Communications and Network, 2013, 5, 681687 http://dx.doi.org/10.4236/cn.2013.53B2122 Published Online September 2013 (http://www.scirp.org/journal/cn) Copyright © 2013 SciRes. CN Asymptotic Design and Analysis of Multiuser Detector for UWB High Data Ra te C h annel Based on Channel Division Multiple Access Houda Chihi, Ridha Bouallegue Higher School of Communications (Sup’Com), Innov’COM, Tunis, Tunisia Email: houda.chihi@gmail.com, ridha.bouallegue@ieee.org Received April 2013 ABSTRACT In this paper, we study an innovative multiple access scheme that exploits the intrinsic properties of the wireless envi ronment to improve the multiuser environment, socalled Channel Division Multiple Access (ChDMA) focusing on spectral efficiency analysis and system performance. In particular, we show that Simultaneous multiuser accessing to a common destination is made possible by considering the channel impulse response (CIR) of each user as a signature. We begin by the assumption of the channel state information available at the receiver. Then, we analyze the perform ance of the ChDMA integration in a random environment over UWB high data rate channel. Next, we discuss the de sign of MMSE and optimal receiver structu re for such a system. Additionally, we show an asymptotic analysis behavior taking into account the channel eigenv alues distribution with the associated spectral efficiency. Keywords: ChDMA; MBOFDM; UWB; Spectral Efficiency; Eigenvalues Distribution; MMSE; Optimal Receiver; PDP 1. Introduction UWB is a promising technology that has attracted con siderable interest in the research and standardization communities for wireless communications. For a very large bandwidth, the capacity derivation for UWB chan nel cannot be the same as for a classical narrowband channel. The main difference results from two frequency dependent attenuation origins: distance and frequency selectivity of channel. Indeed, the attenuation due to dis tance is proportional to the squared in verse of frequency, and cannot therefore be considered constant over the full UWB bandwidth. Besid es, the various realization s of the channel act as linear filters that attenuate randomly the transmitted signal at diff erent frequencies. In this contribution we describe, firstly, the MBOFDM UWB high data rate system. This must be taken into ac count in the capacity computation [1]. Next, we present ChDMA Characteristics with the channel model. We shall examine system spectral efficiency in a random environment with channel state information in the recap tion. The key performance measure here is in particular the eigenvalue distribution of the channel matrix. Then, in section III, we investigate two receiver structures that we will study in different system behavior: the optimal receiver and the linear minimum mean square error (MMSE) detector. The MMSE linear detector is particu larly interesting as it allows maximizing the signalto interference ratio (SIR) among all linear receivers. In section IV, we present the PDP impact in spectral effi ciency. Then, we present the simulation results. 2. System Description 2.1. The Multiband OFDM Solution The multiband technique proposed in the WiMedia Alli ance MBOFDM scheme divides the UWB spectrum into 14 bands of 528 MHz each, as illustrated in Figure 1. The first 12 bands are then grouped into four band groups consisting of three bands each. The last two bands are grouped into a fifth band group [2]. Initially, most of the studies have been performed on the first three sub bands from 3.1 to 4.8 GHz. An OFDM signal can be transmitted on each subband using a 128 point inverse fast Fourier transform (IFFT). Out of the 128 subcarriers used, only 100 are assigned to transmit data. The multi user access is performed with time frequency codes (TFC) which provide frequency hopping from a subband to another at the end of each OFDM symbol. Hence, at a given instant, if we consider a 3 user system, each user occupies one of the first three subbands [ 2]. In the MBOA solution case, the signal generated at the output of the IFFT [2] is:
