hopping sequence is constrained to the set {0,,...,N T }, where N T N c L, so that there is no inter-frame interference (IFI). Due to the high resoution of UWB signas, chip-rate and frame rate samping are not very practica for such systems. In order to have a ower samping rate, the received signa can be correated with tempate signas which enabe symbo rate samping of the output []. The tempate signa for the th path of the incoming signa can be expressed as (i+)nf temp, (t) = d () j p rx (t jt f c () j T c ( )T c ), j=in f s () for the ith information symbo, where we consider user without oss of generaity. In other words, by using a correator for each mutipath component that we want to combine, we can just use symbo rate samping at each branch, as shown in Figure 2. Note that the use of such tempate signas resuts in equa gain combining (EGC) of different frame components. This may not be optima under some conditions (see [2] for (sub)optima schemes). However, it is very practica since it faciitates symbo-rate samping. Since we consider a system that empoys tempate signas of the form (3), i.e. EGC of frame components, it is sufficient to consider the probem of seection of the optima paths just for one frame. Hence, we assume N f =without oss of generaity. Let L = {,..., M } denote the set of mutipath components that the receiver coects (Figure 2). At each branch, the signa is effectivey passed through a matched fiter (MF) matched to the reated tempate signa in (3) and samped once for each symbo. Then, the discrete signa for the th path can be expressed, for the ith information symbo, as 2 (3) r = s T Ab i + n, (4) for =,..., M, where A = diag{ E,..., E K }, b i = [b () i b (K) i ] T and n N(0,). 2 s is a K vector, 2 Note that the dependence of r on the index of the information symbo, i, is not shown expicity.

3 which can be expressed as a sum of the desired signa part (SP) and mutipe-access interference (MAI) terms: s = s (SP) + s (MAI), (5) where the kth eements can be expressed as { [ ] s (SP) α () =, k = and (6) k 0, k =2,...,K { [ ] s (MAI) 0, k = = k d () L d(k) m= α(k) m I (k),m, k =2,...,K, (7) with I (k),m being the indicator function that is equa to if the mth path of user k coides with the th path of user, and 0 otherwise. III. PROBLEM FORMULATION AND OPTIMAL SOLUTION The probem is to choose the optima set of mutipath components, L = {,..., M }, that s the bit error probabiity (BEP) of the system. In other words, we need to choose the best sampes from the L received sampes r, =,...,L,asshownin(4). To reformuate this combinatoria probem, we first define an M L seection matrix X as foows: M of the coumns of X are the unit vectors e,...,e M (e i having a at its ith position and zero eements for a other entries), and the other coumns are a zero vectors. The coumn indices of the unit vectors determine the subset of the mutipath components [ that are seected. ] For exampe, for L =4and M =2, X = 000 chooses the second and third mutipath components. 000 Using the seection matrix X, we can express the vector of received sampes from M mutipath components as r = XSAb i + Xn, (8) where n is the vector of therma noise components n = [n n L ] T, and S is the signature matrix given by S = [s s L ] T, with s as in (5). Using (5)-(7), (8) can be expressed as r = b () i E Xα () + XS (MAI) Ab i + Xn, (9) where S (MAI) is the MAI part of the signature matrix S. Then, the inear MMSE receiver can be expressed as ˆbi = sign{θ T r}, (0) where the MMSE weight vector is given by [3] θ = R Xα (), () with R being the correation matrix of the noise term: R = XS (MAI) A 2 (S (MAI) ) T X T + I. (2) The SINR of the system can be expressed as SINR(X) = E (α () ) T X T ( I + ) 2 XS (MAI) A 2 (S (MAI) ) T X T Xα (). (3) Hence, the optima path/finger seection probem can be formuated as maximize SINR(X), (4) where X has the previousy defined structure. Note that the objective function to be maximized is not concave and the optimization variabe X takes binary vaues, with the previousy defined structure. In other words, two major difficuties arise in soving (4) gobay: nonconvex optimization and integer constraints. Either makes the probem NP-hard. Therefore, it is an intractabe optimization probem in this genera form. IV. CONVENTIONAL ALGORITHM Instead of the soving the probem in (4), the conventiona finger seection agorithm chooses the M paths with argest individua SINRs, where the SINR for the th path can be expressed as SINR = (s (MAI) E (α () ) 2 ) T A 2 s (MAI), (5) + 2 for =,...,L. This agorithm is not optima because it ignores the correation of the noise components of different paths. Therefore, it does not aways maximize the overa SINR of the system given in (3). For exampe, the contribution of two highy correated strong paths to the overa SINR might be worse than the contribution of one strong and one reativey weaker, but uncorreated, paths. The correation between the mutipath components is the resut of the MAI from the other users in the system. V. RELAXATIONS OF OPTIMAL FINGER SELECTION Since the optima soution in (4) is quite difficut, we first consider an approximation of the objective function in (3). When the eigenvaues of σ XS (MAI) A 2 (S (MAI) ) T X T are n 2 consideraby smaer than, which occurs when the MAI is not very strong compared to the therma noise, we can approximate the SINR expression in (3) as foows 3 : SINR(X) E 2 (α () ) T X T ( I ) 2 XS (MAI) A 2 (S (MAI) ) T X T Xα (), (6) 3 More accurate approximations can be obtained by using higher order series expansions for the matrix inverse in (3). However, the soution of the optimization probem does not end itsef to ow compexity soutions in those cases.

