D-BLAST Lattice Codes for MIMO Block Rayleigh Fading Channels Λ

Transcription

1 D-BLAST Lattice Codes for MIMO Block Rayleigh Fading Channels Λ Narayan Prasad and Mahesh K. Varanasi frasadn, University of Colorado, Boulder, CO October 1, 2002 Abstract We roose a class of layered sace-time block codes (STBCs) for the quasi-static multi-inut multi-outut (MIMO) Rayleigh fading channel. The roosed codes use singleinut single-outut (SISO) comonent codes based on algebraic number theory in the so called diagonal BLAST (D-BLAST) architecture. Based on a airwise error robability analysis, we show that the diversity order of the frame error robability () of the roosed codes is NK K(K 1)=2 in a N receive and K transmit antenna (N K) system with K layers. Further, this diversity order is achieved with an average comlexity, at moderate to high signal to noise ratio (SNR), of about O(K 3 ) er symbol interval. The error robability analysis also yields design criteria for the comonent SISO codes. Through simulation results we show that even with a much lower comlexity the erformance of the D-BLAST lattice codes for moderate to high sectral efficiencies and a wide range of SNR is generally close to the erformance offered by the universal lattice codes of [1] and the linear disersion codes [2]. 1 Introduction In [3] it was shown that the D-BLAST architecture could realize a significant ortion of the MIMO outage caacity by emloying SISO comonent codes. The other existing layering schemes include the horizontal BLAST (H-BLAST) [3] and the threaded sace-time (TST) layering []. The common theme is to emloy satially formatted SISO comonent codes where the satial formatting is determined by the layering. The decoder consists of a zeroforcing (ZF) or a minimum mean squared error (MMSE) front-end filter to obtain the soft statistics, followed by SISO decoders with or without decision feedback. By slitting the decoding rocess into the searate decoding of the SISO comonent codes, the decoding comlexity is greatly reduced. Diversity multilexing tradeoff curves for D-BLAST architecture were obtained in [5] 1. Recently in [1], the threaded layering concets of [] and the sace-time rotated constellations of [6] were combined to rovide full rate, full diversity codes referred Λ This work was suorted in art by NSF grant CCR and ARO grant DADD Note that the codes roosed in this aer are fixed rate codes in the terminology of [5]. They yield the maximum ossible diversity order redicted by [5], for fixed rate codes over this architecture when the decoder emloys the ZF filter.

2 to as the universal lattice codes (ULC). The aroach there uses SISO comonent codes based on algebraic theoretic constellations in the sace-time layering of []. These full rate 2 codes guarantee full diversity only if the comonent SISO codes are jointly decoded. The resulting average decoding comlexity (er symbol interval) using the shere decoder [7] for a full rate code (at moderate to high SNR) is roughly O(K 5 ). To reduce this comlexity [1] suggests designing the ULC with fewer layers so that it now transmits L symbols er channel use where L<K. This aroach of reserving full diversity but sacrificing rate for lower comlexity is in line with the code designs of [6] and [8]. However, as noted earlier in [2], ensuring full diversity order is less imortant at moderate to high sectral efficiencies, hence full diversity codes sacrificing rate for comlexity erform oorly at these sectral efficiencies. The linear disersion (LD) codes of [2] were designed using mutual information considerations and in general without any diversity criterion. They have a decoding comlexity comarable to the ULCs and yet do not in general guarantee any transmit diversity. Our codes (henceforth referred to as the D-BLAST lattice codes) reresent a judicious tradeoff between rate, diversity order and comlexity, and are articularly suitable for moderate to high sectral efficiencies. The D-BLAST lattice codes use SISO comonent codes based on algebraic number theoretic constellations in the D-BLAST architecture. The decoding involves ZF or MMSE filtering to obtain the soft statistics, followed by SISO decoding (using the shere decoder) and decision feedback. Coding for the D-BLAST architecture has also been considered in [9, 10]. [9] suggests using a single trellis code which is decoded using ZF or MMSE decision feedback detection couled with Viterbi decoding through the use of er-survivor rocessing. However, a recise diversity order analysis for the seems intractable with the code design and decoding of [9] and obtaining code designs for high sectral efficiencies has not been addressed. [10] on the other hand, derives design criteria based on the airwise error robability (PWEP) between diagonals assuming erfect feedback. Moreover the PWEP analysis assumes maximum likelihood decoding of the diagonals but no efficient code designs and decoding strategy is suggested 3. In this aer we derive the diversity order of the without any erfect feedback assumtion and rovide code design which can be effciently decoded even for systems with high sectral efficiencies. 