Multiple-Symbol Detection of Differential Unitary Space-Time Modulation in Fast-Fading Channels with Known Correlation

Transcription

1 Confeence on Infomation Sciences and Systems, Pinceton Univesity, ach, ultiple-symbol Detection of Diffeential Unitay Space-Time odulation in Fast-Fading Channels with Known Coelation Bijoy Bhukania and Phil Schnite 1 Depatment of Electical Engineeing The Ohio State Univesity 15 Neil Ave, Columbus, OH 41 {bhukanib, schnite}@ee.eng.ohio-state.edu Abstact With a fading channel, standad L detection of diffeential Unitay Space-Time modulation (DUST) leads to an eo floo in the vesus-snr cuve since it is deived unde the assumption that the channel emains constant duing evey consecutive pai of matix-symbols. Assuming knowledge of the channel fading coelations, we design multiple-symbol L diffeential detectos which dastically educe this eo floo, especially fo fastfading channels. ultiple-symbol diffeential detection has the additional benefit of educing the db loss in SNR (elative to coheent detection) that chaacteizes standad one-symbol diffeential detection. To deive these new detectos, we make the simplifying assumption that the channel changes once pe matix-symbol (i.e., block fading) athe than once pe channel use (i.e., continuous fading). Howeve, the block fading assumption is not equied when diagonal codes ae used. The simulation esults show that, in spite of the block-fading assumption, the new eceives fa outpefom standad single-symbol detection unde continuous fading as well. I. Intoduction The capacity of wieless communication systems ove fading channels is significantly enhanced by the use of multiple antennas at the tansmitte and/o at the eceive. Space-time coding is a bandwidth and powe efficient method of communication ove fading channels that ealizes benefits of multiple tansmit antennas [1]. Although the majoity of space-time codes poposed so fa assume knowledge of channel state infomation (CSI) at the eceive [1], the class of unitay space-time codes [] and diffeential unitay space-time codes [, 4] have been poposed fo Rayleigh flat-fading channels when neithe the tansmitte no the eceive knows the fading coefficients. Diffeential unitay space-time modulation (DUST) can be seen as an extension of diffeential phase-shift keying (DPSK), commonly used in single antenna systems when the channel is unknown to the eceive as well as to the tansmitte. The infomation is encoded in the phase diffeence between two consecutive symbols, so that the infomation can be decoded fom phase diffeence between two consecutive obsevations. In DUST, the tansmitted symbols can be consideed unitay matices, whee is the numbe of tansmit antennas. The peviously tansmitted matix-symbol is pe-multiplied by the cuent infomation matix-symbol to fom the cuent tansmitted matix-symbol. The standad diffeential eceive detects one infomation matix-symbol fom each pai of consecutive eceived matices. 1 Coesponding autho Ideally, we desie that the channel expeienced by evey pai of consecutively tansmitted matices is the same. Howeve, this would equie that the channel be foeve constant a equiement which is neve met in pactice. The pefomance of DUST with standad single-symbol maximumlikelihood (L) decoding in continuously-fading channel has been evaluated in [5], whee it has been shown that an eo floo is achieved when the eo due to channel vaiation dominates that due to additive noise. In this pape, we pesent DUST detectos fo use in fast fading channels channels fo which standad single-symbol detection is unsuccessful. In deiving the new detectos, we make the simplifying assumption that the channel changes once pe matix-symbol (i.e., channel uses), and that the eceive knows the fading coelation. We will see, howeve, that the block fading assumption is not necessay when diagonal space-time codes [] ae used. Unde these assumptions, single- and multiple-symbol L diffeential detectos ae deived. Simulation esults show that, in spite of the blockfading assumption, the new detectos exhibit significantly impoved pefomance compaed to the standad single-symbol detecto in fast continuously-fading channels. The notations used in this pape ae as follows: atices will be denoted by capital lettes (e.g., X and X) and vectos by lowe case bold (e.g., x). I N will denote identity matix of size N N. The opeato vec( ), e.g., x n = vec(x n), denotes stacking of the columns of matix X n in column vecto x n. ( ) denotes conjugate tansposition, denotes the Konecke poduct, t( ) denotes the tace opeato, and det( ) the deteminant. X n = 6 4 II. Backgound The system model in a continuously-fading channel is s,n H,n 7 6 H 1,n 7. s 1,n H 1,n Wn... s 1,n (1) whee X n is the N eceived matix duing the n th matixsymbol inteval, and whee and N ae the numbe of tansmit and eceive antennas, espectively. H k,n is the N IO channel esponse matix at the k th time instant in the n th matix-symbol inteval, i.e., at the (n + k) th channel use. S n = [s,n s 1,n... s 1,n] is the n th tansmitted matix-symbol, encoded as S n = V zn S n 1. L = {, 1,..., R 1} is the time-n intege index into matix alphabet A of size R, so that V zn A. Thus R is the numbe of bits pe channel use. W n is a matix of i.i.d. unit vaiance Gaussian enties and is the aveage SNR.