H. CHIHI, R. BOUALLEGUE Copyright © 2013 SciRe s. CN ( ) ( ) ( ) 2() F cp Mjnt iT OFDM n cCP iM Xip t Te St π +∞ +∆− =−∞ − = − ∑∑ (1) ΔF, M and TCP represent respectively the subcarriers spacing, the total number of used subcarriers and the spacing between two consecutive OFDM symbols. Xn(i) is a complex symbol belonging to QPSK constellation and is transmitted by subcarrier n during the ith OFDM symbol. It represents a data, a pilot or a reference symbol, pc(t) is a rectangular window defined by [2] as: ( ) ( ) 1 for 0 0 for c FFT cFFTFFTCP GI pt tT ptTtTT T = ≤≤ =≤≤+ + Transmitted data rates in each subband vary from 53.3 to 480 Mbit/s, which are listed in Table 1 [2]. 2.2. UWB Channel Modeling The proposed model for MBOFDM is 802.15.3a high data rate channel which derived from Saleh Valenzuela model [3]. Mathematically, the impulse response of the multipath model is given by [4]: ()( )( )( ) () 00 ,, kk ZB kk zb kk htXzbzzbtT αδ = = − − ∑∑ (2) where: Xk is the lognormal shadowing of the kth channel re alization, Tk(z) is the dela y of clu ster z, and αk(z, b), τk(z, b) represent respectively the gain and the delay of multipath b withincluster z respectiv ely. Figure 1. UWB spectrum bands in the MBOFDM system. Table 1. MBOFDM UWB data rates [4]. Data rate Coding rate 53,3 1/3 80 1/2 106,7 1/3 160 1/2 200 5/8 320 1/2 400 5/8 480 3/4 2.3. ChDMAUWB System Principle Analysis The idea behind ChDMA is to use the channel impulse response (CI R) of each user as a signature, i.e . , th e sign a ture code is given by the channel and the users are sepa rated as a CDMA system. It is important to note that the signatures are given by the environment and by user’s position, which means that they are uniquely determined. This signature location dependent property provides de centralized flexible multiple access as the codes are natu rally generated by the radio channel [5]. The previous multiple access schema don’t benefit of using low duty cycle transmissions. For this reason, in this paper we consider a multiuser communications system which ex ploits the duty cycle assumption and sends a modulated peaky signal. The new multiple access considered system is based on the integration of ChDMA in the 802.15.3a model. The ChDMA proposition works because UWB channels have a large coherence time (typically about 100 µs) relatively to their delay spread (typically around 15  40 ns, depending on the user environment) [6]. Fol lowing the clustered ray model, each ray is considered as a single replica of the sign al due to the distortion accom panied the environment, yields that each user occupies an appropriate position as presented in Figure 2 [6], where we consider an uplink channel between user K and the destination. For this reason, the transmitted signal identi fication is easily obtained where the receiver benefits of the different position occupied by each user independ ently, as it knows the channel it will be able to detect and to demodulate the received wavefor m. 3. Spectral Efficiency Performance Analysis We analyze the performance of ChDMA UWB system in terms of spectral efficiency using two different receivers: • The optimal receiver. • The minimum mean square error receiver. The spectral efficiency γ is defined as the number of bits per chip summed over the users that can be reliably Figure 2. Channel Impulse Division Multiple Access with three users.
H. CHIHI, R. BOUALLEGUE Copyright © 2013 SciRes. CN transmitted [7]. It is expressed as the bits per second per Hertz (bit/s/Hz) supported by the syst e m. 3.1. Linear Minimum Mean Square Error Receiver The concept of MMSE detector originates from turning the problem of detection of transmitted symbols in a CDMA system into a problem of linear estimation [8]. In the following, we consider the uplink communication, where the SINRk,n at the output of the MMSE detector is given by [9]: ( ) 1 2 , ,,,, HH knkn knknkn SINRhH HIh σ − = + . (3) where n is related to nth symbol from N and k related to the kth user from K, Hk,n is the matrix obtained from H suppressing the column hk,n. The instantaneous spectral efficiency is given by [5]: 2, 1 21 2, ,,, 1 1log (1) 1log (1()). N kn i