2 Channel Model The discrete-time block fading model of a wireless communication system in a flat fading environment with N receive, K transmit antennas and a coherence interval of T symbol eriods is given by Y = 1 HX + V : (1) S Y is the N T received matrix and X is the K T codeword drawn from the STBC. The fading is described by the N K matrix H having indeendent, identically distributed (i.i.d.), zero-mean, unit variance, comlex normal elements. The random matrix H stays constant for T symbol eriods after which it jums to an indeendent value. The N T matrix V reresents the additive noise at the receiver and has i.i.d., zero-mean, comlex normal elements each with variance ff 2. The code matrix X satisfies the ower constraint E[trXX y ] = ST. Thus, with our normalizations, the average received SNR er receive antenna er symbol interval is given by ρ = 1 ff 2. 2 For a system with K transmit antennas with N K, these codes transmit K symbols er channel use. 3 The O(K 2 ) decoding comlexity of [10] accounts only for the satial rocessing comlexity (of the ZF or MMSE filtering) and does not include the comlexity of decoding the comonent SISO codes.

3 3 Code Descrition In order to describe the code construction we adot some terminology and notation from [1,]. A layer [] in a K T matrix (or resource array) is identified by its indexing set LρI K I T, where I K = f1; 2; ;Kg and I T = f1; 2; ;Tg. It has the roerty that the t th symbol on antenna a belongs to the layer iff (a; t) 2 L. Further if (a; t) 2 L and (a 0 ;t 0 ) 2 L then either t 6= t 0 or a = a 0. The sace time block code X is generated by L comonent encoders fl 1 ; ;fl L oerating indeendently. Secifically, the kk length information symbol vector (with either QAM or PAM symbols as its comonents) u 2Y kk, with Y ρ Z[i], where Z[i] reresents the P ring of comlex integers, is artitioned as u =[u T 1 ; ; u T L ]T where u m 2Y kmk L so that k = k m=1 m. u m =[u T m;1; ; u T m;k m ] T 1» m» L is then fed to the m th comonent encoder fl m : Y kmk!s kmk to obtain fl m (u m )=[(M m u m;1 ) T ; ;(M m u m;km ) T ] T where S reresents the outut symbol constellation and M m is the K K non-singular transformation matrix associated with the m th layer. With some abuse of notation, we let fl m (u m;j )=M m u m;j for 1» j» k m and 1» m» L. In this reliminary work no outer code is used over the entire layer so that ffl m (u m;j )g 1» j» k m are essentially indeendent. The role and imortance of M m will be discussed in the following sections. The outut of the m th encoder fl m (u m ) is then fed to the m th satial formatter f m : S kmk! S ~ K T where S ~ = fs; 0g. f m laces the elements of fl m (u m ) in a K T matrix according to L m the index set of the m th layer and sets the off-layer elements to zero. Finally defining X m = fm (fl m (u m )) the codeword X is obtained as X = X X L : (2) The main sace-time layering considered in this aer corresonds to the D-BLAST architecture of [3] with the width of the diagonal set to one. In articular we choose the number of layers as L = min(n;k). For simlicity, we assume that L divides K (say K = rl for some ositive integer r). Then for the D-BLAST layering defining q = r 1,wehavethat μ ν T (m 1)(q +1) k m = 1» m» L; (3) K where b:c is the standard floor oerator. The index set of m th layer, L m is given by L m = f(k g((t (m 1)(q + 1)) mod K)+1;t):1» t (m 1)(q +1)» k m Kg ; () where g(:) is defined as g(n) = ρ K if n =0 n else: (5) It can be noted that the rate of the code in bits er channel use is given by R = kklog(jyj).for T tyical values of the frame length in a quasi-static channel ( a few hundred symbol intervals) the rate loss due to the initial set u and termination can be ignored and we have that kk T ß L. Thus for L = K the code entails almost no loss in sectral efficiency and transmits K symbols er channel use ( i.e., at nearly full rate in the terminology of [1]). Note that even the codes of [1, 2] transmit at most min(n;k) symbols er channel use in order to avoid exonential comlexity in the decoding. By using lattice codes over a larger diagonal width ( i.e. using a Kw Kw transformation Mm instead where w>1 is the width of the diagonal) error roagation effects can be reduced but at the exense of increased decoding comlexity and no additional diversity order gain.

