Multiband Differential Modulation for UWB Communication Systems

Transcription

1 Multiband Differential Modulation for UWB Communication Systems Thanongsak Himsoon, Weifeng Su,andK.J.RayLiu Deartment of Electrical and Comuter Engineering, University of Maryland, College Park, MD 074. Deartment of Electrical Engineering, State University of New York (SUNY at Buffalo, Buffalo, NY Abstract In this aer, we roose a differential encoding and decoding scheme for multiband UWB systems. The roosed scheme incororates frequency-domain differential en/decoding with the hoing multiband OFDM modulation. To cature the effect of multiath-rich clustering roerty of UWB channels, we characterize the airwise error robability erformance of the roosed scheme in terms of cluster and ray arrival rates. It turns out that the diversity advantage does not strongly deend on the random-clustering of UWB channels, and we can achieve the same diversity gain in different channel environments. However, the system erformance relies on the clustering behavior through the coding gain. Simulation results show that the roosed differential scheme achieves good erformance in the shortrange line-of-sight scenarios. In addition, the jointly encoded differential multiband UWB scheme is able to yield suerior erformance to the uncoded coherent multiband UWB system at high SNR. I. INTRODUCTION Ultra-wideband (UWB is an emerging technology that offers great romises to satisfy the growing demand for low cost and high-seed digital wireless home networks. A traditional UWB technology, which occuies the available bandwidth of 7.5 GHz, is based on single-band aroaches that directly modulate data into a sequence of imulse-like waveforms. Recently, multiband UWB schemes were roosed 1, in which the entire UWB frequency band is divided into several subbands, each with a bandwidth of at least 500 MHz. Since many alications enabled by UWB are exected to be in ortable devices, low comlexity becomes a fundamental requirement. This indicates the imortant need of a simle transceiver design. In conventional coherent detection system, it requires channel estimation and hence introduces comlexity to the receiver. An alternative aroach to overcome such roblem is through the use of non-coherent detection techniques. In recent years, non-coherent UWB systems have been roosed, e. g. in. Nevertheless, most of the existing works, are based on single-band imulse radio technology. The current works for multiband UWB mostly focus on coherent detection schemes 3. Differential sace-time modulation 4 has been widely acceted as one of many ractical alternatives that byasses multi-channel estimation and rovides a good tradeoff between erformance and comlexity in frequency-non-selective multile-inut multile-outut (MIMO systems. In order to further imrove the erformance and gain full sace-frequency diversity in wideband systems, differential modulation incororating with MIMO-OFDM transmission has been introduced in 5-7 and references therein. The scheme in 5-6 focuses on time-domain differential encoding, while that in 7 and This work was suorted in art by U.S. Army Research Laboratory under Cooerative Agreement DAAD some art of 5 are based on frequency-domain differential encoding. In this aer, we roose a differential encoding and decoding scheme for UWB systems emloying MIMO multiband OFDM. In the roosed scheme, the information is jointly encoded across satial, temoral, and frequency domains. By differentially en/decoding in the frequency domain, the roosed scheme does not rely on the assumtion that the fading channel stays constant within several OFDM symbol durations. In this way, we are able to exlore the available sace and frequency diversities, richly inherent in UWB channels. More imortantly, it allows us to incororate the differential transmission with hoing multiband OFDM modulation so as to gain the additional diversity from time-domain sreading. In order to cature the unique multiath-rich and randomclustering roerties of UWB channels, we characterize the airwise error robability erformance of the roosed scheme based on the