New Space Time Trellis Codes for Two-Antenna Quasi-Static Channels

Transcription

1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER We can conclude the followng. From ν = λ>0, =1,...,K, t follows that the correspondng constrants are tght,.e., tr(q )= p. Furthermore, Γ = W 1 0 mples that the correspondng constrant s tght,.e., W = K H =1 Q H H + I. Ths confrms the equvalence of problems (4) and (6) and mples that Γ = ( K H =1 Q H H + I) 1 and H h ( K =1 H Q H H + I) 1 H =λi Ψ, =1,...,K. (21) The last equaton s very smlar to the sngle user WF condton. The only dfference s that λ (.e., the nverse of the water level) s not obtaned from a sum power constrant. It s a fxed value so that t determnes the sum power by tself. Snce Ψ 0, from(21),t follows that λ max ({Q }), whch s defned n (15), satsfes λ ({Q })={ λ0 <λ, f Ψ 0 for all =1,...,K λ, otherwse. (22) From the complementary slacness,.e., tr(ψ Q )=0, t follows that Ψ 0 for all =1,...,K mples that Q =0 for all =1,...,K. By lettng Q =0 n (21), t s easy to observe that Ψ 0 f and only f λ>λ max ( H H H ) for all =1,...,K. Physcally, ths stuaton corresponds to a very small water level n (21), whch does not nundate any of the channel egenmodes. Clearly, such λ values cannot be optmal for master problem (5). Thus, the search doman of master problem (5) can be restrcted to λ< max =1,...,K λ max ( H H H ). In such condtons, λ({q })=λ, and therefore, the dual varable that s defned n (16) become an optmal dual varable,.e., Γ({Q })=( K H =1 Q H H + I) 1 = Γ. REFERENCES [1] S. Vshwanath, N. Jndal, and A. Goldsmth, Dualty, achevable rates, and sum-rate capacty of Gaussan MIMO broadcast channels, IEEE Trans. Inf. Theory, vol. 49, no. 10, pp , Oct [2] P. Vswanath and D. N. C. Tse, Sum capacty of the vector Gaussan broadcast channel and upln downln dualty, IEEE Trans. Inf. Theory, vol. 49, no. 8, pp , Aug [3] W. Yu and J. M. Coff, Sum capacty of Gaussan vector broadcast channels, IEEE Trans. Inf. Theory, vol. 50, no. 9, pp , Sep [4] H. Vswanathan, S. Venatesan, and H. Huang, Downln capacty evaluaton of cellular networs wth nown nterference cancellaton, IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp , Jun [5] W. Yu, A dual decomposton approach to the sum power Gaussan vector multple access channel sum capacty problem, n Proc. CISS, Baltmore, MD, Mar , [6] N. Jndal, W. Rhee, S. Vshwanath, S. A. Jafar, and A. Goldsmth, Sum power teratve water-fllng for mult-antenna Gaussan broadcast channels, IEEE Trans. Inf. Theory, vol. 51, no. 4, pp , Apr [7] M. Codreanu, M. Juntt, and M. Latva-aho, Low complexty teratve algorthm for fndng the MIMO-OFDM broadcast channel sum capacty, IEEE Trans. Commun., vol. 55, no. 1, pp , Jan [8] W. Yu, Sum-capacty computaton for the Gaussan vector broadcast channel va dual decomposton, IEEE Trans. Inf. Theory, vol. 52, no. 2, pp , Feb [9] W. Yu, W. Rhee, S. Boyd, and J. M. Coff, Iteratve water-fllng for Gaussan vector multple-access channels, IEEE Trans. Inf. Theory, vol. 50, no. 1, pp , Jan [10] T. M. Cover and J. A. Thomas, Elements of Informaton Theory. New Yor: Wley, [11] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge, U.K.: Cambrdge Unv. Press, [12] W. Yu, prvate communcaton, New Space Tme Trells Codes for Two-Antenna Quas-Statc Channels Y Hong, Member,IEEE,and Albert Gullén Fàbregas, Member,IEEE Abstract New space tme trells codes wth four- and eght-level phase-shft eyng (PSK) and 16-phase quadrature ampltude modulaton (QAM) for two transmt antennas n slow-fadng channels are presented n