Space-time Linear Dispersion Using Coordinate Interleaving

Transcription

1 Space-time Linear Dispersion Using Coorinate Interleaving Jinsong Wu an Steven D Blostein Department of Electrical an Computer Engineering Queen s University, Kingston, Ontario, Canaa, K7L3N6 wujs@ieeeorg an stevenblostein@ueensuca Abstract This paper proposes a general coorinateinterleaving metho for block-base space-time coes or linear ispersion coes, calle space-time coorinate interleaving linear ispersion coes ST-CI, which enables not only symbollevel iversity but also coorinate-level iversity for high rate block-base space-time coe esign This paper analyzes the upper boun iversity orer an provies the analysis results of the upper boun statistical iversity orer an average iversity orer for ST-CI systems Compare with conventional ST- systems, ST-CI systems may show either almost ouble average iversity orer or extra coing avantage in time varying channels With trivial extra complexity over ST- systems, ST-CI systems maintain the iversity performance in uasi-static block faing channels, an significantly improve the iversity performance in rapi faing channels I INTRODUCTION To support high reliability of space-time multiple input multiple output MIMO transmission, space-time coing STC may be applie to improve system performance an achieve high capacity potential Space-time trellis coes 1] have great iversity an coing gain but exponential ecoing complexity, which motivates the esign of low complexity STC Due to their attractive complexity, a number of block-base STC have been propose 2], 3] Recently, Hassibi an Hochwal have constructe a class of high-rate block-base STC known as linear ispersion coes 4], which support arbitrary numbers of transmit an receive antenna channels We treat as a general framework of complex space-time block coe esign A problem in most existing esign criteria of block-base space-time coes, incluing, is that they o not efficiently exploit aitional iversity potential in the real an image parts of coorinates of source ata constellation symbols A techniue to utilize the iversity potential of real an image parts of coorinates is calle coorinate interleaving or component interleaving CI, which was first propose for single stream communications systems 5], 6] Recently, CI has been applie to multiple antenna systems 7] 9] However, current existing approaches to using CI in blockbase space-time coes are low-rate esigns using orthogonal space-time block coes or their variation 7] 9] This paper proposes coorinate-interleaving as a general principle for high-rate block-base space-time coe esign, ie, space-time coorinate interleaving linear ispersion coes ST-CI This paper etermines the the upper boun iversity orer, statistical iversity orer an average iversity orer of ST-CI ST-CI maintains the same iversity orer as conventional ST- However, ST-CI may show either almost ouble average iversity orer or extra coing avantage over conventional ST- in time varying channels Compare with conventional ST-, ST-CI maintains the iversity performance in uasi-static block faing channels, an notably improves the iversity performance in rapi faing channels The following notation is use:re ] an Im ] enote the real an image parts, respectively, T matrix transpose, H matrix transpose conjugate, A] a,b enote the a, b entry element of matrix A, Kronecker prouct, I K enotes K K ientity matrix, an C m n enotes a complex matrix with imensions m n II PROPOSED SYSTEMS A MIMO system moel for in time varying channels In freuency-flat, time non-selective Rayleigh faing channels whose coefficients may vary per channel symbol time slot or channel use, a multi-antenna communication system is assume with transmit an N R receive antennas Assume that an uncorrelate ata seuence has been moulate using complex-value source ata symbols chosen from an arbitrary, eg D-PSK or D-QAM, constellation Each coewor of size T is transmitte uring every T time channel uses from transmit antennas 1 Component matrices in system euations: We now introuce several component matrices uring the k-th space-time coewor transmission The receive signal vector x k ] T x k,1,, x k,t, where x k,t CNR 1, t 1, T is the receive vector corresponing to the t-th row of the k-th coewor, S k The system channel matrix is H k H k,1 0 0 H k,t

