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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009 1359 Effects of Non-Identical Rayleigh Fading on Differential Unitary Space-Time Modulation Meixia Tao, Member, IEEE Abstract—Non-identical fading distribution in a multiple-input both time and space. A number of unitary ST signal sets multiple-output (MIMO) channel, including unequal average have been designed, including orthogonal codes [3]–[5], cyclic channel gains and fade rates, often occurs when antennas are not group codes [6], and Cayley differential codes [7]. co-located. In this paper, we present an analytical study of the effects of non-identical Rayleigh fading on the error performance It is commonly assumed in the design and performance of differential unitary space-time modulation (DUSTM). The analysis of space-time coding that the channels on different fading processes for different transmit-receive antenna pairs are transmit-receive antenna pairs are statistically identical. The assumed to be independent and time-variant. We ﬁnd that the assumption typically holds when antennas in the system maximum-likelihood (ML) differential detector of DUSTM over such channels is involved except for differential cyclic group are co-located and hence the channel path loss, as well as codes. The conventional detector is proved to be asymptotically potential shadowing, experienced by each signaling branch is optimal in the limit of high signal-to-noise ratio (SNR) over static the same to each other. There are many occasions, however, fading channels. Applying the distribution of quadratic forms of that the antennas are not necessarily co-located. For instance, Gaussian vectors, we then derive closed-form expressions for the in distributed antenna systems [8], [9], the antennas are exact error probabilities of two speciﬁc unitary classes, namely, cyclic group codes and orthogonal codes. Simple and useful geographically distributed at different radio ports and are asymptotic bounds on error probabilities are also obtained. Our connected together through high-speed cables. It is natural to analysis leads to the following general ﬁndings: (1) equal power expect different path loss as well as fade rates on different allocation is asymptotically optimal, and (2) non-identical channel links. Similarly, in aeronautical telemetry communications gain distribution degrades the error performance. Finally, we [10], multiple antennas can be placed at different parts of also introduce a water-ﬁlling based power allocation to exploit the transmit non-identical fading statistics. the air vehicle and hence they experience different attenuation during maneuvers. Cooperative communications among mo- Index Terms—Differential detector, error probability analysis, independent and non-identical channels, Rayleigh fading, space- bile nodes in a network is another important scenario. After time modulation. knowing each other’s data to be sent, the cooperating nodes can form a virtual multiple-antenna system and employ space- time coding in a distributed manner [11], [12]. Clearly, the I. I NTRODUCTION different signaling branches in the cooperation phase can have T HE use of multiple antenna elements promises con- siderable diversity and multiplexing gains in wireless communication systems. This motivated enormous develop- unequal fading statistics. In all the aforementioned MIMO (or virtual MIMO) communication settings, the resulting channels can be modeled as independent and non-identically distributed ment of multiple-input multiple-output (MIMO) techniques (i.n.i.d) fading. in the context of space-time (ST) coding and modulation The goal of this paper is to study the effects of non- in the last decade. Existing ST techniques can be broadly identical fading distribution on the performance of existing ST divided into coherent schemes and non-coherent schemes, codes, in particular differential unitary space-time modulation. based on whether or not instantaneous channel knowledge is There are two issues to be addressed. First, whereas uniform needed by the receiver. As channel estimation is waived, non- power allocation in the spatial domain for both coherent and coherent schemes can not only reduce receiver complexity non-coherent ST codes is capacity-achieving in traditional but also lower transmission overhead required for sending independent and identically distributed (i.i.d) fading, it may pilot symbols. Among the non-coherent schemes, differential not be so in i.n.i.d fading. Therefore, it is of interest to unitary space-time modulation (DUSTM) [1], [2] is known investigate the optimal power allocation among the distributed for its good error performance and high spectral efﬁciency. antennas (or cooperating nodes). Second, the conventional DUSTM is often viewed as a multiple-antenna counterpart of differential detector for DUSTM over i.i.d channels may no differential phase-shift-keying (DPSK) modulation, where the longer be optimal in the maximum-likelihood (ML) sense. signal constellation is a set of unitary matrices spread over Hence, optimal differential detector is to be discussed. Paper approved by G. M. Vitetta, the Editor for Equalization and Fading Attempts have been made recently to study the effects of Channels of the IEEE Communications Society. Manuscript received October non-identical channels in MIMO systems from different as- 15, 2007; revised February 8, 2008. This work was presented in part at the IEEE International Conference on pects. The outage probability of mutual information and power Communications, Beijing, China, May 2008. This work is supported in part by control over distributed multiple-input single-output (MISO) the Doctoral Fund of the Ministry of Education of China (No. 20082481002). channels with independent Rayleigh fading are studied in [13]. The author is with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China (e-mail: mxtao@sjtu.edu.cn). The bit error probabilities (BEP) of coherent orthogonal space- Digital Object Identiﬁer 10.1109/TCOMM.2009.05.070534 time block codes (OSTBC) over i.n.i.d Rayleigh/Riciean and 0090-6778/09$25.00 c 2009 IEEE Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. 