ths paper. Unle most of the codes that are reported n the lterature, the proposed codes are specfcally desgned to mnmze the frame error probablty from a unon-bound perspectve. The performance of the proposed codes wth varous memory orders and receve antennas s evaluated by smulaton. It s shown that the proposed codes outperform prevously nown codes n all studed cases. Index Terms Dversty, mult-nput mult-output channels, multple antennas, space tme codes, trells codes. I. INTRODUCTION Space tme trells codes (STTCs) were orgnally proposed n [1] to acheve both dversty and codng gans on mult-nput mult-output (MIMO) fadng channels by combnng codng over multple transmt antennas wth hgh-order sgnal constellatons. Desgn gudelnes based on mnmum ran and determnant were proposed n [1], manly based on worst-case ParWse Error Probablty (PWEP) analyss. Ever snce, multple efforts have been dedcated to further maxmze the codng gan usng the same ran and determnant crtera [2], [3]. An mproved determnant crteron that hghlghts the role of the Eucldean dstance for systems wth medum to large dversty order was presented n [4]. Dependng on the dversty order of the system, ran and determnant or Eucldean dstance crtera have been used n [5], [6] to construct four eght-level phase-shft eyng (8-PSK) STTCs by exhaustve computer search. A specfc ran crteron was developed n [7] for the partcular case of hgh-order sgnal constellatons such as 16-phase quadrature ampltude modulaton (16-QAM). The core of ths crteron s a suffcent condton to select full-ran codes wthout resortng to exhaustve computer search. Unfortunately, the codng gan of the codes n [7] was not optmzed. In [8], mproved 16-QAM STTCs were found by optmzng the codng gan based on the Eucldean dstance crteron. A common feature of all aforementoned code desgn crtera s to mnmze the worst case PWEP. To further mprove performance, Jung and Lee [9] ntroduce a code search method based on the dstance spectrum of the code [10]. Ths method s then used to search a sngle four-state four-level phase-shft eyng (4-PSK) STTC [9]. Manuscrpt receved September 5, 2006; revsed January 10, 2007, February 11, 2007, and February 12, Ths wor was supported by the Australan Research Councl under Grant DP and Grant DP Ths paper was presented n part at the 2006 Australan Communcatons Theory Worshop, Perth, Australa, February 2006, and the 2006 IEEE 63rd Vehcular Technology Conference, Melbourne, Australa, May The revew of ths paper was coordnated by Prof. T. J. Lm. Y. Hong was wth the Insttute for Telecommuncatons Research, Unversty of South Australa, Mawson Laes, SA 5095, Australa. She s now wth DEIS Unversty of Calabra, Rende CS, Italy (e-mal: hong@des.uncal.t). A. Gullén Fàbregas was wth the Insttute for Telecommuncatons Research, Unversty of South Australa, Mawson Laes, SA 5095, Australa. He s now wth the Department of Engneerng, Unversty of Cambrdge, CB2 1PZ Cambrdge, U.K. (e-mal: gullen@eee.org). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TVT /$ IEEE