4 Decoding For notational convenience we consider the scenario where L = K and N K. At the start of a new frame, the diagonal section of layer 1 corresonding to fl 1 (u 1;1 ) is decoded first. The soft statistics for decoding fl 1 (u 1;1 ) can be obtained by either using the the zero-forcing (ZF) or the MMSE filter. We first discuss the ZF aroach. We obtain the QR decomosition [11] of the fading matrix H as H = UQ where Q is a K K lower triangular matrix and the N K matrix U satisfies U y U = I. The received vector at time t, Y t (the t th column of the received matrix Y in (1)) is re-multilied by U y, to obtain Z t = U y Y t. We letx j m = fl m (u m;j ) 1» m» K; 1» j» k m with x j m;l 1» l» K denoting the l th comonent of x j m and define W = Uy V. Then the vector of soft statistics for decoding fl 1 (u 1;1 ) is given by ~z 1 1 = [Z K;1;Z K 1;2 ; ;Z 1;K ] T. Further letting ~w 1 1 = [W K;1;W K 1;2 ; ;W 1;K ] T and G 1 ZF = 1 S diagfq K;K ;Q K 1;K 1 ; ;Q 1;1 gm 1, we can exand ~z 1 1 as ~z 1 1 = G1 ZFu 1;1 + ~w 1 1 : (6) The shere decoder [7] is used on the soft statistics (6) with G 1 ZF as the equivalent generator matrix to obtain ^u 1;1, the decision for u 1;1. As noted in [12, 13], the exected comlexity of the shere decoder (at moderate to high SNR) is roughly O(K 3 ). Further as in the case of [6], due to the form of (6), using the shere decoder with a real transformation matrix M 1 (after equalizing the hases of Q kk ) is about times less comlex than that with a comlex matrix M 1 but as shown in Section 6, the resulting erformance is generally oorer. With ^u 1;1, we obtain M 1 ^u 1;1 which is fed back. The decoding of fl 2 (u 2;1 ); ;fl K (u K;1 ); is done assuming erfect feedback in a similar manner. The rocess continues till the entire frame is decoded. For MMSE filtering, let e m;l = E[jx jm;l j2 ] 1» m; l» K. Note that by our construction, e m;l here is the same for all 1» j» k m.thekmmse Filters F l 1» l» K of size 1 K are obtained using the rocedure in [1] with ff 2 as the noise variance and the aroriate fe m:l g 1» m; l» K as the interfering symbol energies. Then letting C = 1 S H with C j denoting the j th column of C, the soft statistics for decoding fl 1 (u 1;1 ) along with the equivalent generator matrix for the shere decoder are given by ~z 1 1 =» F1 Y 1 ff1 ; F 2Y 2 ff2 ; ; F KY K ffk T and G 1 MM = diag ρ F1 C K ff1 ; ; F KC 1 ffk ff M 1 ; (7) where ff j 1» j» K are given by ff j = ff 2 F j F y j + K X m=k j+2 e m K+j;K m+1 jf j C m j 2 : (8) Note that in (7) and (8) the multile antenna interference (MAI) left after the MMSE filtering has been modelled as Gaussian noise with the aroriate variance. This results in suerior erformance as comared to the case where the MAI is ignored. Note that this issue does not arise in the standard MMSE decision feedback detector. Again, by using a real M 1 a comlexity reduction by a factor of can be obtained but at the cost of some erformance degradation 5. Using ^u 1;1, M 1 ^u 1;1 is fed back and fl 2 (u 2;1 ) is decoded in a similar manner 5 Note that this feature is absent in the ULC, where the comlexity remains the same for both real and comlex transformation matrices.