2 Unde the assumption that the channel emains constant fo one matix-symbol inteval, we have H n = H,n = = H 1,n, which changes (1) to X n = SnHn + Wn () Even without this assumption, the use of diagonal space time codes S n implies that the continuous fading model (1) simplifies to X n = SnH(c) n + W n () whee the k th ow of the equivalent continuous fading channel matix H n (c) is the k th ow of H k,n, fo k =,..., 1. If the IO fading pocess H k,n is independent between antennas, then the equivalent fading pocess H n (c) is also independent between antennas; the pocess H n (c), howeve, is -fold faste than H k,n. Keeping this in mind, H n (c) shall be denoted as H n fom hee onwads, and () will be used as the system model. Note that () is an appoximate model when non-diagonal codes ae used in continuous fading channel. Since X n 1 = p Sn 1Hn 1 + Wn 1, on ealizing that S n = V zn S n 1, and assuming H n = H n 1, we can wite X n = V nx n 1 + W n V zn W n 1 Ŵ n (4) Since V zn is unitay, Ŵ n contains i.i.d. Gaussian enties with vaiance twice of those in W n. Fom (4) it is staight-fowad to show that the L detection ule fo is [] ẑ n = ag max R[t{XnV zn X n 1}] L Due to inceased noise vaiance in Ŵn, this detecto loses db in pefomance compaed to coheent detection. Unde the assumption of H n = H n 1, the Chenoff uppe bound on P e fo the detecto in conveges to zeo as []. Howeve if the channel changes fom one matix-symbol inteval to the next, i.e., if H n = H n 1 + H, then (4) becomes X n = V zn X n 1 + Sn H +W n V zn W n 1 X n whee the tem X n ceates additional noise that induces an eo floo in cuve. Note that the detecto in ignoes channel vaiation and is theefoe suboptimal in a fading envionment. In following sections we deive detection ules that exploit knowledge of the autocoelation of the time-vaying channel coefficients. III. One-Symbol L Diffeential Detection Hee we deive the L detecto fo given X n and X n 1 that exploits knowledge of the coelation between H n and H n 1 (athe than assuming H n = H n 1 as in standad detecto []). Denoting h n = vec(h n),x n = vec(x n), and w n = vec(w n), xn x n 1 x n = + IN S n wn w n 1 I N S n 1 hn h n 1 (6) Assuming h n and w n contain zeo-mean unit-vaiance i.i.d. Gaussian andom vaiables, we have E[h nh n] = I N. Futhemoe, we assume that E[h nh n k] = ζ k I N, k >. Conditioned S n and S n 1, x n is a zeo-mean Gaussian vecto with autocoelation matix R (1) = I N ζ 1I N V ζ 1I N V zn I N + I N (7) whee we have used the facts that S n is unitay and S n = V zn S n 1. Note that the distibution of x n depends on V zn athe than S n and S n 1. Equation (7) can be ewitten R (1) = (1 ζ1 ) + 1 I N " p # + ζ1 h p ζ1 ζ 1 ζ 1 IN V z n ζ 1 ζ 1 IN Vzn i Fom the identity det(i + AB) = det(i + BA) it follows that det(r (1) ) is independent of V zn. Note also that R (1) 1 1 IN = d 1 ζ 1I N V d 1 = ` ζ 1 > ζ1in Vzn IN Thus the L detecto fo given X n and X n 1 (equivalently given x n ) becomes ẑ n = ag max p(x n V zn ) L = ag max e x n R(1) 1 x n L h = ag min t L ζ 1 d 1 X nv zn X n 1 + ζ 1 d 1 X n 1V X n i (8) = ag max R{t[ζ 1XnV zn X n 1]} L While detecto is simila to detecto, it exploits the known fading coelation ζ 1. When ζ 1 is eal and positive, howeve, it does not affect the decision ule and can be emoved, making identical to. Fo easons that will become clea late, we denote a (1),1 = ζ1. In ode to futhe impove pefomance we conside joint diffeential detection of multiple symbols in the next section. IV. ultiple-symbol L diffeential detection ultiple-symbol diffeential detection of -PSK has been poposed as an effective way to educe the db SNR loss incued by one-symbol diffeential detection [6], as well as to enhance pefomance in coelated Rayleigh fading channels [7]. We extend this idea to DUST and incopoate knowledge of fading coelation, assuming the block-fading channel descibed in the pevious section. Fist we deive the - and -symbol joint detectos and late genealize to an abitay numbe of symbols. A. Two-Symbol L Detection We now collect X n, X n 1, and X n, with the intention of