2 3582 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER 2007 In ths paper, we consder desgn gudelnes that am to mnmze a truncated unon bound on the frame error rate (FER) by tang nto account the frst three terms. The PWEP terms depend on the values of the determnants and the correspondng number of codewords. Ths s smlar to the approach that was taen n [11], where the PWEP terms depend on the Eucldean dstances for moderate dversty gans nstead. In our desgn, we construct three complete sets of 4-PSK, 8-PSK, and 16-QAM STTCs for two transmt antennas over slow (quas-statc)-fadng channels. Through smulatons, t s shown that, n all cases, the new codes outperform prevously nown codes [1] [8], [11]. The rest of ths paper s organzed as follows: Secton II ntroduces the system model, and Secton III revews the code desgn crtera. Secton IV ntroduces STTC encoder structures for phase-shft eyng (PSK) and 16-QAM, respectvely. In Secton V, new 4-PSK, 8-PSK, and 16-QAM STTCs are presented together wth the smulaton results. Conclusons are fnally drawn n Secton VI. Notaton: T denotes transpose, and denotes transpose conjugate. Superscrpts I and Q denote the real and magnary parts of a complex number. Z m = {0, 1,...,m 1} denote the rng of the ntegers modulo m, andz m [j] s the rng of Gaussan ntegers modulo m, where each element z Z m [j] has {z = z I + jz Q : z I,z Q Z m }, j = 1. Complex numbers are denoted by C. II. SYSTEM MODEL We consder a quas-statc multple-antenna fadng channel wth n T transmt and n R receve antennas, for whch the receved sgnal matrx Y C n R L s gven by Y = E s HX + N (1) where L s the frame length; X =[x 1,...,x t,...,x L ] C n T L s the transmtted sgnal matrx; x t =[x 1 t,...,x n T t ] T C n T s the sgnal vector at tme 1 t L; H =[h 1,...,h nt ] C n R n T s the channel matrx, whch remans constant durng a frame of L channel uses and vares from one frame to another ndependently; N C nr L s a matrx of complex whte Gaussan nose ndependent and dentcally dstrbuted (..d.) samples N C (0,N 0 );ande s /N 0 s the average sgnal-to-nose rato (SNR) per transmt antenna. The elements of H are assumed to be..d. crcularly symmetrc Gaussan random varables N C (0, 1). We assume perfect channel state nformaton at the recever. We assume dscrete sgnal constellatons, namely, x t X for 1 t L and 1 n T,whereX Cde- notes the PSK or quadrature ampltude modulaton (QAM) sgnal set. We wll refer to a rate R space tme code S X n T L as the set of all 2 LR codewords. III. ERROR PROBABILITY AND CODE DESIGN Assumng that a codeword X S s transmtted, the maxmumlelhood recever mght decde erroneously n favor of another codeword X S, dependng on the fadng or nose realzatons. Followng [1], we defne B = X X as the codeword dfference matrx and A = BB as the codeword dstance matrx. Letnow d =det(a) = r λ =1,whereλ are the r nonzero egenvalues of A and D = {d R : d =det(a), X X S} s the set of all possble determnants of the codeword dstance matrx. Then, the unon bound on the FER becomes [9], [10] where N(d) = 1 b v p max p=p mn N(d, p) (2) mp (3) s the spectrum term, m s the number of bts per symbol of the sgnal constellaton, p mn = v/b +1 s the mnmum length n trells steps of smple error events [5] and x denotes the largest nteger that s smaller than or equal to x, p max s the maxmum allowed length of trells paths [5], N(d, p) s the number of error events of length p and determnant d, v s the memory order of the encoder, and b =log 2 M,log 4 M for M-PSK and M-QAM, respectvely (see [10] for detals). If we further defne { η(p max ) = 1 pmax } N(d, p) d n b v (2) mp R (4) d D p=p mn (2) becomes ( Es ) rnr P FER η(p max ) N 0 and we can now formulate the code desgn crtera. 1) Dversty gan: A has to be full ran for all pars of codewords. 