2 where C N R ], t 1, T with entries H k,t h m,n k,t, m 1,,, n 1,, N R, is m,n a complex Gaussian MIMO channel matrix with zero-mean, unit variance entries corresponing to the t-th row of the k-th coewor, S k, an 0 enotes a zero matrix of size N R The complex Gaussian noise vector is v k ] T v k,1,, v k,t, where v k,t CNR 1, t H k,t 1, T is a complex Gaussian noise vector with zero mean, unit variance entries corresponing to the t-th row of the k-th coewor, S k corre- The encoe complex symbol vector s k spons to the k-th coewor, S k, where s k vec S k 1 2 System moel euation: The system euation for the transmission of the k-th matrix coewor is expresse as x k ρ H k sk +vk 2 where ρ is the signal-to-noise ratio SNR at each receive antenna, an inepenent of B Proceure of space-time inter- coorinate interleaving We propose a new space-time encoing proceure using inter- CI I-CI, calle space-time coorinate interleaving linear ispersion coes ST-CI as follows: Consier a pair of source ata symbol vectors s 1 with the same s 1 1,, s1 Q an s 2 s 2 1,, s2 Q number, Q of source ata symbol symbols, where s i Re s i + jim s i, i 1, 2, an 1,, Q The transmitter first coorinate-interleaves ] s 1 an s 2 into s CI1 T s CI1 1,, s CI1 Q an s CI2 s CI2 1,, s CI2 Q, where an s CI1 s CI2 Re Re s 1 s 2 + jim + jim s 2 s Then, s CI1 an s CI2 are encoe into two coewors of size T, S CI1 an SCI2, respectively Then the transmitter successively sens S CI1 an SCI2 uring two interleave perios such that channels are less correlate We remark that 1 using ifferent permutations, other methos of spacetime inter- CI than 3 an 4 are also possible; 2 The encoing matrices for S CI1 an SCI2 nee not be the same C ST-CI system structure The propose ST-CI system structure is shown in Figure 1 The system structure basically consists of three layers : 1 mapping from ata bits to constellation points, 2 inter- coorinate interleaving, an 3 coing Using the propose layere structure, the only aitional complexity compare with a conventional ST- system is the coorinate interleaving operation Thus, ST-CI system is computationally efficient The motivation of ST-CI is to rener the faing more inepenent of each coorinate of the source ata signals Note that ue to the superposition effects of signals from multiple transmit antennas at the spacetime MIMO receivers, existing esigns cannot guarantee faing inepenence of each coorinate of the source ata signals Compare with ST-, ST-CI introuces coorinate faing iversity at the cost of more ecoing elay using a pair of coewors of the same size III DIVERSITY ANALYSIS Su an Liu 10] recently analyze the iversity of spacetime moulation over time-correlate Rayleigh faing channels A moifie strategy can be use to investigate the iversity of ST-CI systems Consier a ST-CI block C, which consists of two ST- coewors of size T, S k, where k 1, 2 The communication moel for one ST-CI block C can be rewritten as where Y ρ MH + Z, 5 1 the noise vector is Z, Y 1 2 the receive signal vector Y ] ], Y 2 T T, where Y k Y k k 1,, Yk N, Y n R ] ] x k,1,, x k,t, k 1, 2, n,1 n,1 3 M is the channel symbol matrix corresponing to the block C, M iag M 1, M 2, where M 1 an M 2 are the matrices corresponing to the coewor S 1 an S2, respectively, M k I NR iag ] iag S k,, 1,m M k 1 S k,, Mk, M k ] T,m ] m, k 1, 2, an m 1,,, H 1 4 the channel vector H ] ], H 2 T T, where H k an h km,n h T k1,1,, ht k,1,, ht k1,n R,, h T k,n R, h k,1 m,n,, h k,t m,n A irectional pair, enote as X Y, means that a system etects X as Y Consier the irection pair of matrices M an

3 M corresponing to two ifferent ST- blocks C an C The upper boun pairwise error probability 11] is Pr M M 2r 1 r r a1 1 r ρ γ a 6 H, where r is the rank of M M R H M M an R H E H H] H is the correlation matrix of vector H, R H is of size 2 N R T 2 N R T, γ a, a 1,, r are the non-zero eigenvalues of H Λ M M R H M M Then the rank an prouct criteria are 1 Rank criterion: The minimum rank of Λ over all irection pairs of ifferent matrices M an M shoul be as large as possible 2 Prouct criterion: the minimum value of the prouct r γ a over all irection pairs of ifferent M an M a1 shoul be maximize To maximize the rank of Λ, we nee to maximize the ranks of both R H an M M Denote Ω k M k M k, where k 1, 2 Assume that all the possible M k an M k are containe in a set M k, ie, M k, M k M k, where k 1, 2 Then the iversity orer of the ST-CI, r, is r min rank Λ, M M, M M, M M 7 When M situations, M, there are three istinct categories of 