5 assuming erfect feedback and the rocess is continued until the entire frame is decoded 6.It can be verified that the exected imlementation comlexity (er symbol interval) for moderate to high SNR is about O(K 3 ) which comares favourably to the O(K 5 ) comlexity for the codes of [1, 2]. 5 Error Probability Analysis Our figure of merit in this aer is the average frame error robability () which is defined as the robability that at least one of the symbols in a frame is in error and where the average is taken over the fading coefficients. Again, for notational convenience we focus on the case where L = K and N K although the results obtained can be readily extended to the case with L = N layers and K = rl. Further, we assume that the zero-forcing filter is used to obtain the soft statistics which are then fed to the shere decoder. The main result here is that the of D-BLAST lattice codes has a diversity order equal to NK K(K 1). 2 We first multily (1) by U y, where H = UQ is the QR decomosition of H, to obtain Z = U y Y = 1 S QX + W : (9) Using the results from [15], we have that the columns of U are art of a random isotroically distributed unitary matrix and that U is statistically indeendent of Q. Further, since V has comlex normal elements and is indeendent of U, in order to derive the without loss of generality, we can consider instead the model in (9) assuming that the fading is described by Q and W is the indeendent additive noise matrix with i.i.d CN(0;ff 2 ) elements. We denote the average frame error robability by Pr(E) and exand it as Pr(E) = Pr(E 1 [E 2 [E K ) ; (10) where E m denotes the error event for the m th layer ( i.e., the event of at least one symbol of layer m being decoded erroneously) with 1» m» K. Then letting E m;j denote the error event for the j th ; 1» j» k m diagonal of the m th layer, we have E m = [ km E m;j and can exand (10) as Pr(E) = Pr([ k 1 E 1;j [ k 2 E 2;j [ k K E K;j ) (11) = Pr(E 1;1 [E 2;1 [ [E K;1 [E 1;2 ) : Note that no outer code is used over any of the layers and that the diagonals of each layer are decoded indeendently. Further since hard decisions are fed back, using a result from [16] we have, Pr(E) = Pr(E g 1;1 [E g 2;1 [ [E g K;1 [Eg 1;2 ) (12) = Pr(E g 1 [E g 2 [ [E g K ) ; where E g m, E g m;j are the error events for layer m and its jth diagonal under erfect feedback. Further, using (12) we have Pr(E g m )» Pr(E)» K X k=1 Pr(E g k ) : (13) 6 With both the zero-forcing and the MMSE filtering, decoding the diagonal sections corresonding to the frame termination require some straightforward modifications.

6 We first consider the term Pr(Em g ) where 1» m» K. Then under erfect feedback, for the m th layerwelet, Z m = 1 S diag(q)x m + W ; (1) where diag(q) is a diagonal matrix comrising of the diagonal elements of Q. Note that since fl m (u m;l ) 1» l» k m are indeendent and the noise comonents affecting them are indeendent, the searate ML decoding of fl m (u m;l ) 1» l» k m is equivalent to the ML decoding of fl m (u m ). Thus we focus on the airwise error robability (PWEP) between x m = fl m (u m ) and s m = flm (v m ). Then letting x j m;l and sj m;l for 1» l» K ; 1» j» k m denote the ((j 1)K + l) th comonent of fl m (u m ) and fl m (v m ) resectively, it is easily shown that where fi m is given by fi m = jq 11 j 2 Pr(x m! s m )=E jq11 j 2 ; ;jq KK j 2 "Qψr fim 2Sff 2!# ; (15) k m X jx j m;k sj m;k j2 + + jq KK j 2 k m X jx j m;1 s j m;1j 2 : (16) Using the results on the Bartlett decomosition of a Wishart matrix from [15] we have that the coefficients jq ii j 2 1» i» K are indeendent and that jq ii j 2 is a chi-squared random variable with 2(N K + i) degrees of freedom with its robability density function given by f jqii j 2(fl) =fla i 1 e fl (A i 1)! ; (17) where A i = (N K + i). Note that using the techniques develoed in [17], a closed-form exression for the exectation in (15) can be obtained. However, in order to exhibit the the diversity order and retain simlicity, we