NHH knkn knkn i SINR N hHHI h N γ σ = − = + = ++ = ∑ ∑ (4) We must mention that the signal to noise ratio 1/σ2 is related to the spectral efficiency γ following the author in [7] by: Deeper insight on the linear MMSE spectral efficiency performance behavior is obtained byK, N → ∞ with con stant ratio β where β = K/N. The study of the eigenvalue distribution of random matrices has triggered important consideration in many numerous works. For this reason, the performance of MBOFDM ChDMA UWB communication system is determined via the eigenvalues of the covariance matrix . We study the eigenvalues distribution performance of linear MMSE multiuser receiver in ran dom environment. The eigenvalue distribution is de scribed in Figure 3 [10]: The formula for the MMSE demodulator could be de scribed via considering the transformation of CN using the singular value decomposition (SVD). We denote the SVD representation of the channel correlation matrix CN: . where S is a unitary matrix containing the eigenvectors of HHH and D is a diagonal matrix, containing the singu lar values λi with i = 1···N representing the eigenvalues of HHH. Second, following results from random matrix theory [11], it can be deduced that the empirical distribu tion of the eigenvalues of CN converges to some limiting distribution F, we obtain: Figure 3. Eigenvalue distribution. ( ) 1 2 ,, , H knkm ikn SINRhI h λσ − = + (5 ) We assume a large system, i.e., we let K, N → ∞ while K/N is finite and converges to a specific value . Yields in the asymptotic analysis, the spectral efficiency can be expressed in terms of the eigenvalue distribution of HHH and it is given by: 22 1log (1) 1/ m K ms i ei i N λ λσ =++ ϒ= ∑ . (6) We must mention that in [12] (Lemma 9) if we con sider the key performance criterion as SIR of user k un der the MMSE detector we obtain: 11 11 KK kk ii kk SIR SIR λ λ = = = ++ ∑∑ . By this way (6 ) c ould be described by: 22 1log 1/(1 ) mm e Ki i s i SIR IR NS σ = ϒ+ + =∑ . We should point out that (6) converges in probability to: ( ) 22 log (1) N iH i mmse fF λ λσ λ ++ ϒ= ∫ . (7) where defined by the author in [13] by: is the empirical eigenvalue distribution (e.e.d) of the ran dom matrix H, i.e. . Let GA(z) the Stieltjes Transform of as defined by [14]: ( )() 1 N HA d or zzFGz f λ λ + − = ∈ ∫ . where Yields:
H. CHIHI, R. BOUALLEGUE Copyright © 2013 SciRe s. CN ( ) ( ) ( ) ( ) ( ) 2 22 2 2 2 2 log log 2 log . mN ii H oi N H o m N o s H e dF dF dF λσ λλ λσ λσ λ λσ λ ∞ ∞ ∞ ++ = + = + −+ ϒ ∫ ∫ ∫ (8) Following Theorem 6 in [15], we have that F converge if to a distribution function G, we obtain the SNR of the MMSE receiver convergence to: ( ) 21 , ,,,2 1 () . HH kmkm kmkm hH HIhdG σρ λ λσ − +→=+ ∫ (9) Finally we obtain: Additionally, for asymptotically large random matrices such upper bound s are prov ided by r andom matrix th eory. If the channel matrix is composed of independent identi cally distributed random entries. The eigenvalues of HHH are asymptotic a l l y upper bounded by [16] : . By this way the bounds of (8) are given by: () ( ) ( ) ( ) 2 22 22 o log 2λσdG λlog 2(1β)σdG λ ∞ +<++ ∫∫ ( ) () ( ) 2 22 22 o log ()dGλlog (1)σdG λ. λσ β ∞ +< ++ ∫∫ Yields: ( ) ( ) ( ) 22 2 22 2 log 2(1) log (1)d mmse G βσ βσ λ ++ − ++ ϒ< ∫ We obtain: ( ) ( ) 22 222 21 log 1 mmse βσ σ γ β + < + ++ . Following Theorem 2 presented in [17] we have that under the condition of large system the minimal value of eigenvalue is given by: . The minimum bounds of (8) are gi ven by: ( ) () () () 2 22 22 o log 2λσdG λlog 2(1β)σdG λ. ∞ +> −+ ∫∫ ( ) ( ) ( ) 2 22 22 o log ()dGλlog1dGλ. λσβ σ ∞ +>−+ ∫∫ Let the minimal bound is given by: ( ) ( ) 22 222 21 log 1 mmse βσ σ γ β − > + −+ . If we consider the case of 0 < β < 1, and following the author in [18], the density of F(x) is given by: ( ) ( )( ) 22 22 for (((1)(((1)) )/ 0, otherwi 2 s . e. 11 k f β λββ λπβλ ββ λ λ −−+ − = << −+ Then, if we consider the case of 0 < β < ∞, we obtain: ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) 2 2 22 1 2 1 1/01. () 11 1. 