2) Codng gan: η(p max ) has to be mnmzed over all the possble error events n the trells dagram. IV. STTC ENCODER In ths secton, we ntroduce two dfferent STTC encoder structures for PSK and QAM, respectvely. A. M-PSK STTC Encoder We consder the M-PSK STTC encoder wth memory order v and n T transmt antennas shown n Fg. 1(a). The M-PSK STTC encoder conssts of an m-branch shft regster wth total memory order v. At tme t, m bnary nputs c t, =1, 2,...,m, are fed nto the m branches. The memory order of the th branch v s gven by v + 1 v = (6) b where v = v. The m streams of nput bts are smultaneously passed through ther respectve shft regster branches and multpled by the generator vectors g 1 = [( g 1 0,1,g 1 0,2,...,g 1 0,n T ),..., ( g 1 v1,1,g 1 v 1,2,...,g 1 v 1,n T )]. g m = [( g m 0,1,g m 0,2,...,g m 0,n T ),..., ( g m vm,1,g m v m,2,...,g m v m,n T )] where g q, Z M, =1, 2,...,m; q =0, 1, 2,...,v ; = 1, 2,...,n T. Fnally, the encoder output w t Z M, t =1,...,L; =1,...,n T, can be computed as (5) ( P FER N(d) d n R d D ) ( Es N 0 ) rnr (2) ( m v ( ) ) wt = g q,c t q mod M. (7) =1 q =0

3 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER Fg. 1. (a) M-PSK and (b) 16-QAM STTC encoder wth n T transmt antennas. The STTC encoder can also be descrbed n generator polynomal form. The bnary nput stream c can be represented as c =c 0 + c 1D + c 2D c td t + (8) where D represents a unt delay operator. The generator matrx for antenna can be represented as G 1 G 2 G = (9). G m where G =g 0, + g 1,D + + g v,d v (10) s the th branch generator polynomal for transmt antenna. The coded symbol sequence that was transmtted from antenna s gven by ( m ) w = c G mod M. (11) =1 These outputs are then mapped onto M-PSK symbols x t, t =1,...,L; =1,...,n T, whch are labeled by the ntegers from 0toM 1. B. 16-QAM STTC Encoder The 16-QAM STTC encoder wth memory order v and n T transmt antennas s shown n Fg. 1(b). At tme t, the nput mapper converts the nput bts (c 1 t,c 2 t ) and (c 3 t,c 4 t ) nto two components u 1 t,u 2 t Z 4, respectvely, through natural mappng. The two components go through a two-branch shft regster wth total memory order v. The memory order of the th branch v, =1, 2, s also gven by (6).

4 3584 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER 2007 The two streams of the components are multpled by the coeffcent vectors,.e., [ ((a ) ( )) g 1 = 0,1,a1,Q 0,1,..., a 0,n T,a 1,Q 0,n T,..., (( ) ( )) ] a v 1,1,a1,Q v 1,1,..., a v 1,n T,a 1,Q v 1,n T (12) [ ((a ) ( )) g 2 2,I = 0,1,a2,Q 0,1,..., a 2,I 0,n T,a 2,Q 0,n T,..., (( ) ( )) ] a 2,I v 2,1,a2,Q v 2,1,..., a 2,I v 2,n T,a 2,Q v 2,n T (13) where a,i q,,a,q q, Z 4,anda,I j, + ja,q j, Z 4[j], =1, 2; q = 0, 1, 2,...,v. The encoder output w t, t =1,...,L; =1,...,n T, can be computed as wt = ωt, I + jω Q t, ( 2 v ( ) ) ( 2 v ( ) ) = a,i q, u t q + j a,q q, u t q mod 4 =1 q =1 =1 q =1 where w t Z 4 [j]. In generator polynomal form, the nput components u, =1, 2, can be represented as u =u 0 + u 1D + u 2D u td t +. (14) Let us defne the th branch generator polynomal for the transmt antenna as where a,i G =a,i =a,i 0, + a,i 1, a,q 0, + a,q 1, +ja,q, =1, 2 (15) D + + a,i v, Dv and a,q = D + + a,q v, Dv. The generator matrx for antenna can be represented as [ ] [ G 1 G = a G 2 = ] [ a 1,Q ] a 2,I + j a 2,Q. (16) The coded symbol sequence transmtted from antenna s gven by ( 2 ) w = u G mod 4. (17) =1 The coded symbol sequence s mapped from Z 4 [j] to a 16-QAM sgnal set by a lnear translaton mappng [10],.e., x t =2w t (3 + 3j). V. N EW STTCS AND SIMULATION RESULTS In ths secton, we present new sets of 4-PSK, 8-PSK, and 16-QAM STTCs for two transmt antennas over slow-fadng channels. The SNR per receve antenna s defned as SNR = n T E s /N 0. We assume that each frame conssts of L = 130 symbols and L =66symbols, for quadrature PSK and 16-QAM, respectvely. Ths corresponds to a total of 260 and 264 nformaton bts/frame. The codng gan of 4-PSK, 8-PSK, and 16-QAM STTCs s optmzed by tang nto account the frst three terms of the dstance spectrum n the truncated unon bound (5) [10]. The codng gan η(p max ) only provdes an estmate of performance due to the fact that the length p max that s consdered n the dstance spectrum s sgnfcantly less than the frame length. Therefore, after the code search, some codes wll have the same dversty and gan parameters. The presented codes are the ones n ths reduced set that show the best numercal performance. Smlar approaches can be found n [5], [6], and [11]. A. 4-PSK and 8-PSK STTCs Generator coeffcents are determned through exhaustve search for 4-PSK and 8-PSK sgnal sets. Snce the encoder structure cannot guarantee geometrcal unformty of the code, the search was based on all possble parwse error events. To reduce the complexty of the code search, we use the determnants of nown codes n [5] as benchmars. The complexty of the code search s the same as that for prevously nown codes reported n the lterature [5], [11]. Tables I and II lst the new 4-PSK and 8-PSK STTCs wth bandwdth effcency of 2 and 3 bts/s/hz, respectvely. Prevously nown codes are also reported for comparson. We use the standard conventon of denotng the codes by the ntals of the authors who proposed them. For example, we refer to the codes n [1] as TSC. All these codes have a full ran of r =2. The codes are descrbed by memory order v; generator coeffcents g 1, g 2 ; the frst three mnmum determnants d 1,d 2,d 3 D; the assocated weghts N(d 1 ), N(d2 ), and N(d 3 ); and the term η(p max ) wth p max =7. Fnally, SNRs at a FER of 10 4 wth n R =1, 2 are gven. In the case of n T =2, some of the nown codes n Tables I and II were specfcally desgned for very low dversty order,.e., n T n R < 4. We only report the correspondng SNRs for those. In all cases, we observe that the new codes have the lowest SNRs that are requred to acheve the FER of B. 16-QAM STTCs A specfc QAM ran crteron based on the lnear translaton mappng was frst proposed n [7]. Ths crteron s used to determne the full-dversty 16-QAM STTCs n the Z 4 [j] doman rather than usng the complex 16-QAM sgnal set, so that the code search s smplfed (see [7] for detals). A specal case of ths ran crteron s descrbed n [7, Prop. 10]. In our desgn, both generator matrces G 1 and G 2 n (9) have the specal structure that was descrbed n [7, Prop. 10]. Hence, the problem of determnng full transmt dversty n our code search can be smplfed n two steps. Step 1) Based on [7, Prop. 10], we frst chec the nonsngularty of generator matrx G 1 by determnng whether the polynomal (a 1 a2,q 1 ) (a 2,I 1 a1,q 1 ) mod 4 has at least one odd coeffcent. All the possble nonsngular generator matrces for G 1 can be also used for G 2. Remar that permutatons of G 1 and G 2 are not needed snce they yeld an equvalent code. Step 2) Based on [7, Prop. 10], for nonequvalent Σ o -coeffcent sets {α 1,α 2 } that are defned n [7, Props. 6 8], let G be a lnear combnaton of the generator matrces where G = α 1 G 1 +α 2 G 2 [ĝ ] [ĝ1,q ] = + j mod 4 ĝ 2,I ĝ 2,Q ĝ,i =α 1 a +α 2a 2,I mod 4, ĝ,q =α 1 a 1,Q +α 2 a 2,Q mod 4 and =1, 2. All the possble nonsngular generator matrces for [G 1 G 2 ] T can be obtaned by checng whether the polynomal (ĝ ĝ 2,Q ) (ĝ 1,Q ĝ 2,I ) mod 4 has at least one odd coeffcent. If the generator matrces satsfy both condtons that were prevously descrbed, the code acheves full transmt dversty [7], and t s further