1 M 1 M 1 an M 2 M 2, 2 M 1 M 1 an M 2 M 2, 3 M 1 M 1 an M 2 M 2 Note that when R H is full rank, 1 in the above Situations 1 an 2, the upper boun of rankλ is N R T, 2 in the above Situation 3, the upper boun of rankλ is 2N R T Thus ST-CI oes not further increase the iversity orer over ST- in terms of the conventional efinition 7 However, ST-CI oes increase r over ST- for the above-mentione thir situation, which is not the conventional iversity orer of the STC an may significantly impact system performance It is necessary to introuce a new concept to uantify this effect as follows, Definition 1: Statistical iversity orer, r s, is the rank of Λ achieve with a certain probability α, mathematically written as rank Λ r s, Pr M M, M, M α 8 M, Then, we have the following theorem Theorem 1: A ST-CI is constructe through coorinate interleaving across a pair of component coewors Both component encoers are able to generate ifferent coewors for ifferent input seuences The iversity orers of the component s are r 1 an r 2, respectively Suppose that R H is full rank The coebook sizes of the two component s are the same value, N a 1 The iversity orer of this ST-CI, r, is min r 1, r2 2 Assuming that all irectional pairs M an M are eually probable, the statistical iversity orer of this ST-CI, r s, is r 1 + r 2 with probability Na Na 2 2 α Na Na Na + N 2 2 a 2 The proof of Theorem 1 is omitte ue to space limitations A problem of the above iscussion is that the analysis is purely base on pairwise error probability However, system performance is normally expresse as average error probability AEP We further introuce a iversity concept base on AEP Definition 2: Denote AEP of the communications system with the coewor block set M at average receive SNR ρ as AEP M, ρ Assume that AEP M, ρ is ifferentiable at ρ Denote fρ log 10 AEP M, ρ an gρ log 10 ρ The average iversity orer, r a, at the average signal-tonoise ratio SNR of each receive antenna, ρ, is efine as a ifferential r a fρ gρ 9 Note that AEP cannot be generally erive Thus we propose an analysis of the iversity performance of CI-ST base on the error union boun EUB, an upper boun on the average error probability, is an average of the pairwise error probabilities between all irection pairs of coewors Due to space limitations, the EUB base analysis is not provie in etail The result of this analysis is that the average iversity orer of CI-ST can be approximate as either min r 1, r2 or r 1 + r 2, the choice of which epens on the value of SNR ρ an the coebook size N a In the case of r a min r 1, r2, the merit of CI appears as an extra coing avantage

4 Note that except for the trivial extra computational loa of coorinate interleaving, for the same size of encoing matrices, the complexity per coewor of the ST- CI system is almost the same as that of conventional systems However, the upper boun achievable average iversity orer of a ST-CI system is almost twice that of conventional block-base space-time coe BSTC systems if the two component s in the ST-CI have similar iversity features It is worth mentioning that using nonlinear sphere or ML ecoing, the conventional BSTC systems nee much higher complexity to reach an average iversity orer comparable to ST-CI We remark the scope of this approach is not limite to Other block-base space-time coe esigns may also be improve using the propose space-time inter- coorinate interleaving approach Further, the pair of coewors use in ST-CI coul be viewe as a single specially esigne coewor of size 2T Thus ST-CI systems coul be viewe as extensions of systems using ifferent esign criteria A Simulation setup IV PERFORMANCE Perfect channel knowlege amplitue an phase is assume at the receiver but not at the transmitter Assume the number of receive antennas is eual to the number of transmit antennas Channel symbols are estimate using MMSE estimation Data symbols use QPSK moulation in all simulations The signal-to-noise-ratio SNR reporte in all figures is the average symbol SNR per receive antenna The matrix channel is assume to be constant over ifferent integer numbers of channel uses or symbol time slots, an ii between blocks We enote this interval as the channel change interval CCI Three space-time block coes, Coe A, Coe B, an Coe C, are use as component coing matrices of ST-CI systems in the simulations Coe A is chosen from E 31 of 4], a class of rate-one suare of arbitrary size