1 1/ 1 N H N H k H k Nk for for whe dt n F Ff F λ β β β λβ β β λβλ λβ λ λλ − +=− << =− +<< = > −+ + ∫ 3.2. Optimal Receiver The optimal receiver is defined as the receiver that mi nimizes the probability of symbol error among all re ceiver structures. It is based on the analysis of the post erior probabilities of the transmitted signal [19]. The spectral efficiency is defined by: ( ) ( ) ( ) 2 2 1/*1/ H N Nlog detIHH γσ = + . (10) where: σ2 is the noise variance and N the channel vector length. Equation (10) can also be represented like the author in [6]: ( ) ( ) 2 1/ *. H N Nlog detIHH γρ = + (11) where ρ is defined in [6] by: . In fact, the asymptotic analysis allows providing a good understanding of the ChDMA limiting behavior in MBOFDMUWB channel. The author in [6] allows us to write the spectral effi ciency represented in (10) in terms of the eigenvalues (λ1···λm) of HHH like:
H. CHIHI, R. BOUALLEGUE Copyright © 2013 SciRes. CN ( ) ( ) () log 1.1/ * NH ii H NH ρλ γ += ∑ (12) Following, the author in [20], when N → +∞, the spectral efficiency can be represented in terms of the eigenvalue distribution of HHH yields to: . (13) We conclude that the spectral efficiency is determinis tic and only depends on a few macroscopic system pa rameters in asymptotic regime. 4. Power Delay Profile Following the Toeplitz structure of the matrix CN, the followin g eigendecomposi tion hol ds when N → ∞ [21]: where FN being the Fourier matrix and D could be de fined following the decomposition in [22] according to its elements Dn,n. where: PDP refers to power delay profile representing the ex ponential decay of each cluster; also it reflects the decay of the total cluster power with the delay. The detailed definition is given following the author in [23]: , 2 ,0 kl l lk T l Eee τ γ ξβ − − Γ = Ω . (15) with Ω0 represents the average energy of the first path of the first cluster, Γ and γ are, respectively, defined as con stants that characterize the exponential decay of each cluster and each ray. • Wc repre s ents the frequency resoluti on. • Tl is the arrival time of the first path of the lth cluster and L is the number of multipath. We evaluate the impact of a power delay profile (PDP) on the spectral efficiency. In fact, if we refer to (14) and replace PDP with his previous expression in (15), we can represent the results of spectral efficiency with a random generation of the channel by employing the diagonal matrix D .We assume, respectively, the number of fre quency samples N, the frequency resolution Wc, the number of multipath L, 128,40 MHz and 100. Figure 4 presents the impact of the PDP on the spec tral efficiency. As the SNR increase, the spectral effi ciency of the simulated channel increase and this depend intimately to the system parameters. The result shows that the energy is equally spread over all the bandwidth. In fact, PDP has been shown to be a useful measure, and we have employed it to identify the optimum number of parameters to represent the performance. This increase of spectral efficiency is also due to the attractive choice of OFDM for UWB communication because it can capture the multipath energy efficiently. The obtained result shows that the cluster s are disjoint because we have an increase of spectral efficiency. How ever, we can find generally some overlap between the lth and (l + 1)th clusters. 5. Results In this section, we present the simulation r esults obtained by averaging over 500 Monte Carlo iteration, we consid er the mode I of the MBOFDM employing the first three subbands of 528 MHz. Additionally, each realization of the channel model related to each user is generated inde pendently and assumed to be time invariant during the transmission of a frame. We simulated the MMSE spec tral efficiency using the CM1UWB channel model where the parameters are present ed in Table 2 [24] which is pre sented in Figur e 5. In Figure 6 we present a comparison of the spectral efficiency of optimal receiver performance results over the CM1 UWB model. The obtained results show that the performance of the MBOFDM ChDMAUWB high data rate system is critically dependent of the receiver struc ture as the number of users increase, the spectral effi ciency increases as well. The increase in spectral effi ciency comes from the multiuser diversity effect provided by ChDMA system; if we consider many users transmit ting independently, we could find a user with strong 0510 1520 25 0 10 20 30 40 50 60 70 80 90 100 SNR S pc tral eff i ciency Figure 4. Impact of the power delay profile. ( ) () () ( ) ( ) ( ) ( ) ( )( ) 2 c lclcl 22 N H n 1 ,n PDP*L/W*TwithNW T*l1/Ln1NW T*l/ F L 1? D 0.Otherwise ()for 11 k f dt β λ β λβ λ λβ β − − −≤ −≤ =+< < = + ∫ (14