1360 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009 Nakagami fading channels are analyzed in [14] and [15], antenna elements at both the transmitter side and receiver respectively. In [16], the authors derived the BEP of differ- side are not necessarily co-located. At each time block k, ential OSTBC, i.e., the orthogonal-design based DUSTM [3], a set of log2 L information bits are mapped onto a data [5], over independent and semi-identically distributed (i.s.i.d) matrix D[k] ∈ V, where V = {Di , 0 ≤ i < L} denotes Rayleigh channels, where the non-identical fading occurs at a unitary space-time signal constellation with cardinality L. the receiver side only. The study in [16] shows that in i.s.i.d Each element of V is an M × M unitary matrix, satisfying channels the ML differential detector (DD) for differential Di DH = IM , for 0 ≤ i < L. For the special case of i OSTBC is still on a per symbol basis but should weight differential cyclic group codes [2], [6], the constellation set V the output from each receive antenna according to its fading forms a group under matrix multiplication and each element statistics. Moreover, the ML detector signiﬁcantly outperforms of it is a diagonal matrix. In the case of differential OSTBC, the conventional one at high signal-to-noise ratio (SNR) region each element Di is a linear mapping of a set of P M-ary PSK when the channel ﬂuctuates rapidly over time. modulated information symbols, denote as {sp = ejθp }P , p=1 In this paper, we extend the previous work in [16] to 1 P and is given by D[k] = √P p=1 (Φp cos θp + jΨp sin θp ). a general framework of DUSTM over i.n.i.d time-varying Here the set of encoding matrices {Φp , Ψp }P are chosen p=1 Rayleigh fading channels. We ﬁrst show that for a general subject to certain orthogonality constraints [17]. unitary space-time constellation the ML differential detector Let S[k − 1] denote the M × M dimensional code matrix needs to perform joint optimization of the current data matrix at the (k − 1)-th time block. The data matrix D[k] is then and the previously transmitted signal matrix. However, for differentially encoded as cyclic group codes, it is independent of the previous signals and differs from the conventional DD only by appropriate S[k] = D[k]S[k − 1], weights. The conventional DD is shown to be asymptotically optimal in the limit of high SNR over static fading channels. where the initial code matrix S[0] is an arbitrary unitary We then apply the well-established distribution of quadratic matrix. The actual signal matrix to be transmitted at time block forms of Gaussian variables to derive the error performance k over M antennas is given by for two speciﬁc unitary classes: orthogonal codes and cyclic group codes. For cyclic group codes, closed-form expressions X[k] = Es S[k]Σ1/2 , (1) for the exact pairwise error probabilities (PEP) with both ML and conventional DD at arbitrary channel ﬂuctuation rates where Es is the total transmit power, and Σ1/2 = √ √ are derived. For orthogonal codes, closed-form expressions diag{ ε1 , . . . , εM } is the diagonal power allocation matrix. for the exact BEP with conventional DD in static fading The power allocation coefﬁcients εm ’s are subject to the are derived. Furthermore, simple asymptotic bounds on error constraint M εm = M and to be optimized. m=1 probabilities for both codes are obtained. These bounds lead to Since the transmission is on a per block basis, we assume several useful ﬁndings applied to any DUSTM design. Lastly, the channel is block-wise time-varying with each block con- we propose a water-ﬁlling based power control to exploit taining M symbol intervals. Let H[k] denote the M × N the transmit non-identical fading statistics. This is carried channel matrix of the k-th transmission block, where the out by minimizing the Chernoff bound of approximate error (m, n)-th entry hmn [k] represents the fading coefﬁcient from probabilities under a total power constraint. the m-th transmit antenna to the n-th receive antenna. Each The rest of the paper is organized as follows. In Section {hmn [k]}k is modeled as a complex Gaussian wide-sense II we present the system model of DUSTM over i.n.i.d stationary random process with zero mean and autocorrelation time-varying ﬂat Rayleigh fading channels. The optimal and function 2Rmn [l] = E[hmn [k]h∗ [k − l]], and is independent mn suboptimal detectors are presented in Section III. The analysis for different m and n. The channel variance and block 2 of error probabilities is presented in Section IV, followed by correlation coefﬁcient are denoted as σmn = 2Rmn [0] and 2 the derivation of transmit power control in Section V. Some ρmn = Rmn [1]/Rmn [0], respectively. The parameters {σmn } numerical examples are illustrated in Section VI. Finally, and {ρmn } represent the unequal average channel gains and Section VII offers some concluding remarks. unequal channel ﬂuctuation rates, respectively. Notations: E[·] denotes expectation over the random vari- Let Y[k] denote the M × N received signal matrix from N ables within the brackets. Tr(A) and ReTr(A) stand for the antennas at the k-th transmission block. It is modeled as trace and the real part of the trace of matrix A, respectively. · 2 represents the squared Frobenius norm. IM is the M ×M Y[k] = X[k]H[k] + W[k], (2) identity matrix. diag(a1 , . . . , aM ) is the diagonal matrix with element am on the m-th diagonal. Superscripts (·)T , (·)∗ , where W[k] is the complex-valued additive white Gaussian and (·)H denote transpose, conjugate, and conjugate trans- noise matrix whose entries are i.i.d with zero mean and pose, respectively. Notation and ⊗, respectively, represent variance N0 . the Hadamard product and Kronecker product. Res[f (x), p] denotes the residue of function f (x) at pole x = p. III. D IFFERENTIAL D ETECTION II. S YSTEM M ODEL Detection techniques of DUSTM over time-varying fading Consider a communication system with M transmit and channels have evolved from traditional one-shot differential N receive antennas over a ﬂat Rayleigh fading channel. The detection based on two consecutive blocks of received signals Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. TAO: EFFECTS OF NON-IDENTICAL RAYLEIGH FADING ON DIFFERENTIAL UNITARY SPACE-TIME MODULATION 1361 [2] to more advanced sequence detection which jointly pro- over i.n.i.d channels: cesses multiple blocks, e.g., [18]–[22]. While these multiple- N symbol based sequence detectors are able to attain coherent- ˆ DML = arg max max H˜˜ ˜ yn SCn SH yn . (7) like performance, they are difﬁcult to analyze. To quantita- D∈V S−1 n=1 tively study the effects of non-identical fading statistics, we focus on the one-shot differential detection for the convenience The complexity of the ML detector is proportional to the of analytical tractability. In this section, we ﬁrst discuss a product of the constellation size L and the total number of general structure of the one-shot optimal differential detector all possible previous code matrices. For those constellations of DUSTM