will comute the Chernoff bound instead. Further, it can be verified that the diversity order obtained using the Chernoff bound coincides with the diversity order of the exact PWEP. We can now simlify (15) as which, using (17), can be comuted as Pr(x m! s m )» E jq11 j 2 ; ;jq KK j 2»ex fi m Sff 2 (18) Pr(x m! s m )» 1+ P km N jx j m;1 sj m;1 j2 Sff P km jx j m;k sj m;k j2 : N K+1 (19) Sff 2 From (19) it is evident that if P k m jxj m;l sj m;l j2 6= 0 1» l» K for every air (x m ; s m ), then the diversity order achieved for Pr(E g m ), denoted by D m,isgivenby K(K 1) D m = NK 2 : (20) In order to ensure that this diversity order is achieved, we select the K K transformation matrix M m as the generator matrix obtained through the canonical embedding of an algebraic

7 number field [6,18,19] such that M m offers full modulation diversity over Z[i] K, i.e., for any v; w 2 Z[i] K such that v 6= w and letting z = M m v and y = M Q K m w,wehave jz i y i j > 0. Further, since the PWEP analysis is valid for 1» m» K, if each layer emloys a full modulation diversity lattice, we have that the diversity order of each Pr(Em g ); 1» m» K is given by (20). Then using (12), since Pr(E) is uer and lower bounded by terms of the same diversity order, the diversity order of Pr(E), D is given as D = NK K(K 1). Also, 2 it can be easily verified that for the case with L = min(n;k) = N layers where K = rl, the diversity order achieved for the using D-BLAST lattice coding (with full modulation diversity lattices) is given by r(nl L(L 1) ). For the secial case of L = 1, D-BLAST 2 lattice codes reduce to the diagonal algebraic sace time codes of [6]. In order to obtain design criteria for constructing a good full modulation diversity generator matrix for layer m we consider the worst case PWEP under erfect feedback for layer m which is obtained by maximizing Pr(x m! s m ) over all airs (x m ; s P k m ). Further since m jxj m;l sj m;l j2 jx 1 m;l s 1 m;lj 2 1» l» K, it suffices to consider just one diagonal. Then the design criteria (which we refer to as the coding gain achieved by M m ) can be written as, min x 1 m =Mmu m;1 ;s1 m =Mmv m;1 u m;1 6=v m;1 ;u m;1 ;v m;1 2Y K ψ 1 + jx1 m;1 s 1 m;1j 2 Sff 2 N 1 + jx1 m;k s 1 m;kj 2 Sff 2 N K+1! 1=Dm (21) The design criteria for selecting M m is to maximize (21) among the class of generator matrices that yield full modulation diversity. Again due to the symmetry resulting from the D-BLAST layering considered here, the airwise error analysis conducted above is valid for all layers and hence the design criteria for choosing the otimum generator matrix (by maximizing (21)) remains the same across all layers. An error robability analysis with the MMSE filter aears intractable. However, as will be seen in the simulation results, emloying the MMSE filtering in general results in a suerior erformance. In this aer as a choice for generator matrices we consider the real and comlex rotations from [18, 19]. These choices ensure full modulation diversity but are not otimal with resect to the coding gain criteria (21). Obtaining generator matrices that are otimal with resect to (21) is still an oen roblem and is under consideration. Thus the simulation results resented here serve as an uer bound on the best attainable error rates through the D-BLAST lattice coding scheme. 6 Code Design Examles and Simulation Results ffl In Figure 1 we lot the of a D-BLAST lattice code for a two transmit and receive antenna (N = K =2) examle with two layers. Here each information symbol belongs to a QAM constellation and the frame length T is equal to 200 symbol intervals. Hence the rate is (almost) bits er channel use (PCU). The 2 2 comlex cyclotomic rotation [18] is chosen as the full modulation diversity generator matrix for both the layers. The s achieved by using the ZF as well as the MMSE filter to obtain the soft statistics, are lotted. Also lotted is the achieved by the corresonding two transmit, two receive antenna full rate ULC of [1] which also emloys the same cyclotomic rotation. Fig. 2 lots the same examle but with the total rate being 8 bits PCU. Note that the diversity order of the of the D-BLAST lattice codes with the ZF aroach is 3 and that of the ULC is. However as seen from Figs. 1 and 2 at the of 10 2 the D-BLAST lattice code with the MMSE filter erforms as well as the ULC but with a much lower decoding