4 which can be expressed as SINR(X) E 2 { (α () ) T X T Xα () 2 α () X T XS (MAI) A 2 (S (MAI) ) T X T Xα () }. (7) Note that the approximate SINR expression depends on X ony through X T X. Defining x =[x x L ] T as the diagona eements of X T X, x = diag{x T X}, wehavex i =if the ith path is seected, and x i =0otherwise; and L i= x i = M. Then, we obtain, after some manipuation, SINR(x) = E { q T x 2 x T Px }, (8) where q = [(α () )2 (α () L )2 ] T and P = diag{α () α () L }S(MAI) A 2 (S (MAI) ) T diag{α () α () L }. Then, we can formuate the finger seection probem as foows: x T Px x T q subject to x T = M, x i {0, }, i =,...,L. (9) Note that the objective function is convex since P is positive definite, and that the first constraint is inear. However, the integer constraint increases the compexity of the probem. The common way to approximate the soution of an integer constraint probem is to use constraint reaxation. Then, the optimizer wi be a continuous vaue instead of being binary and the probem (9) wi be convex. Over the past decade, both powerfu theory and efficient numerica agorithms have been deveoped for noninear convex optimization. It is now recognized that the watershed between easy and difficut optimization probems is not inearity but convexity. For exampe, the interior-point agorithms for noninear convex optimization are highy efficient, both in worst case compexity (provaby poynomia time) and in practice (very fast even for a arge number variabes and constraints) [4]. Interior-point methods sove convex optimization probems with inequaity constraints by appying Newton s method to a sequence of equaity constrained probems, where the Newton s method is a kind of descent agorithm with the descent direction given by the Newton step [4]. We consider two different reaxation techniques in the foowing subsections. A. Case-: Reaxation to Sphere Consider the reaxation of the integer constraint in (9) to a sphere that passes through a possibe integer vaues. Then, the reaxed probem becomes x T Px x T q subject to x T = M, (2x ) T (2x ) L. (20) Note that the probem becomes a convex quadraticay constrained quadratic programming (QCQP) [4]. Hence it can be soved for goba optimaity using interior-point agorithms in poynomia time. B. Case-2: Reaxation to Hypercube As an aternative approach, we can reax the integer constraint in (9) to a hypercube constraint and get x T Px x T q subject to x T = M, x [0, ] L, (2) where the hypercube constraint can be expressed as x 0 and x, with y z meaning that y z,...,y L z L. Note that the probem is now a ineary constrained quadratic programming (LCQP), and can be soved by interior-point agorithms [4] for the optimizer x. C. Dua Methods We can aso consider the dua probems. For the reaxation to the sphere considered in Section V-A, the Lagrangian for (20) can be obtained as ( ) L(x,λ,ν)=x T 2 P +4νI x x T (q λ +4ν) Mλ, (22) where λ Rand ν R +. After some manipuation, the Lagrange dua function can be expressed as g(λ, ν) = 4 [q +(λ +4ν)]T ( ) P +4νI [q +(λ +4ν)] Mλ, (23) Then, the dua probem becomes ( [q +(λ +4ν)]T 4 2 P +4νI) [q +(λ +4ν)]+Mλ (24) subject to ν 0, (25) which can be soved for optima λ and ν by interior point methods. Or, more simpy, the unconstraint probem (24) can be soved using gradient descent agorithm, and then the optimizer ν is mapped to ν = max{0, ν}. After soving for optima λ and µ, the optimizer x is obtained as ( x = 2 2 P +4ν I) [q +(λ +4ν )]. (26) Note that the dua probem (24) has two variabes, λ and ν, to optimize, compared to L variabes, the components of x, in the prima probem (20). However, an L L matrix needs to be inverted for each iteration of the optimization of (24). Therefore, the prima probem can be preferred over the dua probem in this case.