3 detecting 1 and. Staightfowad extension of (6) yields 4 x n x n 1 x n x n p 4 5 = 4 I N S n w n w n 1 w n 5 + I N S n 1 I N S n 5 4 h n h n 1 h n Unde the channel and noise assumptions of Section III, the eceived vecto x n conditioned on S n, S n 1 and S n is zeomean Gaussian with an autocoelation matix R () that depends only on V zn 1, V zn, ζ 1, ζ and. As befoe, det(r() ) can be shown to be independent of V zn and V zn 1. In this case, R () 1 1 = 4 d whee, IN,1 IN Vz n 1, IN V 1 V d = ` + 1 ` `, = (ζ 1,1 =,1 =, =,1 IN V 1 1, 1,1 IN IN V, IN Vzn V zn 1 1,IN Vzn, IN 5 ` + 1 ( ζ 1 + ζ )+ 1 ζ + ζ1ζ ), = ` ζ 1 ` ζ 1, = 5 (1) ` + 1 ζ 1 ` ζ 1ζ (11) ` + 1 ζ ` ζ 1 (1) Hence the detection ule is given by {ẑ n, ẑ n 1} = ag max x 1 n R() xn 1, L = ag max Rˆt{,1 X nv zn X n 1 1, L + 1, X n 1V zn 1 X n +, X nv zn V zn 1 X n } The main diffeence between detecto and one-symbol detecto is that makes use of additional channel paametes. Thus we hope fo impoved pefomance in fastfading channels. It is woth noticing that when the channels in subsequent blocks ae independent (i.e., ζ 1 = ζ = ), then,1 = a(), = a() 1, =, implying that diffeential encoding cannot be used an intuitively satisfying obsevation. B. Thee-symbol L detection We now give the stuctue of the -symbol diffeential detecto as a pelude to the geneal m-symbol case and to demonstate that closed-fom expessions fo the paametes a (m) become quite lengthy when m. As in pevious sections, we compute the invese of the autocoelation matix of x n = [x n x n 1 x n x n ], conditioned on {S n, S n 1, S n, S n }, and find that its deteminant is independent of {V zn, V zn 1, V zn }. This leads to the L detecto {ẑ n, ẑ n 1, ẑ n } = ag max R[t{, 1, L whee, X n V zn X n + 1, X n 1V zn 1 X n +,1 X nv zn X n 1 + 1, X n 1V zn 1 V zn X n +, X nv zn V zn 1 X n +, X nv zn V zn 1 V zn X n }], =,1 = 1, = b 5 ` b 1 ` ζ 1 b 1 ζ 1ζ ` ζ 1ζ + ` 1, =, = b 5 b 1, = b 1 and b 6 ζ1 + ζ ζ 1 ζ 1ζ ζ ` (ζ ζ 1ζ ζ ` ζ 1 ζ ` b 1 = ` 1 ζ + ζ 1 ζ ζ 1 ζ 1 ) ` `1 + (ζ 1 ζ + ζζ ) + ζ 1 b 5 = + ` R(ζ 1ζζ ) ` ζ1 + ζ + ζ ` b 6 = + ` R `ζ 1ζ ζ + Note that is a staightfowad extension of, except that the coefficients { } ae diffeent fom {a() }. As befoe, if ζ 1 = ζ = ζ =, then = k l, implying that diffeentially-encoded symbols cannot be detected. When ζ k = 1 k, i.e., the channel is fixed, the detection ules and do not depend on SNR, since in that case,1 = a(), > and a(),1 = a() 1, = a(), = a(), >. Though we have not deived L detection ules fo nondiagonal DUST codes in continuously-fading channels, the multiple-symbol detection ules and, deived fo the block-fading channel, should still show impovement ove the standad detecto in a continuous-fading envionment. The coefficients {a (m) i,j } used in this case would be ecomputed with ζ k defined such that E[h,nh,n k] = ζ k I N fo h,n = vec(h,n). The simulation esults in Section VI confim that detectos and significantly outpefom the standad single-symbol detecto unde fast continuous fading. C. m-symbol L Detection The L joint detection ule fo m symbols can be shown to be {ẑ n, ẑ n 1,..., ẑ n m+1} = ag max, 1,..., m+1 L " R t ( m 1 X k= m k 1 X i= a (m) i,i+k+1 X n i (15)! )# i+k Y V zn j X n i k 1 whee Q m 1 j= (Aj) = AA1 Am 1. Fom the pevious two subsections, we see that deiving closed-fom expessions fo j=i

4 the coefficients {a (m) } is quite difficult when m >. These coefficients can, howeve, be computed numeically by fist constucting an (m + 1)N (m + 1)N autocoelation matix as an extension of (7), but with the additional simplification that V zn j = I j, then computing its invese. In the case of a fixed channel it can be shown that a (m) i,j = a (m) > i, j, k, l {,..., m} & i j, k l. As a esult, the detecto (15) becomes independent of coefficients a (m) i,j. It is impotant to note that thee does not appea to exist a Vitebi-like algoithm fo m-symbol L detection that has complexity linea in m. Obseve that it is not possible to wite the detection metic as a sum of tems that contain stict subsets of the symbol set {V zn,..., V zn m+1 }; thee is always one `Q tem with the fom t Xn m 1 j= V j Xn m. Thus, maximization of the quantity in (15) can only accomplished using a bute foce seach ove all symbol combinations, yielding a complexity exponential in m. V. Suboptimal Sequence Detection Pactical applications