8 comlexity. ffl In Figure 3 we consider a three transmit and one receive antenna examle with the total rate being R = 2 bits PCU and comare the D-BLAST lattice code (with one layer) with the corresonding linear disersion code from [2] (with Q = ). Note that since L = 1 here the D-BLAST lattice code reduces to the corresonding diagonal algebraic sace-time code of [6]. The 3 3 comlex cylotomic rotation from [18] is used as the generator matrix 7 and the frame length is 20 symbol intervals. The diversity order of the of the D-BLAST lattice code with the ZF filter is 3. At the of 10 2 the D-BLAST lattice code erforms as well as the linear disersion code but with a lower comlexity. ffl In Figure, we consider a three transmit and three receive antenna examle with the total rate being R = 6 bits PCU and a frame length of 200 symbol intervals, and comare the D-BLAST lattice code (with three layers) with the corresonding full rate and full diversity ULC from [1]. We lot the achieved by the D-BLAST lattice code with both the ZF and the MMSE filter. Further, for each choice we lot the obtained by using the real 3 3 full modulation diversity rotation from [19] as well as the comlex cylotomic rotation of [18]. The ULC uses the comlex cyclotomic rotation. Here, the diversity order achieved by the D-BLAST lattice code with the ZF filter and either choice of the generator matrix is 6. Attheof10 2, we see that the ULC gains about 1. db and 2.3 db comared to the D-BLAST lattice codes with MMSE filtering along with the comlex and real rotations, resectively. ffl In Figure 5 we consider a four transmit and four receive antenna examle with the total rate being R = 16 bits PCU and a frame length of 200 symbol intervals. We lot the of the D-BLAST lattice code with four layers and for both the ZF and MMSE filters. The comlex cyclotomic rotation of [18] is used as the full modulation diversity generator matrix. Also lotted is the corresonding ULC [1] which emloys the same cyclotomic rotation with two layers. Note that the decoding comlexity (er symbol interval) of the ULC with two layers is around the same as that of the D-BLAST lattice code with four layers. At the of 10 1, we see that the D-BLAST lattice code (with either the ZF or MMSE filtering) gains about 6 db comared to the ULC. Here, the diversity order achieved by the D-BLAST lattice code with the ZF filtering is 10 and that of the ULC is 16. This examles demonstrates that at high sectral efficiencies, it is much more feasible to obtain a reduction in decoding comlexity at the exense of diversity gain rather than through sacrificing rate. ffl In Figure 6 we consider a eight transmit and four receive antenna examle with the total rate being R = 16 bits PCU and a frame length of 20 symbol intervals. We lot the of the D-BLAST lattice code with four layers and for both ZF and MMSE filtering. The 8 8 comlex cyclotomic rotation of [18] is used as the full modulation diversity generator matrix. The diversity order achieved by the D-BLAST lattice code with the ZF filtering is 20. This examle demonstrates that D-BLAST lattice codes can be easily constructed for systems with large number of transmit antennas and high sectral efficiencies and with a feasible decoding comlexity 8. 7 For this cyclotomic rotation, the information symbols are taken to be from Z[j] where j = ex(i2ß=3). 8 Note that the corresonding linear disersion code of [2] for these system arameters involves a shere decoding over 32 comlex dimensions whereas our examle involves searate shere decoders over 8 comlex dimensions each.