5 Simiary, the dua probem for the reaxation in Section V-B can be obtained from (2) as Optima Conventiona Sphere Hypercube 2 4 (q + µ ν λ)t P (q + µ ν λ)+mλ+ ν T (27) subject to µ, ν 0. (28) It is observed from (27) that there are 2L + variabes and aso L L matrix inversion operations for the soution of the dua probem. Therefore, the simper prima probem (2) is considered in the simuations. D. Seection of Finger Locations After soving the approximate probem (9) by means of integer reaxation techniques mentioned above, the finger ocation estimations are obtained by the indices of the M argest eements of the optimizer x. Both the approximation of the SINR expression by (6) and the integer reaxation steps resut in the suboptimaity of the soution. Therefore, it may not be very cose to the optima soution in some cases. However, it is expected to perform better than the conventiona agorithm most of the time, since it considers the correation between the mutipath components. However, it is not guaranteed that the agorithms based on the convex reaxations of optima finger seection aways beat the conventiona one. Since the conventiona agorithm is very easy to impement, we can consider a hybrid agorithm where the fina estimate of the convex reaxation agorithm is compared with that of the conventiona one and the one that s the exact SINR expression in (3) is chosen as the fina estimate. In this way, the resuting hybrid suboptima agorithm can get coser to the optima soution. VI. SIMULATION RESULTS The simuation resuts are performed to evauate the performance of different finger seection agorithms for an IR- UWB system with N c = 20 and N f =. There are 5 equa energy users in the system (K = 5) and the users TH and poarity codes are randomy generated. We mode the channe coefficients as α = sign(α ) α for =,...,L, where sign(α ) is ± with equa probabiity and α is distributed ognormay as LN (µ,σ 2 ). Aso the energy of the taps is exponentiay decaying as E{ α 2 } =Ω 0 e λ( ), where λ is the decay factor and L = E{ α 2 } = (so Ω 0 =( e λ )/( e λl )). For the channe parameters, we have λ [ =0., σ 2 =0.5 and µ can be cacuated from µ =0.5 n( e λ ) λ( ) 2σ ],for 2 =,...,L. e λl We average the overa SINR of the system over different reaizations of channe coefficients, TH and poarity codes of the users. In Figure 3, we pot the average SINR of the system for different noise variances when M =5fingers are to be chosen out of L =5mutipath components. As is observed from the figure, the convex reaxations of optima finger seection resut in SINR vaues reasonabe cose to those of the optima Average SINR (db) Eb/No (db) Fig. 3. Average SINR versus E b /N 0 for M =5fingers. The channe has L =5mutipath components and the taps are exponentiay decaying. The IR-UWB system has N c =20chips per frame and N f =frame per symbo. There are 5 equa energy users in the system and random TH and poarity codes are used. exhaustive search scheme. Note that the gain by using the proposed agorithms over the conventiona one increases as the therma noise decreases. This is because when the therma noise get ess significant, the MAI becomes dominant, and the conventiona technique gets worse since it ignores the correation between the MAI noise terms when choosing the fingers. Next, we pot SINR of the proposed suboptima and conventiona techniques for different finger numbers in Figure 4, where there are 50 mutipath components and E b /N 0 = 20. The number of chips per frame, N c,issetto75, and a other parameters are kept the same. In this case, the optima agorithm takes a very ong time to simuate since it needs to perform exhaustive search over many different finger combinations (therefore not impemented). The improvement using convex reaxations of optima finger seection over the conventiona technique decreases as M gets arge since the channe is exponentiay decaying and the most of the significant mutipath components are aready combined by a the agorithms. Finay, we consider a MAI-imited scenario, where there are 0 users with E = and E k = 0 k, and a the parameters are as in the previous case. Then, as shown in Figure 5, the improvement by using the suboptima finger seection agorithms increase significanty. The main reason for this is that the suboptima agorithms consider, athough approximatey, the correation caused by MAI whereas the conventiona scheme simpy ignores that. VII. CONCLUDING REMARKS Optima and suboptima finger seection agorithms for MMSE-SRake receivers in an IR-UWB system are consid-