equie the detection of N symbols. Yet, as we have seen, the joint L diffeential detecto fo N symbols has a complexity that is exponential in N. Thus, we ae motivated to conside suboptimal N-symbol detection using some combination of m-symbol L detectos fo, say, m. It is instuctive to note the diffeence between the N- symbol L detecto and any suboptimal N-symbol detecto constucted fom m-symbol L detectos (m < N). Fom (15), we see that the suboptimal sequence detecto uses detection metics that linealy combine tems based on subsets of {V zn,..., V zn m+1 } fo n {m 1,..., N 1}. The combining coefficients ae taken fom the set {a (m) }. In contast, the optimal N-symbol detection metic linealy combines these tems with additional tems based on subsets of {V zn 1,..., V z } that ae ignoed by the suboptimal detecto. In addition, the N-symbol combining coefficients ae {a (N) }, which ae, in geneal, diffeent fom {a (m) }. Thus, the task of constucting a good N-symbol detecto that employs (at most) m-symbol optimal detections (m < N) can be consideed equivalent to the appoximation of {a (N) } by a spase coefficient set. Fo the simulations in Section VI, we divide the (N + 1)- matix obsevation sequence into consecutive subsequences of length m + 1, each of which ovelaps its neighbo by one matix-obsevation. Then, m-symbol detection is pefomed on each subsequence. This is equivalent to eplacing the coefficients {a (N) } with {a(m) } whee defined, and eplacing the est with zeos. Schemes in which neighboing obsevation subsequences ovelap by moe than one matix-obsevation lead to diffeent appoximations of the coefficient set {a (N) } and ae cuently unde investigation. VI. Simulations We evaluate the pefomance of the detectos in two types of channel: the block fading channel () and the continuous fading channel (). The coelation between fading coefficients k symbols apat is given by J (πf DT sk) [8] in continuous fading channels, whee f DT s is the nomalized Dopple fequency. In block fading channel, coelation between channel coefficients m matix-symbols apat is given by J (πf DT sm) fixed channel fixed channel SNR [db] Fig. 1: Diagonal codes in continuous fading channel with f DT s =.1. As shown in Section II, the use of diagonal codes in continuous-fading channels yield the same model as geneal codes in block-fading channels, and hence detecto pefomance with diagonal codes is identical fo these two channel types. Theefoe, we fist pesent the pefomance of the detectos and in continuous-fading channels using diagonal codes. The simulations assume two tansmit and two eceive antennas with R = 1 and the constellation specified in []. As descibed in Section V, detection is accomplished by detecting non-ovelapping potions of the tansmitted symbol sequence using 1-, -, and -symbol L diffeential detectos,, and. Results will be pesented fo detectos with exact knowledge of fading coelations, as well as fo detectos which assume that the channel is fixed (maked by fixed in the figues). In the figues shown hee, solid, dashed, and dotted lines coespond to 1-, - and - symbol detection, espectively. Fig. 1, whee f DT s =.1, clealy illustates the advantage of detectos which jointly detect multiple symbols and incopoate channel fading paametes. Note that detecto, which ignoes the fading coelation, succumbs to a vey high eo floo. Detecto, designed to incopoate fading coelation into single-symbol detection, pefoms same as does, since Rayleigh flat-fading model used in ou simulations leads to ζ 1 >. Thus, both foms of single-symbol detection pefom vey sub-optimally in the fading envionment. The -symbol detecto which incopoates fading coelation exhibits consideably impoved pefomance, although still succumbing to an eo floo. eanwhile, the -symbol detecto which ignoes fading coelation pefoms even wose than the one-symbol detecto. Pefomance inceases damatically with the -symbol detecto that incopoates fading, and deceases with the -symbol detecto that ignoes fading. As hinted by the plot, even the good -symbol detecto will succumb to an eo floo at high-enough SNR. The impotant point, howeve, is that the eo floo has been pushed outside of the expected opeating ange. Figues and, coesponding to nomalized Dopple fe-