in the ML sense over the considered channel with group structure, S−1 also belongs to the signal set V and model. Simpliﬁed detectors under certain constraints are then hence the complexity is in the order of L2 . For constellations discussed. For notation brevity, we deﬁne γ0 = Es /N0 as without group structure, such as OSTBC, the total number of 2 the total transmit SNR, and deﬁne γmn = εm σmn γ0 as the possible S−1 can grow rapidly as L increases. SNR on the branch between transmit antenna m and receive For the semi-identical channel [16] and with equal power antenna n, for 1 ≤ m ≤ M and 1 ≤ n ≤ N . In our high allocation, we have γmn = γn and ρmn = ρn , for all m. SNR assumption, all γmn ’s approach inﬁnity as γ0 → ∞, but It follows that both Cn0 and Cn1 become scaled identity the ratios between one another are kept constant and ﬁnite. matrices. Therefore, the ML detector no longer depends on We also omit the time index k in the data matrix D[k] and S−1 and is simpliﬁed to rewrite S[k − 1] as S−1 hereafter as only one data matrix is processed at one time. N I ˆ DML,semi = arg max wn yn H I DH yn , D∈V D n=1 A. ML Detection of a General Constellation T where wn = γn ρn /[(1+γn)2 −(ρn γn )2 ]. This detector differs Deﬁne Y = Y [k − 1], Y [k] . Given D and S−1 , T T from the conventional detector for i.i.d channel [2] at the it can be shown easily that the column vectors of the weights wn ’s only. The weights exploit the knowledge of sufﬁcient statistics Y, denoted as yn for n = 1, . . . , N , fading statistics and total transmit SNR, and are optimal for are mutually independent Gaussian vectors with zero mean any unitary constellation including OSTBC [16]. and covariance (see (3) at top of next page). Here, Kni ’s, for i = 0, 1 and n = 1, . . . , N , are M × M diagonal matrices deﬁned as Kn0 = diag{γ1n , . . . , γMn } and Kn1 = B. ML Detection of Cyclic Group Codes diag{ρ1n γ1n , . . . , ρMn γMn }. Applying the formula for the determinant of a partitioned matrix [23], we can show that It is shown in [6] that every full-diverse unitary constellation the determinant of the covariance matrix Λn is independent having a group structure can be made equivalent to a cyclic of the data matrix D and the previous code matrix S−1 . The (also called diagonal in [2]) group for odd M , and either a inverse of Λn , however, in general depends not only on D, but cyclic group or dicyclic group for even M . Therefore, the also on S−1 . Using the Sherman-Morrison-Woodbury formula cyclic group constellations are of particular interest to us. for the inverse of the matrix of the form A + BCD [23] and Because of the diagonal structure inherent in cyclic groups, utilizing the diagonal structure of Kni , we obtain the inverse the code matrix S−1 is always diagonal as long as the of Λn as initial matrix S[0] is diagonal, say IM . Since multiplication commutes for diagonal matrices, we have S−1 Cni SH = Cni , −1 1 ˜ Cn0 Cn1 ˜ for all i and n. Therefore, the ML detector for cyclic group (Λn )−1 = 1 N0 I2M − N0 S SH , (4) Cn1 Cn0 codes reduces to: ˜ Cn N I ˆ DML,c = arg max yn H I Cn1 DH yn (8) where matrices Cni , for i = 0, 1 and n = 1, . . . , N , are also D∈V Cn1 D n=1 diagonal, whose m-th diagonals, for m = 1, . . . , M , are given by which can be further expressed as: γmn [1 + γmn (1 − ρ2 )] ˆ DML,c = arg max ReTr (Y[k − 1] W)H DH Y[k] , (9) Cn0 = mn , (5) D∈V m (1 + γmn )2 − (ρmn γmn )2 γmn ρmn Cn1 = . (6) where W is an M ×N matrix with the (m, n)-th entry formed m (1 + γmn )2 − (ρmn γmn )2 by the m-th diagonal of Cn1 and repeated as: The ML differential detector of D is to choose the candidate γmn ρmn ˆ D ∈ V that maximizes the joint likelihood function of the wmn = . (10) (1 + γmn )2 − (ρmn γmn )2 received signal matrix Y over all possible S−1 . Note that the dependence of ML detection on S−1 also arises in transmit- It is clear from (9) that the ML DD for cyclic group codes correlated channels as mentioned in [24] and [25]. But no resembles the conventional DD but applies a weight wmn to explicit ML decision metric is given therein due to the lack the (m, n)-th element of Y[k − 1]. The resulting Hadamard of closed-form expression for (Λn )−1 . Applying (4), we have product Y[k−1] W behaves as the equivalent channel matrix the quadratic-form based ML differential detector of DUSTM as in coherent receivers. Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. 1362 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009 Λn = E[yn yn ] H S−1 0 Kn0 Kn1 SH −1 0 = N0 · · + N0 I2M . (3) 0 DS−1 Kn1 Kn0 0 SH DH −1 ˜ S ˜ SH C. Asymptotically Optimal Detection of a General Constella- where the pairwise decision variable zij is deﬁned as tion in Static Channels N In static fading channels the channel coefﬁcients are as- zij = yn Ωij,n yn , H (14) sumed to remain unchanged over the duration of two trans- n=1 mission blocks. Therefore it has ρmn = 1 for all m and n. In with Ωij,n given by the limit γmn → ∞ for all m and n, the matrices Cn0 and Cn1 deﬁned in (5) (6) all approach (1/2)IM . Applying these 0 Cn1 (Di − Dj )H Ωij,n = (15) into (7), we obtain the asymptotically optimal detector: Cn1 (Di − Dj ) 0 N for ML detection and I ˆ DAO = arg max yn H I DH yn . (11) 0 (Di − Dj )H D∈V D Ωij,n = Ωij = , ∀n (16) n=1 (Di − Dj ) 0 This is identical to the conventional DD [2]. Hence we for conventional detection. Since each vector yn is inde- conclude that the conventional DD is suboptimal in i.n.i.d pendent and zero-mean complex Gaussian distributed, the time-varying channels but asymptotically optimal (high SNR) pairwise decision variable zij is a quadratic form of Gaussian in i.n.i.d static channels. vectors. Therefore, the evaluation of PEP can be carried out by using the well-established techniques in, e.g., [24, Appendix IV. E RROR P ROBABILITY A NALYSIS A]. We summarize the results in the following proposition. In this section, we derive the error performance of DUSTM Proposition 1: The exact pairwise error probability Pe,ij with two speciﬁc constellation designs: cyclic group codes of differential cyclic group codes over i.n.i.d time-varying and orthogonal codes. Through the analysis, we obtain several Rayleigh fading channels is (17) at the bottom of the page, general ﬁndings that are applicable to an arbitrary DUSTM where design. (ρmn γmn )2 amn = dij,m (18) (1 + γmn )2 − (ρmn γmn )2 A. Pairwise Error Probability for Cyclic Group Codes with dij,m being the m-th diagonal entry of the difference matrix (Di − Dj )(Di − Dj )H , and Since the exact bit or block error probability of a cyclic group code V with L > 2 elements is usually not computable, 1, ML detector we resort