Saleh-Valenzuela (S-V fading model. Finally, the merit of our roosed scheme is shown through comuter simulations. II. SYSTEM MODEL We consider a eer-to-eer multiband UWB system equied with M t transmit and M r receive antennas. Within each subband, OFDM modulation with N subcarriers is used at each transmit antenna. The modulated OFDM symbols can be time-interleaved across several subbands as secified in 1. According to the IEEE a standard 8, the fading channels for UWB systems are based on the S-V model for indoor channels 9. The mathematical model of the channel imulse resonse from the i th transmit antenna to the j th receive antenna during the k th OFDM block is given by 8 h k ij(t = αij(c, k lδ(t T c τ c,l, (1 where i =1,,M t and j =1,,M r. In each i j transmission link, αij k (c, l denotes the multiath gain coefficient of the l th arrival in the c th cluster at time k. The time duration T c reresents the arrival time of the c th cluster, and τ c,l is the delay of the l th ath in the c th cluster relative to the cluster arrival time T c. The cluster arrivals and the ath arrivals within each cluster are modelled by Poisson rocess with rate Λ and λ (λ >Λ, resectively. The ath amlitude αij k (c, l may follow the log-normal, Nakagami or Rayleigh distributions 8, whereas the hase αij k (c, l is uniform random variable over 0, π. In this aer, αij k (c, l is modeled as Rayleigh distribution, i.e., αij k (c, l are zero-mean comlex Gaussian random variables with variances 8 Ω c,l =E α(c, l ( =Ω 0,0 ex T c Γ τ c,l, ( γ 3789

2 where Ω 0,0 is the mean energy of the first ath of the first cluster, Γ is the cluster decay factor, and γ is the ray decay factor. The channel coefficients are assumed to be satially uncorrelated and the owers of all indeendent delay aths are normalized such that C L Ω c,l =1. The channel model arameters corresonding to different scenarios are rovided in 1. We denote x k i (n as a differentially encoded data symbol to be transmitted on the n th subcarrier at the i th transmit antenna during the k th OFDM symbol eriod. At the receiver, after cyclic refix removing and OFDM demodulating, the received signal at the n th subcarrier at the j th receive antenna during the k th OFDM block is given by yj k (n = M t ρ x k i (nhij(n+w k j k (n, (3 i=1 where ρ is the average signal to noise ratio er receiver, and Hij(n k = αij(c, k lex jπn f(t c + τ c,l (4 is the subchannel gain. Here, f =1/T s is the inter-subcarrier sacing, and T s is the OFDM symbol eriod. The additive noise wj k (n is modeled as indeendent comlex Gaussian random variable with zero mean and unit variance. III. THE PROPOSED DIFFERENTIAL SCHEME FOR MULTIBAND UWB SYSTEMS We roose in this section a frequency-domain differential scheme for multiband UWB system. In addition, we exloit the additional diversity from band hoing inherently in multiband transmission by jointly encoding across K OFDM blocks and transmitted the K OFDM symbols on different subbands. In each OFDM block, we exloit subcarrier interleaving strategy as in 7. A. Transmit Signal and Differential Encoding Structures We introduce a differential multiband UWB scheme based on a transmit signal structure roosed in 3. Particularly, X is a jointly design KN M t sace-time-frequency code structure in which it consists of stacking sace-frequency signal X k, each of dimension N M t,fork OFDM symbols. To reduce comlexity of the design, we divide X k into serval submatrices or grous. By introducing a fixed integer G (1 G N as a number of jointly encoded subcarriers, X k at each OFDM symbol is artitioned into P = N/(GM t submatrices as follows 3: X k = (X k 1 T (X k T (X k P T (0 N PGMt T T, (5 for k =1,,, K and T denotes the matrix transosition. The (N PGM t M t matrix 0 N PGMt reresents a zero adding matrix to be inserted if N cannot be divided by GM t. Each of the GM t M t submatrix X k,fork =1,,K, and =1,,...,P, is modeled as X k = diag(x k,1 x k, x k,m t, (6 where diag( denotes diagonal oeration that laces all vectors or scalar elements at the main diagonal matrix, and x k,i, for i =1,,...,M t,isag 1 vector: T = 1 X 1 1 s,1 0 s,1 