propose by Hassibi an Hochwal Coe B is chosen from Design A of full iversity full rate FDFR coes propose by Ma an Giannakis 12] Coe C is a non-rate-one high rate coe for the configuration of 4,T 6,Q 12, propose by Hassibi an Hochwal 4] B Performance comparison The performance comparison of coe A is shown in Figures 2, 3 an 4 The performance comparison of coe B is shown in Figure 5 The performance comparison of coe C is shown in Figure 6 In block faing channels, ie, when the 4 4 MIMO channels are constant over the pair of ST- coewors an coe A is use, ST-CI obtains the same performance as that of ST- as shown in Figure 3 However, as shown in Figures 2, 4 5, an 6, ST-CI significantly outperforms ST- at high SNRs in rapi faing channels Thus, the ST-CI proceure may be applie to both rate-one an slightly lower rate coes Observing Figures 2 an 5, the performances of coe A an coe B are similar in rapi faing channels Thus, even though coe A is not esigne uner a iversity criterion, coe A appears to possess goo iversity properties V CONCLUSION This paper has propose a general space-time inter- coorinate interleaving proceure, ST-CI, which may be applie to either rate-one information lossless or slightly lower rate block-base space-time coing systems This enables not only symbol-level iversity but also coorinatelevel iversity The upper boun iversity orer of ST-CI is analyze, an the analysis results of the upper boun statistical iversity orer an average iversity orer of ST- CI are provie Compare with conventional ST-, ST-CI show either much higher average iversity orer or extra coing avantage in time varying channels Compare with conventional block-base STCs, ST-CI maintains iversity performance in uasi-static block faing channels, an significantly improves the iversity performance in the high SNR regions of rapi faing channels This work was supporte by the Natural Sciences an Engineering Research Council of Canaa Grant an Communications an Information Technology Ontario REFERENCES 1] VTarokh, NSeshari, an ACalerbank, Space-time coes for high ata rate wireless communications: performance criterion an coe construction, IEEE TransInformTheory, vol 44, pp , Mar ] SAlamouti, A simple transmitter iversity scheme for wireless communications, IEEE JSelectAreas Commun, pp , Oct ] VTarokh, HJafarkhani, an ARCalerbank, Space-time block coe from orthogonal esigns 3, IEEE TransInformTheory, vol 45, pp , July ] B Hassibi an B M Hochwal, High-rate coes that are linear in space an time, IEEE TransInformTheory, vol 48, no 7, pp , July ] KBoulle an JCBelfiore, Moulation schemes esigne for the Rayleigh channel, in Proc CISS 1992, 1992, pp ] BDJelicic an SRoy, Cutoff rates for coorinate interleave QAM over Rayleigh faing channels, IEEE TransCommun, vol 44, no 10, pp , Oct ] Y-H Kim an MKaveh, Coorinate-interleave space-time coing with rotate constellation, in Proc IEEE VTC, vol 1, Apr 2003, pp ] MZAKhan an BSRajan, Space-time block coes from co-orinate interleave orthogonal esigns, in Proc IEEE ISIT 2002, 2002, pp ] MZAKhan, BSRajan, an M H Lee, Rectangular co-orinate interleave orthogonal esigns, in Proc IEEE Globecom 2003, vol 4, Dec 2003, pp ] W Su, ZSafar, 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5 Input Data Bits Digital Moulation Constellation Mapping 2 ST-Inter- coorinate interleaving ST- Encoing ST- Encoing Transmitte first Transmitte secon Multiple transmit antennas Performance comparision ST I CI vs ST 2, N R 2, T 2, Q 4, CCI 1 ST ST I CI Channel Output Data Bits Digital De-Moulation Constellation De-Mapping 2 ST-Inter- coorinate einterleaving ST- Decoing ST- Decoing Receive first Receive secon Multiple receive antennas SNR B Fig 1 Space-time inter- coorinate interleaving system structure Fig 4 performance comparison of ST-CI vs using coe A, CCI 1, 2, N R 2, T 2, Q 4 Performance comparision ST I CI vs ST 4, T 4, Q 16, CCI 1 Performance comparision ST I CI vs ST 4, T 4, Q 16, CCI 1 ST ST I CI ST ST I CI SNR B Fig 2 performance comparison of ST-CI vs using coe A, CCI 1, 4, T 4, Q 16 Performance comparision ST I CI vs ST 4, T 4, Q 16, CCI 8 ST ST I CI SNR B Fig 5 performance comparison of ST-CI vs using coe B, CCI 1, 4, T 4, Q 16 Performance comparision ST I CI vs ST 4, T 6, Q 12, CCI 1 ST ST I CI SNR B SNR B Fig 3 performance comparison of ST-CI vs using coe A, CCI 8, 4, T 4, Q 16 Fig 6 performance comparison of ST-CI vs using coe C, CCI 1, 4, T 6, Q 12