to the union bound by summing up pairwise error bmn = . (19) wmn in (10), conv. detector probabilities. In speciﬁc, the block error probability (BkEP) for equiprobable elements is bounded by In the case of ML differential detection, an alternative expres- sion is given by L−1 L−1 1 π/2 N M −1 Pe,UB = Pe,ij , (12) 1 amn L Pe,ij = 1+ dθ. (20) i=0 j=0 π 0 n=1 m=1 4 sin2 θ j=i where Pe,ij denotes the PEP of deciding in favor of data Proof: See Appendix A. matrix Dj given that Di is sent. In the following we derive Although the expression for Pe,ij is exact and in closed the exact expressions for Pe,ij and the asymptotics. form, it does not offer much insight on the effects of channel Based on the quadratic form of the ML and suboptimal parameters. Therefore, useful asymptotic bounds are desirable. detectors given in (8) and (11) respectively, the PEP can be We ﬁrst consider the asymptotic Pe,ij over static channels expressed as with ρmn = 1, for all m, n. Then we conduct the error ﬂoor analysis in time-varying channels. The results are summarized Pe,ij = P (zij < 0|Di ) , (13) in the following two corollaries. ⎧ ⎫ ⎨ 1 1 1 1 ⎬ Pe,ij = − Res , s = bkl + + (17) ⎩s M N 1 1 − s − 1 2 2 4 akl ⎭ 1≤k≤M m=1 n=1 amn 4 + amn bmn 2 1≤l≤N Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. TAO: EFFECTS OF NON-IDENTICAL RAYLEIGH FADING ON DIFFERENTIAL UNITARY SPACE-TIME MODULATION 1363 Corollary 1: The asymptotic PEPs of differential cyclic Corollary 2: The pairwise error ﬂoor of differential cyclic group codes at high SNR with ML and conventional differen- group codes with ML differential detection over i.n.i.d time- tial detection over i.n.i.d static Rayleigh fading channels are varying Rayleigh fading channels is independent of the un- the same and given by 2 equal average channel gains {σmn } but depends on the fading 2M N − 1 M −N γgm −M N correlation coefﬁcients {ρmn } only. It is given by (24) at the lim Pe,ij = dij,m , bottom of the page. In the case where ρm,n = ρ and ρ ≈ 1 γ0 →∞ MN m=1 2 (21) but ρ = 1, the pairwise error ﬂoor is simpliﬁed as where γgm is the geometric mean of {γmn }, given by lim γ0 →∞ Pe,ij = 1/MN ρmn =ρ≈1 , and 2MN −1 denotes the N M γgm = n=1 m=1 γmn MN M −N −MN binomial coefﬁcient. 2M N − 1 ρ2 dij,m (25) . Proof: Assuming static fading channels with ρmn = 1 MN 1 − ρ2 m=1 and taking the limit γ0 → ∞, one ﬁnds from (18) and (10) that amn → γmn dij,m /2 and wmn → 1/2. As a result, the Proof: The proof of the ﬁrst equation in the M N poles, where the residues are evaluated in Proposition 1, corollary is straightforward by observing that amn → all approach the constant 1 for ML DD and the constant 1/2 ρ2 dij,m /(1 − ρ2 ) when γ0 → ∞ in (17) (with bmn = 1). mn mn for conventional DD. Using the residue equation of a function If ρm,n = ρ and ρ ≈ 1 but ρ = 1, the M N positive poles f (x) at a pole p of multiplicity v all approach 1/2. Using the formulas (22) and (23) again, we prove the second equation in the corollary. 1 dv−1 Res [f (x), p] = lim [(x − p)v f (x)] (22) Corollary 2 concludes that the irreducible error ﬂoors (v − 1)! x→p dxv−1 achieved over i.n.i.d channels (with ML detection) and tra- and applying the formula ditional i.i.d channels are the same, as long as their fade rates are the same (ρm,n = ρ, ∀m, n). Moreover, the error ﬂoor dm (−1)m (m + k − 1)! (x + p)−k = (x + p)−(m+k) (23) decreases exponentially with M N when ρ is very close to dxm (k − 1)! but not equal to one. This condition on ρ typically holds if in (17), we arrive at the asymptotic results in (21) for both the normalized Doppler frequency of the channel is much less ML and conventional DD. than one. Corollary 1 leads to several insights. First, the result that the asymptotic PEPs of ML and conventional DD are the same is consistent with the ﬁnding in Section III-C that the B. Bit Error Probability for Orthogonal Codes conventional DD is in fact asymptotically optimal for a general In this subsection we derive the error performance of constellation without assuming a speciﬁc signal structure. differential OSTBC. Only the conventional DD and static Second, the traditional diversity product design criterion1 for channels are considered. The analysis for time-varying fading i.i.d channels [1], [2] still applies to i.n.i.d channels. That is, or ML detection is so far not tractable. Since the data matrix M the minimum of m=1 dij,m over all distinct pairs (Di , Dj ) D is a linear combination of P information symbols as should be maximized. Using the arithmetic-geometric-mean mentioned in Section II, the differential detector (11) reduces N M inequality (i.e. γgm ≤ γam = 1/M N n=1 m=1 γmn , to P independent symbol-by-symbol detectors. The details are where the equality holds if and only if γmn ’s are all the same), given in [5] or [16, eq.(12)]. Hence, instead of PEP, BEP is we can see that the non-identical channel distribution will de- derived. grade the error performance compared with the identical case As shown in [16, eq.(15)-(17)], the BEP conditioned on if the total received SNR is kept the same. Furthermore, after symbol sp is the same for all p, and can be expressed as rewriting γgm as γ0 ( m=1 εm )1/M ( n=1 m=1 σmn )1/MN , 2 M N M where εm is the power allocation coefﬁcient deﬁned in (1) Pb (α) = P (zp (α) < 0|sp = 1), (26) subject to the constraint M εm = M , it follows that γgm m=1 where the decision phasor zp (α) is deﬁned as is maximized when εm = 1 for all m. Therefore, equal power allocation is asymptotically optimal in static channels. N zp (α) = yn Ωp yn H (27) 1 Assume the full-rank criterion is already satisﬁed. n=1 ⎧ ⎫ ⎨ 1 1 1 1 − ρ2 ⎬ lim Pe,ij = − Res , s= + + kl (24) γ0 →∞ ⎩s M N ρ2 dij,m mn 1 + 1−ρ2 mn − s− 1 2 2 4 ρ2 dij,k ⎭ kl ρmn <1 1≤k≤M m=1 n=1 1−ρ2 4 ρ2 dij,m 2 mn mn 1≤l≤N ⎧ ⎫ ⎨ 1 cγkl 1 + 2γkl ⎬ Pb (α) = − Res , s= 1+ 1+ 2 2 (28) ⎩s N M 1 + 2cγmn s − (1 + 2γmn )s2 1 + 2γkl c γkl ⎭ 1≤k≤M n=1 m=1 1≤l≤N Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. 