5 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER TABLE I 4-PSK STTCS TABLE II 8-PSK STTCS consdered n the code selecton. Table III lsts new full-dversty 16-QAM STTCs wth a bandwdth effcency of 4 bt/s/hz. Agan, prevously nown codes are reported for comparson. The codes are descrbed n the same manner as those n Tables I and II. We can agan observe that the new 16-QAM codes have the lowest SNRs that are requred to acheve the FER of Let l be the computaton complexty of (a a 2,Q ) (a 2,I a 1,Q ) mod 4 n step 1). Let n be the number of nonequvalent Σ o -coeffcent sets {α 1,α 2 }, where n =2 (2b 1)(n T 1) (2 n T 1) = 24 [7], provded that b =2 and n T =2. Then, n step 2), the computaton complexty wll be n l. In our desgn, the code search provdes savng costs of 73% and 71% of the full search by smply applyng step 1) for 16-state and 64-state 16-QAM STTCs, respectvely. By usng Step 2), the code search provdes another 8.5% and 10.5% savng costs. Note that the full-ran code search n [8] s based on Step 2) only. Hence, to search for full-ran codes, the computaton complexty of our approach s only ((73% l +8.5% n l)/(81.5% n l)) = 14.12% and ((71% l +10.5% n l)/(81.5% n l)) = 16.43% of that n [8] for 16-state and 64-state 16-QAM STTCs, respectvely. C. Smulaton Results Fg. 2 compares the performance of the 4-PSK, 8-PSK, and 16-QAM STTCs n a multple-antenna channel wth n T =2 and n R =1. We can see that the proposed 32-state 4-PSK code outperforms

6 3586 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER 2007 TABLE III 16-QAM STTCS Fg. 2. Performance of 4-PSK, 8-PSK, and 16-QAM STTCs (2Tx, 1Rx). Fg. 3. Performance of 4-PSK, 8-PSK, and 16-QAM STTCs (2Tx, 2Rx). the best prevously nown codes by 0.2 db at the FER of 10 4.Both the new 16-state 8-PSK code and 64-state 16-QAM code outperform the current best nown codes by 0.4 db at the FER of Fg. 3 plots the performance of 4-PSK, 8-PSK, and 16-QAM STTCs wth two receve antennas over slow-fadng channels. It s shown that the new 16-state 4-PSK and 8-PSK codes outperform the best prevously nown codes by 0.85 and 0.4 db at the FER of 10 4.It s also shown that the new 64-state 16-QAM code outperforms the current best code by 0.16 db at the FER of VI. CONCLUSION Three complete sets of 4-PSK, 8-PSK, and 16-QAM STTCs over quas-statc two-antenna channels are proposed. To mnmze the frame error probablty, the new codes are constructed by 1) guaranteeng the codeword dstance matrx to be full ran over all pars of codewords and 2) mnmzng the gan term η(p max ). Based on these desgn crtera, new 4-PSK and 8-PSK codes are found based on exhaustve search over the code generators. For 16-QAM STTCs, the search for full-ran codes s smplfed by applyng a specal case of Σ o ran crteron of [7] that saves over 80% wth respect to exhaustve search. In all settngs, t s shown through numercal smulatons that the proposed codes outperform all prevously nown codes. REFERENCES [1] V. Taroh, N. Seshadr, and A. R. Calderban, Space tme codes for hgh data rate wreless communcaton: Performance crteron and code constructon, IEEE Trans. Inf. Theory, vol. 44, no. 2, pp , Mar [2] S. Baro, G. Bauch, and A. Hansmann, Improved codes for space tme trells-coded modulaton, IEEE Commun. Lett., vol. 4, no. 1, pp , Jan