6 Average SINR (db) Conventiona Sphere Hypercube Number of Fingers Fig. 4. Average SINR versus number of fingers M, fore b /N 0 = 20dB, N c =75and L =50. A the other parameters are the same as those for Figure 3. Average SINR (db) Conventiona Sphere Hypercube Number of Fingers Fig. 5. Average SINR versus number of fingers M. There are 0 users with each interferer having 0dB more power than the desired user. A the other parameters are the same as those for Figure 4. ered. Since UWB systems have arge number of mutipath components, ony a subset of those components can be used due to compexity constraints. Therefore, the seection of the optima subset of mutipath components is important for the performance of the receiver. We have shown that the optima soution to this finger seection probem requires exhaustive search which becomes prohibitive for UWB systems. Therefore, we have proposed approximate soutions of the probem based on the Tayor series approximation and integer constraint reaxations. Using two different integer reaxation approaches, we have introduced two convex reaxations of the optima finger seection agorithm. Impementing these suboptima agorithms on top of the conventiona scheme, we can get cose to the optima soution, with much ower compexity. The two contributions of the paper are the formuation of the optima probem and the convex reaxations. In the first, the formuation is gobay optima but the soution methods for non-convex noninear integer constrained optimization must use heuristics to get to a ocay optima soution because otherwise computationa oad for goba optimaity is too much. In the second, the formuation is reaxed, but the interior-point methods efficienty computes the gobay optima soution for these reaxations. REFERENCES [] U. S. Federa Communications Commission, FCC 02-48: First Report and Order. [2] M. Z. Win and R. A. Schotz, Impuse radio: How it works, IEEE Communications Letters, 2(2): pp , Feb [3] M. Z. Win and R. A. Schotz, Utra-wide bandwidth time-hopping spread-spectrum impuse radio for wireess mutipe-access communications, IEEE Transactions on Communications, vo. 48, pp , Apri [4] F. Ramirez Mireess, On the performence of utra-wideband signas in gaussian noise and dense mutipath, IEEE Transactions on Vehicuar Technoogy, 50(): pp , Jan [5] R. A. Schotz, Mutipe access with time-hopping impuse moduation, Proc. IEEE Miitary Communications Conference (MILCOM 993), vo. 2, pp , Boston, MA, Oct [6] D. Cassioi, M. Z. Win and A. F. Moisch, The utra-wide bandwidth indoor channe: from statistica mode to simuations, IEEE Journa on Seected Areas in Communications, vo. 20, pp , Aug [7] D. Cassioi, M. Z. Win, F. Vataaro and A. F. Moisch, Performance of ow-compexity RAKE reception in a reaistic UWB channe, Proc. IEEE Internationa Conference on Communications (ICC 2002), vo. 2, pp , New York, NY, Apri 28-May 2, [8] E. Fisher and H. V. Poor, On the tradeoff between two types of processing gain, IEEE Transactions on Communications, to appear. [9] S. Gezici, H. Kobayashi, H. V. Poor and A. F. Moisch, Performance evauation of impuse radio UWB systems with puse-based poarity randomization in asynchronous mutiuser environments, Proc. IEEE Wireess Communications and Networking Conference (WCNC 2004), vo. 2, pp , Atanta, GA, March [0] Y.-P. Nakache and A. F. Moisch, Spectra shape of UWB signas infuence of moduation format, mutipe access scheme and puse shape, Proc. IEEE 57th Vehicuar Technoogy Conference, (VTC Spring), vo. 4, pp , Jeju, Korea, Apri [] A. F. Moisch, Y. P. Nakache, P. Orik, J. Zhang, Y. Wu, S. Gezici, S. Y. Kung, H. Kobayashi, H. V. Poor, Y. G. Li, H. Sheng and A. Haimovich, An efficient ow-cost time-hopping impuse radio for high data rate transmission, Proc. IEEE 6th Internationa Symposium on Wireess Persona Mutimedia Communications (WPMC 2003), Yokosuka, Kanagawa, Japan, Oct. 9-22, [2] S. Gezici, H. Kobayashi, H. V. Poor, and A. F. Moisch, Optima and suboptima inear receivers for time-hopping impuse radio systems, Proc. IEEE Conference on Utra Wideband Systems and Technoogies (UWBST 2004), Kyoto, Japan, May 8-2, [3] S. Verdú. Mutiuser Detection, Cambridge University Press, Cambridge, UK, 998. [4] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004.