1364 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009 with the Hermitian matrix Ωp given by where E = Di − Dj . Given that Di is sent, substituting (2) into (30) yields 0 cos αΦH + j sin αΨH Ωp = p p , zij = ˆ ˆ 2ReTr{Es HH DH EH} cos αΦp − j sin αΨp 0 i ˆ + 2ReTr{ Es H D EW[k − 1]} H H i and α is some angle that depends on the symbol modulation η1 scheme. For BPSK, the exact BEP is obtained by letting α = 0, and for QPSK with Gray mapping we have α = −π/4. + 2ReTr{ ˆ Es WH [k]EH} Proposition 2: The exact bit error probability Pb (α) of η2 differential OSTBC over i.n.i.d static Rayleigh fading channels + 2ReTr{W [k]EW[k − 1]}, H (31) with conventional differential detection is√(28) (see the bottom η3 of the previous page), where c = cos α/ P . Proof: See Appendix B where H = S−1 Σ1/2 H with H being the channel matrix in ˆ At high SNR, all the M N positive poles, where the residues static fading. It can be easily shown that, conditioned on H, ˆ are evaluated, approach the constant c. Therefore, we obtain the noise terms η1 and η2 are independent and real-valued the asymptotics of Pb (α) as follows by applying the deﬁnition zero-mean Gaussian variables with variance 2N0 Es EH 2 ˆ of residue as in the proof of Corollary 1. for each. By neglecting the second-order noise term η3 , which Corollary 3: The asymptotic BEP of differential OSTBC has diminishing effect at high SNR (i.e., Es /N0 1), over i.n.i.d static Rayleigh fading channels with conventional and noting DH E + EH Di = EH E, the pairwise decision i differential detection is variable zij can be approximated as a Gaussian variable with mean Es EH 2 and variance 4N0 Es EH 2 . As a result, the ˆ ˆ −MN 2M N − 1 2 cos2 α conditional probability of zij < 0 can be expressed in the form lim Pb (α) = γgm , (29) of standard Q-function [26]. Further, applying the inequality γ0 →∞ MN P Q(x) ≤ 1 exp (−x2 /2), we obtain the Chernoff bound of the 2 where γgm is the geometric mean of {γmn }. approximate PEP as The implications in Corollary 3 are the same as those 1 γ0 2 in Corollary 1. Therefore, we readily extend the following Pe E exp − ˆ EH . 2 8 remarks to the general DUSTM. Remarks: (1) Non-identical fading degrades the error perfor- Obtaining the distribution of EH 2 is difﬁcult in general if ˆ mance compared with the identical case given the same total E is the difference matrix of an arbitrary unitary constellation. received SNR; (2) Equal power allocation is asymptotically Fortunately, by utilizing the diagonal structure of cyclic group optimal in i.n.i.d static fading channels. codes, it is clear that EH 2 can be expressed as a weighted ˆ sum of absolute squares of M N i.i.d complex Gaussian 2 variables with weights given by εm σmn dij,m . Hence, the V. T RANSMIT P OWER C ONTROL above expectation can be evaluated as N M 2 −1 Given the unequal channel gain distribution among different 1 γ0 εm σmn dij,m Pe,ij 1+ . (32) transmit antennas, it is intuitive to use power control to 2 n=1 m=1 8 improve the error performance, especially when the total We now ﬁnd the optimal power allocation coefﬁcients εm ’s transmit power is small. To simplify investigation, we consider to minimize the bound in (32) for a dominant error pair, static channels only in this section. Moreover, as shown in which consequently provides a good result in minimizing the Section VI and [16], the conventional detector performs almost overall block error probability. The dominant error pair of a the same as the ML detector in i.n.i.d static fading channels. cyclic group code is the data matrix pair that has the small- Hence, we assume conventional DD here. M M est m=1 dij,m [2], denoted as ζ = min m=1 dij,m . Both the exact PEP in Proposition 1 for cyclic group codes 0≤i<j<L and the exact BEP in Proposition 2 for orthogonal codes However, there can be multiple pairs in the code that result are difﬁcult to minimize directly. We resort to minimizing a in the same ζ, and they may differ dramatically in dij,m , for simple but useful approximate bound of them. In the following m = 1, . . . , M . Take the cyclic group code with M = 4 we present the derivation of transmit power control for the and L = 16 for example, given by V4,16 = {Dk = two codes separately for the ease of presentation, though the diag(ejkπ/8 , ej3kπ/8 , ej5kπ/8 , ej7kπ/8 ), k = 0, . . . , 15} [27, approaches are very similar. Table I]. The data matrix pair (D0 , D1 ) is a dominant error pair and it has d1 = 0.1522, d2 = 1.2346, d3 = 2.7654 and d4 = 3.8478. On the other hand, the pair (D0 , D3 ) is also A. Power Control for Cyclic Group Codes a dominant error pair but it has d1 = 1.2346, d2 = 3.8478, d3 = 0.1522, and d4 = 2.7654. The sets of power allocation The pairwise decision variable in (14) for the conventional coefﬁcients minimizing the two pairwise error probabilities DD can be rewritten as are obviously different. To overcome this problem, we take the mean of dij,m over all dominant error pairs for each m, denote zij = 2ReTr{Y[k]H EY[k − 1]} (30) ¯ it as dm , and replace all dij,m with it in the PEP bound (32). Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. TAO: EFFECTS OF NON-IDENTICAL RAYLEIGH FADING ON DIFFERENTIAL UNITARY SPACE-TIME MODULATION 1365 0 Then, by using the monotonic property of logarithm function, 10 PEP with conv. decoder (17) the power allocation problem can be formulated as PEP with ML decoder (17) PEP for i.i.d −1 Asymptotic PEP bound (21) N M σ 2 dm γ0 ¯ 10 Pairwise error floor (24) max log 1 + εm mn . (33) M m εm =M 8 Pairwise Error Probability n=1 m=1 −2 In the case of N = 1, we obtain the water-ﬁlling based 10 ρ=0.99 closed-form expression for the optimal power control as [28] + −3 8 10 εm = μ− ¯m 0 2 d γ , (34) σm where (x)+ = max{0, x}, and the Lagrange multiplier μ can −4 10 ρ=1 M be determined by the constraint m=1 εm = M . If there are N > 1 number of receive antennas, closed- −5 form expressions of optimal power coefﬁcients are difﬁcult to 10 0 5 10 15 20 25 30 35 40 ﬁnd. Here we propose a suboptimal approach. Applying the total transmit SNR γ [dB] 0 inequality [29, eq.(25)] Fig. 1. PEP performance of differential cyclic group code V2,4 over the N dominant error pair at M = 2 transmit and N = 1 receive antenna. (1 + xi ) ≥ (1 + xgm )N , i=1 Correspondingly, a water-ﬁlling based sub-optimal power con- xi )1/N , we can reformulate (33) as N where xgm = ( i=1 trol that minimizes the bound is given by M 2 ¯ + σm,gm dm γ0 2P max log 1 + εm , εm = μ− . (36) M m εm =M 8 2 cos2 (α)σm,gm γ0 m=1 2 2 where σm,gm = ( n=1 σmn )1/N . Hence, the solution in (34) N VI. N UMERICAL R ESULTS 2 2 still applies after replacing σm with σm,gm , and is given by In this section we present some numerical examples to con- + ﬁrm our analytical ﬁndings in previous sections. We ﬁrst verify 8 the error probability analysis using a system with M = 2 εm = μ− 2 ¯ . (35) σm,gm dm γ0 transmit antennas and N = 1 or N = 2 receive antennas. Then we demonstrate the performance of the proposed transmit In summary, the proposed transmit power control aims power allocation in a system with M = 4 transmit antennas to minimize the Chernoff bound of an approximate PEP of and N = 1 receive antenna. dominant error pairs in the constellation. It has a water-ﬁlling In our ﬁrst set of examples, we assume equal fade rates on structure, and