0 s, OFDM 1 1 T s, 0 s, 0 0 s, 0 0 X = MOD 1 OFDM 0 s,3 0 s, Post MOD 1 0 s,4 0 s Multily, s,4 1 X X S K =, G =, Mt = KGMt = 8 Fig. 1: Examle of differential encoded signal matrix and transmit signal structure for the UWB system emloying multiband OFDM, K =, G =, and M t =. T x k,i = s k,(i 1G+1 sk,(i 1G+,iG sk, (7 in which all s k,m, m = 1,,...,GM t, are differentially encoded symbols that will be secified later. We will differentially encode across K OFDM symbols within each grou, and our desired transmit signal structure for the th grou after differentially encoding is KGM t M t matrix: X = (X 1 T (X T (X K T T, (8 in which the i th column contains encoded symbols to be transmit at the i th transmit antenna. We now secify information matrices to be differentially encoded as follows. Let V denote a KGM t KGM t unitary information matrix having diagonal form as V = diag(v,1 v, v,kgmt T, (9 in which v,m is an information symbol. We will jointly design the data within each information matrix V, but indeendently design the matrices V s for different. Let S be a KGM t KGM t differentially encoded signal matrix. We recursively construct S by 4 { V S S = 1, 1 I KGMt, =0. (10 Due to the diagonal structure of V, S can be exressed as S = diag(s 1,1,,s 1,GM t,,s K,1,,s K,GM t T, (11 where s k,m is the differentially encoded comlex symbol to be transmitted at subcarrier ( 1GM t + m during the k th OFDM block. In order to transform S into (8, we introduce a KGM t M t multilicative maing matrix ˆΦ = 1 K Φ, (1 where 1 K denotes a K 1 vector of all ones, denotes the Kronecker roduct 10, Φ =φ 1 φ φ Mt is the GM t M t maing matrix in which φ i = e i 1 G is a GM t 1 vector, and e i is an M t 1 unit vector whose its i th comonent is one and all others are zeroes. We ost-multily S by ˆΦ, resulting in the desired KGM t M t matrix X = S ˆΦ (13 as secified in (8. For better understanding the concet of the roosed scheme, we show in Figure 1 an examle of differentially encoded signals in case of K =,G =, and M t =. B. Differentially Decoding The received signal vector corresonding to the transmitted matrix X is given by y = ρ (I Mr D(X h + w, (

3 where D(X denotes an oeration on an KGM t M t matrix X that converts each column of X into a diagonal matrix and results in an KGM t KGM t M t matrix, exressed by D(X =D(x,1 x,mt = diag(x,1 diag(x,mt. (15 The matrix h = (h,1 T (h, T (h,mr T T is a channel matrix constructed from KGM t M t 1 matrix: h,j = (h 1,1j T (h K,1j T (h 1,M tj T (h K,M tj T, (16 where h k,ij = H k ij (( 1GM t H k ij (GM t 1 T is a channel gain vector of size GM t 1. The received signal matrix y = (y,1 T (y, T (y,mr T T is a KGM t M r 1 matrix constructed from the KGM t 1 receive signal vector y,j = (y 1,j T (y,j T (y K,j T T, in which y,j k = yj k(( 1GM t yj k(gm t 1 T is a GMt 1 matrix. The noise matrix w is in the same form as y with y,j and y,j k relaced by w,j and w,j k, resectively. By substituting (13 into (14, we can reformulate y as y = ( ρ I Mr D(S ˆΦ h + w. (17 To simlify (17, we first observe from (1 that ˆΦ can be re-exressed as ˆΦ = φ 1 φ φ Mt, where φ i = 1 K φ i. Therefore, D(S ˆΦ can be given by D(S ˆΦ = diag(s φ1 diag(s φmt. (18 According to (16 and (18 for each j, we have D(S ˆΦ h,j = M t i=1 diag(s φ i h,ij which can be simlified to M t D(S ˆΦ h,j = S φ i h,ij S h,j, (19 i=1 where the last term on the right hand side results from using the roerty of Hadamard roduct 10. The KG 1 channel matrix h,j can be obtained by substituting (16 into (19 as h,j = ( h 1,1j T ( h K,1j T ( h 1,M tj T ( h K,M tj T T, where h k,ij =Hij(n k 0,i Hij(n k 1,i Hij(n k G 1,i T (0 is of size G 1, and n g,i =(i 1G +( 1GM t + g for g =0, 1,...,G 1. By denoting a KGM t M r 1 channel gain vector: h = ( h,1 T ( h, T ( h,mr T T, (1 and using (19 for all j, we obtain an equivalent exression (I Mr D(X h =(I Mr S h. ( Finally, from ( we can simlify (17 to y = ρ (I Mr S h + w. (3 For notation convenience, let us define S (I Mr S and V (I Mr V such that S =(I Mr V S 1 = V S 1. (4 Accordingly, using (3-(4 and after some maniulations, we can write the received signal as y = V y 1 + w, (5 where w = 1 w V w 1 is a noise vector whose each element is indeendent comlex Gaussian random variable with zero mean and unit variance. Without acquiring channel state information, the detector follows the maximum likelihood (ML decision rule 4 ˆV = arg min y V y 1 F, (6 V V where F denotes the Frobinius norm 10. Even though the decoding comlexity increases exonentially with RKGM t where R is the transmission rate, the decoding comlexity can be reduced to olynomial in KGM t by lattice reduction algorithm 11. IV. PAIRWISE ERROR PROBABILITY In this section, we rovide an aroximate PEP formulation based on the results in 113. We first note that the channel matrix in (1 can be reexressed as h = h 1 + h, where h reresents the channel mismatch between h and h 1. For analytical tractability, this section confines the analysis to the case when h is negligible, i.e., h 1 h. Such erformance formulation rovides us a benchmark for subsequent erformance comarisons. Later in Section V, we will show from the numerical results how the channel mismatch affects the system erformance. For secific values of T c and τ c,l the PEP uer bound is given in (1, roosition 7. The average PEP can be obtained by averaging over Poisson distributions, however, it is difficult if not ossible to obtain the average PEP. In what follows, we use the aroximation aroach as in 13. Suose that V and ˆV are two different information matrices, the asymtotic PEP can be aroximated as P a V ˆV «! ν 1 νy 1 ρ ν β ν,m, (7 m=1 where ρ is an average signal-to-noise ratio er symbol, ν is the rank and β,m s are the non-zero eigenvalues of the matrix Ψ S 1 Σ h S H 1( V ˆV H ( V ˆV, (8 in which = E h Σ h hh denotes the correlation matrix of channel vector h. To simlify the exression for matrix Ψ in (8, we evaluate the channel correlation matrix as follows. Due to the Σ h band hoing, the K OFDM symbols in each signal matrix are sent over different subbands. With an ideal band hoing, we assume that the signal transmitted over K different frequencybands undergo indeendent fading. Assuming also that the MIMO channel is satially uncorrelated, we can find that = I KMr E h k,j ( h k,j H, and it can be simlified to Σ h Σ h = I KMr diag (R,1,, R,Mt, (9 where R,i E h k,ij ( h k,ij H denotes the correlation matrix and it is the same for all j s. From (0, we can see that the diagonal elements, i. e., the (u, u th elements, of R,i are C L R u,u,i = E H k ij(n u,i = E Ω c,l =1. (30 The off-diagonal comonents, i.e., the (u, v th for u v comonents, of R,i can be exressed as,i = E Hij(n k u,i ( Hij(n k v,i H = C L E Ω c,l e jπ f(nu,i nv,i (Tc+τ c,l. (

4 Observing that n u,i nv,i re-exress (31 as,i =Ω 0,0 = u v and using (, we can E e g( Γ 1,u,vTc g( 1 γ,u,vτ c,l. (3 where g(a, u, v = a + jπ(u v f. According to the Poisson distribution of the multiath delays, T c and τ c,l can be modeled as summations of identically indeendent distributed (iid exonential random variables with arameter Λ and λ, resectively. Therefore, averaging (3 over the distribution of T c and τ c,l, we arrive at,i =Ω Λ+g( 1 Γ,u,v λ + g( 1 γ,u,v 0,0 g( 1 Γ,u,v g( 1 γ,u,v. (33 Since,i is the same for all i s and s, we denote R R,i, which allow us to further simlify (9 to = I Σ h KMtM r R. (34 Substituting (34 into (8 and alying the roerty of tensor roduct (A 1 B 1 (A B (A 3 B 3 = (A 1 A A 3 B 1 B B 3, we obtain Ψ = I Mr Θ, (35 in which Θ = S 1 (I KMt RS H 1, (36 and = ( V ˆV H ( V ˆV. Hence, by (35, the PEP in (7 can be exressed as P a V ˆV (! Mr rmr 1 ry ρ rmr λ rm,m. r m=1 (37 where r is the rank of Θ and λ,m s are the non-zero eigenvalues of Θ. To quantify the maximum diversity order which is the exonent of ρ/ in (37, we observe from (36 that S 1 and V are of size GKM t GKM t, and the correlation matrix R is of size G G. Therefore, the maximum diversity gain is = M r max rank(θ = GKM tm r. (38 G max d min V k ˆV k Note that R is of full rank if G is less than the total number of multiath comonents (C + 1(L + 1. Due to the large bandwidth of UWB waveform, the received signal tyically contains a significant number of resolvable multiath comonents. Consequently, the