9 7 Other Layering Schemes In this section we demonstrate the advantages of using the lattice codes in the D-BLAST architecture by comaring with the other well known layering schemes. We again focus on the scenario with L = K and N K. The soft statistics for each layering scheme considered are obtained using the ZF filter which are then fed to the shere decoder followed by decision feedback. 7.1 H-BLAST Layering In this scheme [3] there is essentially no sace time formatting and the outut stream fl j (u j ) 1» j» K is transmitted by transmitter j over the T symbol intervals. The following result which is readily roved, gives the diversity order achieved through this layering. ffl The diversity order of the achieved for a N receive K transmit antenna system, by using full modulation diversity lattice codes in the H-BLAST layering is N K Threaded Sace-Time Layering (TST) Another layering scheme is the TST of []. A simle examle of the TST layering is given by defining the index set of the m th layer as We state the following conjecture. L m = f((t + m 2)mod K +1;t):1» t» T g (22) ffl The diversity order of the achieved for a N receive, K transmit antenna system, by using full modulation diversity lattice codes in the TST layering which are decoded using the ZF filter, is N K +1. The conjecture can be analytically roved for the secial case of N 2 and K = 2 (L = 2). Further we have verified the it through simulations for various values of N and K. Asinthe H-BLAST case the diversity order of the is limited by the diversity order achieved for the first layer, since obtaining the soft statistics for the first layer involves nulling out K 1 interferers in each time interval. 8 Conclusions and Future Work We roosed a class of layered STBC for the quasi-static MIMO Rayleigh fading channels. These D-BLAST lattice codes emloy full modulation diversity lattice codes as comonent SISO codes in the D-BLAST architecture. For a N receive and K transmit system with K layers it was shown that the diversity order of the obtained by the D-BLAST lattice codes with a ZF front end is NK K(K 1)=2 with the average decoding comlexity (er symbol interval) at moderate to high SNR being roughly O(K 3 ). The codes were shown to have a erformance comarable to some existing sace-time codes but with a much lower decoding comlexity. An immediate area for future work would be to develo full modulation diversity generator matrices which also otimize the coding gain.

10 10 0 Rate, 2 transmit, 2 receive antennas 10 0 Rate 8, 2 transmit, 2 receive antennas D BLAST Z F D BLAST MMSE ULC Figure 1: Rate 2, 3 transmit, 1 receive antenna D BLAST Z F D BLAST MMSE ULC Figure 2: Rate 6, 3 transmit, 3 receive antennas 10 3 D BLAST LD Figure 3: D BLAST Z F real D BLAST MMSE real D BLAST Z F cycl. D BLAST MMSE cycl. ULC Figure : References [1] H. El Gamal and M. O. Damen, Universal sace time coding, submitted to IEEE Trans. Inform. Theory, Jan [2] B. Hassibi and B. M. Hochwald, High-rate codes that are linear in sace and time, IEEE Trans. Inform. Theory, vol. 8, no. 7, , July [3] G. J. Foschini, Layered sace time architecture for wireless communication in fading environments when using multile antennas, Bell Labs Tech. J., vol. 1, no. 2,. 1 59, Autumn [] H. El Gamal and A. R. J. Hammons, A new aroach to layered sace time coding and signal rocessing, IEEE Trans. Inform. Theory, vol. 7, no. 6, , Set [5] L. Zheng and D. N. C. Tse, Diversity and multilexing: A fundamental tradeoff in multile antenna channels, submitted to IEEE Trans. Inform. Theory, Jan [6] M. O. Damen, K. Abed-Meraim, and J.-C. Belfiore, Diagonal algebraic sace time block codes, IEEE Trans. Inform. Theory, vol., no. 3, , Mar [7] M. O. Damen, A. Chkeif, and J.-C. Belfiore, Lattice code decoder for sace time codes, IEEE Commun. Letters, vol., no. 5, , May [8] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Sace time block codes from orthogonal designs, IEEE Trans. Inform. Theory, vol. 5, no. 5, , July [9] G. Caire and G. Colavole, On sace-time coding for quasi-static multile-antenna channels, submitted to IEEE Trans. Inform. Theory, 2001.

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