hence inherits the two distinguishing properties all transmit-receive antenna pairs and illustrate the effects of of water-ﬁlling principle. First, when the total transmit power non-identical channel gain distribution. The unequal average is low, the transmit antennas with smaller geometric mean channel gains are generated using the Kronecker model [16]. of average channel gains should be turned off. Second, when 2 In speciﬁc, the M N × M N diagonal matrix Δ with σmn the total transmit power is high enough, the power tends to on the [(n − 1)M + m]-th diagonal is decomposed as Δ = be equally distributed among all the antennas. The second ΔT ⊗ ΔR , where ΔT and ΔR are, respectively, the M × M property is consistent with the ﬁnding from Section IV that and N × N diagonal matrices inducing non-identical fading equal power allocation is asymptotically optimal. parameters at the transmitter and receiver. The sum of the average channel gains is normalized so that Tr{ΔT } = M B. Power Control for Orthogonal Codes and Tr{ΔR } = N . In the system with two transmit antennas, we specify ΔT = diag( 1 , 9 ). For one receive antenna, the 5 5 The transmit power allocation for DUSTM with orthogonal 2 2 average channel gains are given by σ1 = 1/5 and σ2 = 9/5. codes is similar to that for DUSTM with cyclic group codes. 1 9 For two receive antennas, we let ΔR = diag( 5 , 5 ), and the The decision phasor zp (α) (27) can also be expressed as 2 2 set of average channel gains is given by {σ11 = 1/25, σ12 = (31), except that E should be deﬁned as E = cos αΦH + p 2 2 σ21 = 9/25, σ22 = 81/25}. j sin αΨH . Using the orthogonal code structure, we can easily p In Figs. 1-3, we show the performance of differential cyclic show that the distribution of zp (α) can be approximated group codes. The exemplary cyclic group code for M = 2 as Gaussian with mean 2 cos(α)Es Σ1/2 H 2 and variance √ P and L = 4 at rate 1-bit/s/Hz [27, Table I], denoted as V2,4 , is 1/2 2 4Es N0 Σ H . Thus, we obtain the Chernoff bound of the chosen. Figs. 1 and 2 show the analytical PEP of the dominant approximate BEP for differential OSTBC as error pair (D0 , D1 ) using one and two receive antennas, N M −1 respectively. The exact PEP results over i.i.d channels are also 1 cos2 (α) 2 plotted for reference, which are obtained using (20) by letting Pb (α) 1+ εm σmn γ0 . 2 2 n=1 m=1 2P σmn = 1, ∀m, n. Several useful observations can be made Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. 1366 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009 0 0 10 10 PEP with conv. decoder (17) −1 −1 10 PEP with ML decoder (17) 10 PEP for i.i.d Asymptotic PEP bound (21) −2 Pairwise error floor (24) −2 10 10 Pairwise Error Probability Bit Error Probability −3 −3 10 10 −4 −4 10 10 −5 −5 10 ρ=0.99 10 Analytical BEP for i.n.i.d (28) −6 ρ=1 −6 Simulated BEP for i.n.i.d 10 10 Asymptotic bound for i.n.i.d (29) BEP for i.s.i.d [16] −7 −7 BEP for i.i.d [30] 10 10 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 total transmit SNR γ [dB] total transmit SNR γ [dB] 0 0 Fig. 2. PEP performance of differential cyclic group code V2,4 over the Fig. 4. BEP performance of differential OSTBC with QPSK modulation at dominant error pair at M = 2 transmit and N = 2 receive antennas. M = 2 transmit and N = 2 receive antennas in static fading. 0 10 Union bound with conv. decoder Simulated BkEP with conv. decoder The overall BkEP performance of this cyclic group code −1 Union bound with ML decoder obtained via simulation is shown in Fig. 3 and compared 10 Simulated BkEP with ML decoder with the BkEP union bound obtained analytically using (12). The block-wise time-varying fading channel is generated −2 10 using Jakes model with autocorrection function 2Rmn [k] = Block Error Probability 2 σmn J0 (2πfd Ts M k), where J0 (·) is the zeroth order Bessel −3 10 function of the ﬁrst kind and fd Ts is the normalized Doppler frequency. In our simulation we set fd Ts = 0.02, which results in ρ = 0.98427. It is observed that the analytical BkEP union bound serves as a tight upper bound on the actual BkEP with −4 10 both conventional and ML detectors. This further validates our −5 10 theoretical analysis on the exact PEP in Proposition 1. The BEP results of differential OSTBC over the i.n.i.d −6 channel with two transmit and two receive antennas are 10 0 5 10 15 20 25 30 35 40 depicted in Fig. 4. The orthogonal code for two transmit total transmit SNR γ [dB] 0 antennas with P = 2 and QPSK modulation at rate 2- Fig. 3. BkEP performance of differential cyclic group code V2,4 at M = 2 bit/s/Hz is used. The analytical BEPs are from (28) and are transmit and N = 2 receive antennas in time-varying fading with fd Ts = validated by simulations. For the i.s.i.d channel, the non- 0.02 (ρ = 0.98427). identical fading occurs at the receiver side only with ΔT = I2 and ΔR = diag( 1 , 9 ). Its exact BEP curve is obtained from 5 5 [16, eq.(27)]. The exact BEP for i.i.d channels is from [30]. As from the two ﬁgures. First, the pairwise error performance expected, the i.n.i.d channel yields the worst performance and achieved using the conventional DD is almost the same as that the best performance is achieved over i.i.d channels. Note that achieved by the ML detector at all SNR when ρ = 1 as well this conclusion only holds when the sum of average channel as at low SNR in time-varying fading with ρ = 0.99. Second, gains is the same. the ML detector considerably reduces the pairwise error ﬂoor Next, we illustrate the effects of unequal channel ﬂuctuation in fast fading compared with the conventional detector. In rates among different signalling branches on the error ﬂoors particular, the pairwise error ﬂoor of the ML detector with as γ0 → ∞. Fig. 5 shows the irreducible dominant PEP of two receive antennas is two order of magnitude lower than the differential cyclic group code V2,4 in the system with two that of the conventional detector. Moreover, the error ﬂoors transmit antennas and one receive antenna. It is observed that approach those in i.i.d channels and match very well with under the same averaged fading correlation coefﬁcient, i.e., the ﬂat lines predicted by the analytical result in (24). This ρ = (ρ1 + ρ2 )/2, the error ﬂoor reduces as the difference on observation conﬁrms our analytical ﬁnding from Corollary 2. the fade rates between the two antennas increases. From the ﬁgures we also observe that the simple asymptotic In all the above ﬁgures, equal power allocation is assumed. PEP bound (21) is very tight when γ0 is large enough. Finally, We now illustrate in Figs. 6 and 7 the performance of the compared with i.i.d channels, the non-identical