correlation matrix R is generally of full rank. Therefore, the maximum diversity order of GKM t M r can be achieved by using a set of roer designed codeword matrices V. The result in (38 leads to some interesting observations as follows. First, the differential multiband UWB system achieves the same diversity gain under different channel environment. This imlies that the clustering roerty of UWB channel does not strongly affect the diversity gain of differential multiband system. On the other hand, the coding gain which is a function of r m=1 λ,m is severely affected by the multiath arrival rates and decay factors through the correlation matrix R. Second, by incororating the frequency-domain differential scheme with the multiband transmission, we are able to achieve the diversity gain of GKM t M r, regardless of the channel time-correlation roerty. This is different from the use of differential STF coding in the conventional MIMO- OFDM systems, e.g. in 5, where the maximum achievable diversity gain is only GM t M r due to the requirement of almost constant channels over several OFDM blocks. V. SIMULATION RESULTS We erformed simulations for a multiband UWB system with N = 18 subcarriers and each subband occuies bandwidth of 58 MHz. The channel model arameters followed those for CM 1 and CM 8. The data matrix V in (9 were constructed by jointly coding across G, K, and M t using existing cyclic grou codes 4. In case of reetition based coding, the codeword is given by V = I K v, where v is a GM t GM t jointly encoded diagonal matrix. Figure deicts the erformances of single-antenna multiband UWB system with different number of G and K. Forfair comarison, the sectral efficiency is fixed at R =1b/s/Hz for all cases. The erformances are simulated under CM 1. For uncoded differential system (G =1and K =1, we can see that the erformance loss is more than 3 db comared to the coherent detection, and an error floor can be observed. This is due to the effect of the channel mismatch between adjacent subcarriers. By jointly encoding across two OFDM symbols (G =1and K =, the diversity gain is increased, hence resulting in significant erformance imrovement. As shown in Figure, the erformance gain is more than 7 db at the of 10. By further jointly encoding across two subcarriers (G =and K =, the roosed scheme obtains additional 4 db gain at a of. This observation is in accordance with our theoretical result in (37 that the erformance can be imroved by increasing the number of jointly encoded subcarriers or the number of jointly encoding OFDM symbols. Moreover, at high SNR, the roosed jointly encoding differential scheme outerforms the uncoded multiband UWB system with coherent detection. We observe about 1 db gain when G =1and K =, and about 3 5 db gain when G =and K =at between 10. In Figure 3, we comare the erformance of the roosed differential scheme under CM 1 and CM. The information is transmitted reeatedly across K = 1,, and 3 OFDM symbols, hence the transmission rate is 1/K b/s/hz. We can see that the erformance of the roosed scheme under CM 1 is better than that under CM for all cases. This is due to the fact that the multiath comonents in CM are more random than that in CM 1, which imlies that comared with CM 1, CM results in larger channel mismatch, and hence worse erformance. For each channel model, the erformance imroves as the number of encoded OFDM symbols increases which confirms our theoretical analysis. Figure 4 deicts the erformances of differential UWB- MIMO systems. The number of jointly encoded OFDM symbolsisfixedatk =1, and the sectral efficiency is R = 1 b/s/hz for all cases. From Figure 4, we can observe the erformance imrovement as the number of antennas increases. When using two transmit and one receive antennas and encoding across one subcarrier and one OFDM symbol, the roosed scheme yields 7 db imrovement over the single antenna system. When we further jointly encode across two 379