7 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 6, NOVEMBER [3] R. S. Blum, Some analytcal tools for the desgn of space tme convolutonal codes, IEEE Trans. Commun., vol. 50, no. 10, pp , Oct [4] D. M. Ionescu, K. K. Muavll, and Z. Yuan, Improved 8- and 16-state space tme codes for 4-PSK wth two transmt antennas, IEEE Commun. Lett., vol. 5, no. 7, pp , Jul [5] J. Yuan, Z. Chen, B. Vucetc, and W. Frmanto, Performance and desgn of space tme codng n fadng channels, IEEE Trans. Commun., vol.51, no. 2, pp , Dec [6] Y. Hong, J. Yuan, Z. Chen, and B. Vucetc, Space tme turbo trells codes for two, three and four transmt antennas, IEEE Trans. Veh. Technol., vol. 53, no. 2, pp , Mar [7] Y. Lu, M. P. Ftz, and O. Y. Taeshta, A ran crteron for QAM space tme codes, IEEE Trans. Inf. Theory, vol. 48, no. 12, pp , Dec [8] A. Wong, J. Yuan, J. Cho, S. R. Km, I.-K. Cho, and D.-S. Kwon, Desgn of 16-QAM space tme trells codes for quas-statc fadng channels, n Proc. IEEE Veh. Technol. Conf., Mlan, Italy, Oct. 2004, pp [9] Y. S. Jung and J. H. Lee, New measure of codng gan for space tme trells codes, n Proc. IEEE Int. Symp. Inf. Theory, Washngton, DC, Jun , 2001, p [10] D. Atas and M. P. Ftz, Dstance spectrum analyss of space tme trells-coded modulatons n quas-statc Raylegh-fadng channels, IEEE Trans. Inf. Theory, vol. 49, no. 12, pp , Dec [11] C. Lao and V. K. Prabhu, Improved code desgn crtera for space tme trells codes over quas-statc flat fadng channels, n Proc. IEEE Int. Worshop Sgnal Process. Advances Wreless Commun., New Yor, Jun. 2005, pp Geometrcally Based Statstcal Channel Models for Outdoor and Indoor Propagaton Envronments Le Jang and Soon Ym Tan, Member,IEEE Abstract Ths paper presents a geometrcally based statstcal channel model wth scatterers that are randomly dstrbuted around the base staton wthn a crcle that s determned by the coverage area of the base-staton antenna. The jont probablty densty functon (pdf) of tme of arrval (TOA)/angle of arrval (AOA), the margnal pdf of AOA, and the margnal pdf of TOA are derved for a general dstrbuton of scatterers around the base staton. Raylegh and exponental dstrbutons are selected as specal cases for dscusson. Comparsons between our theoretcal calculatons and the emprcal results, as well as the measurement data that are reported n the lterature, show that the Raylegh dstrbuton scatterer model can be appled to an outdoor mcrocell propagaton envronment, whle the exponental dstrbuton scatterer model gves accurate results for an ndoor offce/laboratory propagaton envronment. Index Terms Angle of arrval (AOA), channel model, probablty densty functon (pdf), tme of arrval (TOA). I. INTRODUCTION In moble communcaton systems, the fluctuaton of the multpath propagaton sgnal nduces the fadng and dstorton of the receved sgnal. To mtgate these undesrable effects, a multelement antenna s Manuscrpt receved January 12, 2005; revsed March 5, 2006, July 24, 2006, November 27, 2006, and January 21, The revew of ths paper was coordnated by Prof. A. Abd. L. Jang was wth the School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty, Sngapore She s now wth the Fraunhofer Insttute for Telecommuncatons, Henrch-Hertz-Insttut, Berln, Germany (e-mal: le.jang@hh.fraunhofer.de). S. Y. Tan s wth the School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty, Sngapore (e-mal: esytan@ ntu.edu.sg). Dgtal Object Identfer /TVT employed at the recever. For a smart antenna communcaton system, the spatal and temporal propertes of the channel have an enormous mpact on the performance of the system. Furthermore, n multplenput multple-output systems, antenna arrays are used for both the base staton and the moble recever, and the system capacty can sgnfcantly be ncreased by