5 fixed channel fixed channel 1 4 fixed channel fixed channel SNR [db] Fig. : Diagonal codes in continuous fading channel with f DT s = SNR [db] Fig. : Diagonal codes in continuous fading channel with f DT s =.5. quencies of f DT s =.5 and.5, espectively, mimic the esults of Fig. 1 but in a less ponounced fashion. Again we see the impovement associated with multiple-symbol L diffeential detectos that incopoate fading coelation. Next, the pefomance of the detectos has been evaluated in continuously-fading channels with non-diagonal codes to illustate the pefomance loss due to appoximation in the system model (). The non-diagonal codes ae geneated by ight multiplying the diagonal codes by a fixed non-diagonal unitay matix. Because such an opeation does not change the poduct distance of the constellation [], the compaison of diagonal to non-diagonal codes is fai. Figues 4 and 5 illustate the pefomance of the detectos with non-diagonal codes in continuously-fading channel with f DT s =.5 and.5, espectively, and compaes them with pefomance of -symbol detecto with diagonal codes. Although the pefomance loss due to neglecting the channel vaiation within the matix-symbol inteval is significant when f DT s =.5 and negligible when f DT s =.5, in both cases - and -symbol detectos that incopoate knowledge of fading coelations pefom bette than the single symbol detecto in tems of educing the eo floo. The above obsevations lead us to impotant conclusions about the detectos descibed in this pape. Fist, multiplesymbol detection is essential to combat fading channels since the detectos and ae often equivalent. Inceasing the numbe of symbols being jointly detected impoves the pefomance in tems of SNR loss and inceases the obustness against fading channels when knowledge of fading coelations is popely incopoated into the detection ule. Geneally, the faste the fading, the moe symbols ae equied in joint detection to push the eo floo out of the opeating ange. VII. Conclusions In this pape, we have demonstated the efficacy of multiplesymbol L diffeential detection that incopoates channel fading paametes in combating the eo floo exhibited by standad one-symbol L detection of DUST. In fact, we have shown that multiple-symbol (vesus single-symbol) detection is essential to pefomance enhancement. Ou multiplesymbol detection ules, which assume the channels to be blockfading when non-diagonal codes ae used, have been shown to impove pefomance in continuously-fading channels as well. We ae cuently investigating the obustness of these detectos to impefect knowledge of fading coelation and SNR as well as educed complexity implementation of the multiplesymbol detectos fo lage m. In addition, we ae woking to deive theoetical eo bounds fo these detectos. Refeences [1] A. Naguib, N. Seshadi, and A.R. Caldebank, Inceasing data ate ove wieless channels, IEEE Signal Pocessing agazine, vol. 17, pp. 76-9, ay. [] B.. Hochwald and T.L. azetta, Unitay space-time modulation fo multiple-antenna communications in Rayleigh flat fading, IEEE Tans. on Infomation Theoy, vol. 46, no., pp , a.. [] B.. Hochwald and W. Sweldens, Diffeential unitay spacetime modulation, IEEE tans. on Communications, vol. 48, no. 1, pp. 41-5, Dec.. [4] B.L. Hughes, Diffeential space-time modulation, IEEE Tans. on Infomation Theoy, vol. 46, no. 7, pp , Nov.. [5] C. Peel and A. Swindlehust, Pefomance of unitay spacetime modulation in continuously changing channel, in IEEE Intenat. Conf. on Acoustics, Speech, and Signal Pocessing, 1. [6] D. Divsala and.k. Simon, ultiple-symbol diffeential detection of PSK, IEEE tans. on Communications, vol. 8, no., pp. -8, ach 199. [7] P. Ho and D. Fung, Eo pefomance of multiple symbol diffeential detection of PSK signals tansmitted ove coelated Rayleigh fading channels, IEEE Inten. Conf. on Communications, vol., pp , [8] W.C. Jakes, icowave obile Communications, IEEE pess, Piscataway, NJ, 199.

6 fixed fixed diagonal codes SNR [db] Fig. 4: Non-diagonal codes in continuously-fading channel with f DT s = fixed fixed diagonal codes SNR [db] Fig. 5: Non-diagonal codes in continuously-fading channel with f DT s =.5.