5 Differential : K Differential : K =, G = 1 Differential : K =, G = Coherent : K = 1, M r =, M r =, M r = 1, G = =, M r =, G = Coherent : M t = 1, M r E b /N 0 (db Fig. : Performance under CM1, M t =1,M r =1,R =1b/s/Hz. 10 K = 3, G = 1 K K =, G = 1 CM CM E b /N 0 (db Fig. 3: Performance under CM1 and CM, M t = 1, M r = 1, R = 1/K b/s/hz. subcarriers, additional erformance gain of about 4 db can be observed at a of. However, slightly error floors can still be observed when the data is encoded across multile transmit antennas since the chance of channel mismatch is higher in this case. On the other hand, increasing the number of receive antennas imroves the diversity gain without the tradeoff in the channel mismatch. In articular, an additional erformance gain of 6 db is observed when two receive antennas are emloyed. VI. CONCLUSIONS We roose in this aer a frequency-domain differential scheme for multiband UWB systems. By a technique of band hoing in combination with jointly coding across satial, temoral and frequency domains, The roosed scheme is able to exlore the available satial and multiath diversities, richly inherent in UWB environments. The analysis reveals that the roosed differential scheme achieves the same diversity advantage under different channel environments. However, the clustering behavior of UWB channels affects the erformance through the coding gain. For single antenna multiband UWB system, simulation results show that the roosed differential multiband scheme yields suerior erformance to the conventional differential encoding scheme, articularly under very short-range line-of-sight scenario, e.g. in CM 1. We obtain E /N (db b o Fig. 4: Performance comarison of the roosed differential scheme under CM1 emloying SISO and MIMO rocessing, K =1and R =1b/s/Hz. about 7 db gain at a of 10 when jointly encoding across one subcarrier and two OFDM symbols. Moreover, at high SNR range, the roosed jointly encoded differential scheme outerforms the uncoded coherent detection scheme of about 3 5 db at between 10. In case of multiband UWB system with multile transmit antennas, while slightly error floor occurs due to the effect of channel mismatch, additional diversity can be observed when number of transmit antennas is increased. However, increasing the number of receive antennas imroves the diversity gain without tradeoff in erformance due to the effect of channel mismatch. REFERENCES 1 A. Batra, Multi-band OFDM Physical Layer Proosal for IEEE P80.15 Task Grou 3a, Mar M. Ho, V. S. Somayazulu, J. Foerster, and S. Roy, A differential detector for an ultra-wideband communications system, IEEE Vehicular Technology Conf., vol. 4, no. 9, , May W. P. Siriwongairat, W. Su, M. Olfat, and K. J. R. Liu Sacetime-frequency coded multiband UWB communication systems, IEEE Wireless Commun. and Networking Conf., vol. 1, , Mar B. M. Hochwald and W. Sweldens, Differential unitary sace-time modulation, IEEE Trans. Commun., vol. 48, , Dec Q. Ma, C. Teedelenlioğlu, and Z. Liu, Full diversity block diagonal codes for differential sace-time-frequency coded OFDM, IEEE Global Telecommun. Conf., vol., , Dec. 1-5, T. Himsoon, W. Su, and K. J. R. Liu, Single-block differential transmit scheme for frequency selective MIMO-OFDM systems, IEEE WCNC, vol. 1, , Mar W. Su and K. J. R. Liu,, Differential sace-frequency modulation for MIMO-OFDM systems via a smooth logical channel, IEEE Global Telecommun. Conf., Dec J. Foerster, et. al, Channel modeling sub-committee reort final, IEEE /490, Nov. 18, A. A. M. Saleh and R. A. Valenzuela, A statistical model for indoor multiath roagation, IEEE J. on Selected Areas in Commun., vol. 5, no., , Feb R. A. Horn and C. R. Johnson, Toics in Matrix Analysis, NewYork: Cambridge Univ. Press, K. L. Clarkson, W. Sweldens, and A. Zheng, Fast multile antenna differential decoding, IEEE Trans. Commun., vol. 49, , Feb M. Brehler and M. K. Varanasi, Asymtotic error robability analysis of quadratic receivers in Rayleigh-fading channels with alications to a unified analysis of coherent and noncoherent sace-time receivers, IEEE Trans. Inform. Theory, vol. 47, no.6, , Se W. Pam Siriwongairat, W. Su, K. J. R. Liu, Characterizing erformance of multiband UWB systems using Poisson cluster arriving fading aths, IEEE SPAWC, , Jul