channel gain proposed transmit power allocation in a system with four distribution degrades the PEP performance. This conﬁrms the transmit antennas and one receive antenna. An exponentially analytical ﬁnding from Corollary 1. decaying average channel gain proﬁle is used and character- Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. TAO: EFFECTS OF NON-IDENTICAL RAYLEIGH FADING ON DIFFERENTIAL UNITARY SPACE-TIME MODULATION 1367 −1 0 10 10 ρ =ρ =ρ 1 2 Equal PA, δ = 3 conv. detector ρ = ρ − 0.004, ρ = ρ + 0.004 Proposed PA, δ = 3 1 2 Equal PA, δ = 1 ρ = ρ − 0.008, ρ = ρ + 0.008 1 2 −1 10 Proposed PA, δ = 1 Irreducible Pairwise Error Probability −2 10 Bit Error Probability −2 10 ML detector −3 10 −3 10 −4 10 −4 −5 10 10 0.93 0.94 0.95 0.96 0.97 0.98 0.99 0 5 10 15 20 25 30 35 Fading correlation coefficient ρ total transmit SNR γ [dB] 0 Fig. 5. Dominant pairwise error ﬂoor of differential cyclic group code V2,4 Fig. 7. BEP performance of differential OSTBC with transmit power control at M = 2 transmit and N = 1 receive antennas at M = 4 transmit and N = 1 receive antenna in static fading. 0 10 Equal PA, δ = 3 Proposed PA, δ = 3 power allocation is more signiﬁcant than in the cyclic group Equal PA, δ = 1 10 −1 Proposed PA, δ = 1 code case. This is because the power allocation for orthogonal codes aims at minimizing the bound of the overall bit error probability directly, whereas the one for cyclic group codes is Block Error Probability 10 −2 obtained only through minimizing the bound of the dominant pairwise error probability with certain approximations. 10 −3 VII. C ONCLUSION The effects of non-identical fading statistics in MIMO chan- 10 −4 nels on the performance of DUSTM were investigated. Con- trary to the detectors for the traditional i.i.d fading model, we found that the ML differential detector of DUSTM generally 10 −5 requires joint optimization of the current data matrix and the 0 5 10 15 20 25 total transmit SNR γ [dB] 30 35 40 previously transmitted signal matrix. However, for DUSTM 0 with cyclic group design, the ML detector is much simpliﬁed Fig. 6. BkEP performance of differential cyclic group code V4,16 with and is similar to the conventional detector but applies fading transmit power control at M = 4 transmit and N = 1 receive antenna in statistics-dependent weights. Based on the analysis of exact static fading. and asymptotic error probability for both cyclic group codes and orthogonal codes, we obtained several useful ﬁndings. Along with numerical results, we conclude that while the ML 2 ized by σm = e−δ(m−1) , for 1 ≤ m ≤ 4, in which δ ≥ 0 is detector can signiﬁcantly reduce the error ﬂoor over rapidly the decay factor. time-varying fading channels, the conventional detector is Fig 6 shows the simulated BkEP of the differential cyclic near-optimal at all SNR in static fading and low SNR in time- group code V4,16 with M = 4 and L = 16 at rate 1-bit/s/Hz varying fading. In addition, the non-identical channel gain [27, Table I]. The conventional detector is employed. It is seen distribution degrades the error performance compared with that the proposed power allocation (34) cannot outperform the identical distribution for a same total received SNR. To equal power allocation when the average channel gains are exploit the non-identical fading parameters at the transmitter, only slightly unbalanced with δ = 1. On the other hand, we also presented a water-ﬁlling based transmit power control. for highly unbalanced average channel gains with δ = 3, It was shown to provide considerable improvement in error the water-ﬁlling based power allocation can save 2 ∼ 3 dB probability at low to moderate SNR region when the average total transmit power at a given BkEP around 10−2 . But the channel gains are highly unbalanced. At sufﬁciently high SNR, gain diminishes as the target BkEP reduces. This observation equal power allocation is still optimal. conﬁrms our analytical ﬁnding in Section IV that equal power allocation is asymptotically optimal. The BEP performance of the differential orthogonal code A PPENDIX A with M = 4, P = 3 and QPSK modulation at rate 1.5-bit/s/Hz P ROOF OF P ROPOSITION 1 based on the analysis (28) is presented in Fig. 7. We see that By applying the result in [31], the characteristic function the gain of the proposed power allocation (36) over equal (CF) of the pairwise decision variable zij in the quadratic Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. 1368 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009 form of Gaussian vectors (14) can be written as [26] symbols in the data matrix D, nor the previously transmitted 1 signal matrix S−1 . They are: φzij (s) = E e−szij = N ( . 37) n=1 det(I + sΛn Ωij,n ) γmn cos(α + θp ) √ λn,i = Substituting (3) and (15) (or (16)) into (37), we obtain P 2 γmn cos2 (α + θp ) N M 2 −1 ± + 2γmn + 1 (43) 1 1 s 1 P φzij (s) = amn + − − , n=1 m=1 4 amn bmn 2 where λn,i < 0 for 1 ≤ i ≤ M and λn,i > 0 for M + 1 ≤ (38) i ≤ 2M . Substituting (43) into (42) and after some algebra, where amn and bmn are given in (18) and (19), respectively. we obtain the closed-form expression of Pb (α) in Proposition After inverting the Laplace transform, we express Pe,ij deﬁned 2. in (13) as [24, Appendix A] 1 j∞+η φzij (s) R EFERENCES Pe,ij = ds, (39) 2πj −j∞+η s [1] B. L. Hughes, “Differential space-time modulation,” IEEE Trans. In- form. Theory, vol. 46, no. 7, pp. 2567–2578, Nov. 2000. where η > 0 is within the region of convergence. This integral [2] B. M. Hochwald and W. Sweldens, “Differential unitary space-time can be solved using Cauchy’s theorem in terms of residues: modulation,” IEEE Trans. Commun., vol. 48, no. 12, pp. 2041–2052, Dec. 2000. φzij (s) [3] V. Tarokh and H. Jafarkhani, “A differential detection scheme for Pe,ij = − Res , s = pi , (40) transmit diversity,” IEEE J. Select. Areas Commun., vol. 18, no. 7, pp. pi >0 s 1169–1174, 2000. [4] M. Tao and R. S. Cheng, “Differential space-time block codes,” in Proc. where pi ’s are all the positive poles of φzij (s). Finally, sub- IEEE Global Telecommunications Conference (GLOBECOM), 2001. [5] G. Ganesan and P. Stoica, “Differential detection based on space-time stituting (38) into (40) yields Pe,ij expressed more explicitly block codes,” Wireless Personal Comm., vol. 21, pp. 163–180, 2002. in (17). [6] B. L. Hughes, “Optimal space-time constellations from groups,” IEEE In the case of ML differential detection with bmn = 1, we Trans. Inform. Theory, vol. 49, no. 2, pp. 401–410, Feb. 2003. [7] B. Hassibi and B. M. Hochwald, “Cayley differential unitary space-time can choose η = 1/2 for the integration contour in (39). Then, codes,” IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1485–1503, June with a change of variables, we obtain 2002. j∞ [8] W. Roh and A. Paulraj, “Outage performance of the distributed antenna 1 systems in a composite fading channels,” in Proc. IEEE VTC’02 Fall, Pe,ij = 2002. 