H. CHIHI, R. BOUALLEGUE Copyright © 2013 SciRe s. CN channel yields that the channel is used in the most effi cient manner where the total throughput is maximized. By this way, we obtain a maximized multiuser diver sity gain .We must mention also that QPSK is the best practical choice in the MBOFDM ChDMAUWB chan nel coherent regime when the receiver knows the chan nel. 6. Conclusions In this contribution we aim at providing a very good technical solution to be proposed for UWB physical layer 024 6810 12 14 16 18 20 15 20 25 30 SNR S pc tral eff i ciency opti m al receiver as ymptotic Figure 5. Validation of ChDMAUWB channel model with MMSE detector (Eb/No = 5 dB). Table 2. MBOFDM UWB system parameters. WiMedia Number of data subcarriers 100 number of pilot symbols 12 number of guar d s ymbols 10 Subcarrier frequency spacing (Mhz) 4,125 zero padding duration (ns) 60,61 00.2 0.40.6 0.811.2 1.41.6 1.82 1 2 3 4 5 6 7 8 9 10 11 S pec tral Ef f i ci enc y K/N Figure 6. Comparison between optimal and asymptotic spectral effi ci ency. high data rate wireless applications called ChDMA. This approach, simplifies the transmitter ar chitecture allowing simultaneous access as we assimilate the natural disper sion of the channel to code signature. We have studied the eigenvalues distribution performance analysis of the MMSE and optimal receiver in MBOFDM ChDMA  UWB channel in terms of spectral efficiency assuming that the system size is large. Additionally, we investi gated the impact of power delay profile (PDP) on the spectral efficiency of MBOFDM UWBChDMA high data rate system. As perspectives, we could investigate other receiver structure that will be adapted for the considered system. We are more ambitious in studying mutual information criteria as well channel es timation mismatches. REFERENCES [1] M. Kamoun, S. Chaillouz, J. Gosteau, L. Mazet and M. de Courville Pierre Duhamel, “Data Rate Upper Bounds for UWB Link with IEEE802.15.3a Channel Model,” IEEE International Conference, ICU 2005. [2] A. Stephan, “Resource Allocation Strategies and Linear Precoded OFDM Optimization for UltraWideband Com munications,” Ph.D. Thesis, National Institute of Applied Sciences (INSA), 2008. [3] A. Saleh and R. Valenzuela, “A Statistical Model for Indoor Multipath Propagation,” IEEE Journal on Selected Areas in Communications, Vol. 5, No 2, 1987, pp. 128 137. http://dx.doi.org/10.1109/JSAC.1987.1146527 [4] J. Foerster, “Channel Modeling SubCommittee Report Final,” IEEE Document P802.1502/490r1, Intel R&D, 2003. [5] Raul L. de Lacerda Neto, Aawatif Menouni Hayar and Merouane Debbah, “Channel Division Multiple Access: New Multiple Access Approach for UWB Networks,” European Ultra Wide Band Radio Technology Workshop 2007, Grenoble, France, 1011 May 2007. [6] R. L. De LacerdaNeto, “Exploiting the Wireless Channel for Communication,” Ph.D. Thesis, NiceSophia Antipo lis University, 2008. [7] S. Verdu and S. Shamai, “Spectral Efficiency of CDMA with Random Spreading,” IEEE Transactions on Infor mation Theory, 1999, pp. 622640. http://dx.doi.org/10.1109/18.749007 [8] Z. Xie, R. T. Short and C. K. Rushforth, “A Family of Suboptimum Detectors for Coherent Multiuser Commu nications,” IEEE Journal on Selected Areas in Commu nications, Vol. 8, No. 4, 1990, p. 683. [9] L. Cottatellucci, R. R. Muller, M. Debbah, “On the Ca pacity of Asynchronous CDMA Systems,” Conference Record of the Fo rtyFirst Asilomar 2007. [10] R. Couillet, “Random Matrices in Wireless Communica tions Course 2: System Performance Analysis: Capacity and Rate Regions,” 2009. [11] V. A. Marcenko and L. A. Pastur, “Distribution of Eigen
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