explotng rch multpath scatterng envronments. Hence, the spatal dstrbuton of the multpath components s mportant n determnng the system performance. Therefore, t s necessary to have channel models that can predct the angle of arrval (AOA) of multpath components and the multple delay profle. The lterature has many prevous studes on such requrements [1] [20]. The authors n [1] [7] presented the emprcal models that are derved from measurements for outdoor and ndoor envronments. It s found that a Gaussan probablty densty functon (pdf) matches the azmuth pdf for the outdoor envronment [2], [3], and a Laplace dstrbuton s the best ft for the pdf of AOA for the ndoor envronment [5], [6]. However, emprcal models are only effcent and accurate for envronments wth the same specfc characterstcs as those where the measurements were made. They cannot be used for dfferent envronments wthout modfcatons, and they are even useless when appled to qute dfferent envronments. Although both Cramer et al. [5] and Spencer et al. [6] found that the Laplace dstrbuton s the best ft, the reported standard devatons are qute dfferent,.e., 38 and 25.5, respectvely. These emprcal models need further modfcatons before they can be appled to other ndoor envronments. References [8] [20] presented analytcal channel models, whch are derved by assumng some deal condtons. Among them, the sngle-bounce scatterng geometrc model s most wdely used. Lbert and Rappaport [15] developed a statstcal model for a mcrocell communcaton system, assumng that the scatterers are unformly dstrbuted nsde an ellpse wth foc at the base staton and the moble recever. The crcular scatterng macrocell channel model [16] assumes that the scatterers are unformly dstrbuted wthn a crcle around the moble recever, and the base staton s outsde ths area. Ertel and Reed [12] proposed a more general approach n whch the pdfs for both the ellptcal [15] and crcular [16] scatterng models can be derved usng a common approach. Oleno et al. [8] proposed an analytcal channel model based on the assumpton that omndrectonal scatterers are unformly dstrbuted over a 2-D hollow-dsc geometry. By varyng the hollow dsc s thcness, ths spatal densty degenerates to the well-nown unform-rng or unform-dsc densty [12], [16]. Despte the dfferent geometrcal shapes that are assumed n these models, dstrbutons of the scatterers are all assumed to be unform. The geometrcal-based models for nonunformly dstrbuted scatterers are nvestgated n [13] and [18]. Janaswamy [13] presented a Gaussan scatter densty model (GSDM) whch assumed that the moble staton s surrounded by the scatterers of Gaussan dstrbuton. Expressons for the pdf of the AOA, the power azmuth spectrum, the tme of arrval (TOA), and the tme-delay spectrum are provded. Laurla et al. [18] dscussed the nfluences of dfferent scatterer dstrbutons on the power delay profles and the azmuthal power spectra. However, most of the aforementoned models are proposed to predct ether macrocell or mcrocellular envronment. Janaswamy [13] showed that GSDM s applcable to both macrocell and pcocell envronments by changng the value of the standard devaton (σ s ). However, a comparson wth measurements showed that the performance of the model for the ndoor envronments s not as good as that for the outdoor envronments. A smple Gaussan dstrbuton s nsuffcent to model varous propagaton envronments. In ths paper, we nvestgate a more general stuaton where the dstrbuton of scatterers can be arbtrary. The scatterers are assumed to be dstrbuted /$ IEEE