2πj −j∞ [9] H. Zhang and H. Dai, “On the capacity of distributed MIMO systems,” ds in Proc. CISS’04, 2004. · . [10] M. A. Jensen, M. D. Rice, and A. L. Anderson, “Unitary space-time (s + 1 ) 2 M m=1 N n=1 amn 1 4 + 1 amn − s2 coding for multi-antenna aeronautical telemetry transmission,” to be published. (41) [11] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” Now we let s = jw in (41) and the integration becomes along IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. the real axis. By further letting w = tan θ/2, an alternative [12] S. Yiu, R. Schober, and L. Lampe, “Distributed space-time block coding,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1195–1206, July expression of Pe,ij in the form of ﬁnite integral is obtained in 2006. (20). [13] J. Luo, R. S. Blum, L. Cimini, L. Greenstein, and A. Haimovich, “Power allocation in a transmit diversity system with mean channel gain information,” IEEE Commun. Lett., vol. 9, no. 7, 2005. A PPENDIX B [14] J. He and P. Y. Kam, “On the performance of orthogonal space- P ROOF OF P ROPOSITION 2 time block codes over independent, nonidentical Rayleigh/Ricean fading channels,” in Proc. IEEE GLOBECOM’06, 2006. The CF of the quadratic form of Gaussian vectors zp (α) in [15] H. Zhao, Y. Gong, Y. L. Guan, C. L. Law, and Y. Tang, “Space- (27) is given by time block codes in Nakagami fading channels with non-identical m- distributions,” in Proc. IEEE WCNC’07, Mar. 2007. 1 [16] M. Tao and P. Y. Kam, “Analysis of differential orthogonal space-time φzp (s) = E e−sZp (α) = N K . block codes over semi-identical MIMO fading channels,” IEEE Trans. n=1 i=1 (1 + sλn,i )un,i Commun., vol. 55, no. 2, pp. 282–291, Feb. 2007. [17] G. Ganesan and P. Stoica, “Space-time block codes: a maximum SNR where {λn,i }K are the distinct eigenvalues of Λn Ωp with i=1 approach,” IEEE Trans. Inform. Theory, vol. 47, no. 4, pp. 1650–1656, multiplicity of {un,i }K . Thus, the BEP in (26) can be May 2001. i=1 [18] R. Schober and L. Lampe, “Noncoherent receivers for differential space- obtained as [24] time modulation,” IEEE Trans. Commun., vol. 50, no. 5, pp. 768–777, May 2002. φzp (s) 1 [19] E. Chiavaccini and G. M. Vitetta, “Further results on differential space- Pb (α) = − Res , s=− , (42) s λn,i time modulation,” IEEE Trans. Commun., vol. 51, no. 7, pp. 1093–1101, λn,i <0 July 2003. [20] C. Ling, H. K. Li, and A. C. Kot, “Noncoherent sequence detection where the residues are evaluated at the positive poles of of differential space-time modulation,” IEEE Trans. Inform. Theory, φzp (s)/s, that is −1/λn,i with λn,i being negative. Using a vol. 49, no. 10, pp. 2727–2734, Oct. 2003. similar approach as in the proof [25, Corollary 1], we can [21] C. Gao, A. M. Haimovich, and D. Lao, “Multiple-symbol differential detection for MPSK space-time block codes: design metric and perfor- show that the eigenvalues of Λn Ωp are determined by the p- mance analysis,” IEEE Trans. Commun., vol. 54, no. 8, pp. 1502–1510, th information symbol sp = ejθp , and do not rely on the other Aug. 2006. Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply. TAO: EFFECTS OF NON-IDENTICAL RAYLEIGH FADING ON DIFFERENTIAL UNITARY SPACE-TIME MODULATION 1369 [22] P. Pun and P. Ho, “Fano multiple-symbol differential detectors for dif- Meixia Tao (S’00-M’04) received the B.S. degree ferential unitary space-time modulation,” IEEE Trans. Comun., vol. 55, in Electronic Engineering from Fudan University, no. 3, pp. 540–550, Mar. 2007. Shanghai, China, in 1999, and the Ph.D. degree in [23] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge Electrical and Electronic Engineering from Hong University Press, 1985. Kong University of Science & Technology in 2003. [24] M. Brehler and M. K. Varanasi, “Asymptotic error probability analysis She is currently an Associate Professor at the De- of quadratic receivers in Rayleigh-fading channels with applications to partment of Electronic Engineering, Shanghai Jiao a uniﬁed analysis of coherent and noncoherent space-time receivers,” Tong University, China. From Aug. 2003 to Aug. IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2383–2399, Sept. 2001. 2004, she was a Member of Professional Staff in [25] X. Cai and G. B. Giannakis, “Differential space-time modulation with the Wireless Access Group at Hong Kong Applied eigen-beamforming for correlated MIMO fading channels,” IEEE Trans. Science & Technology Research Institute Co. Ltd, Signal Processing, vol. 54, pp. 1279–1288, 2006. where she worked on the design of wireless local area networks. From [26] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw-Hill, Aug 2004 to Dec. 2007, she was with the Department of Electrical and 2001. Computer Engineering at National University of Singapore as an Assistant [27] A. Shokrollahi, B. Hassibi, B. M. Hochwald, and W. Sweldens, “Rep- Professor. Her research interests include MIMO techniques, channel coding resentation theory for high-rate multiple-antenna code design,” IEEE and modulation, dynamic resource allocation in wireless networks, and Trans. Inform. Theory, vol. 47, no. 6, pp. 2335–2367, Sept. 2001. cooperative communications. [28] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Dr. Tao is an Editor of the IEEE T RANSACTIONS ON W IRELESS C OM - Cambridge University Press, 2004. MUNICATIONS . She served as Track Co-Chair for IEEE ICCCN’07 held in [29] M. K. Byun and B. G. Lee, “New bounds of pairwise error probability August 2007 at Hawaii, USA, and IEEE ICCCAS’07 held in July 2007 at for space-time codes in rayleigh fading channels,” IEEE Trans. Com- Fukuoka, Japan. She also served as a Technical Program Committee member mun., vol. 55, no. 8, pp. 1484–1493, Aug. 2007. for various conferences, including IEEE ICC (2006, 2007, 2008), IEEE [30] T. P. Soh, P. Y. Kam, and C. S. Ng, “Bit error probability for WCNC (2007, 2008), IEEE GLOBECOM (2007), and IEEE VTC (2006-Fall, orthogonal space-time block codes with differential detection,” IEEE 2008-Spring). Trans. Commun., vol. 53, no. 11, pp. 1795–1798, Nov. 2005. [31] G. L. Turin, “The characteristic function of hermitian quadratic forms in complex normal variables,” Biometrika, vol. 47, pp. 199–201, 1960. Authorized licensed use limited to: Shanghai Jiao Tong University. Downloaded on May 26, 2009 at 21:06 from IEEE Xplore. Restrictions apply.

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