Outage Analysis and Optimization of a Stacked Orthogonal Space-Time Architecture and Near-Outage Codes

Transcription

1 Outage Analysis and Optimization of a Stacked Orthogonal Space-Time Architecture and Near-Outage Codes Narayan Prasad and Mahesh K Varanasi Electrical & Computer Engineering Department University of Colorado, Boulder, CO Abstract We propose the stacked orthogonal space-time architecture for the quasi-static, multi-input multi-output MIMO Rayleigh fading channel where the K transmit antennas are divided into K/ groups Each group employs the Alamouti design as the inner code with the indeterminates being uncoded symbols or symbols from some outer code Successive group interference suppression strategies based on the decorrelating and minimum mean-squared error MMSE filters are used to decode the component codes in some fixed or channel dependent order These strategies exploit the specific structure of the inner codes to yield high diversity orders while preserving the decoupling property thereby enabling the use of single-input, single-output SISO outer codes Due to the special structure of the effective channel matrix induced by the inner codes, the problems of finding exact frame error probability FEP or outage probability of the interference suppression schemes even for a fixed order of decoding are challenging Nevertheless, these analysis problems are solved for the decorrelating case and the resulting FEP and the outage probabilities are optimized over the rate-power allocations across antenna groups Alternatively, the outage probability is minimized over channel dependent ordering rules via a greedy algorithm for any given rate and power tuples We also demonstrate the intimate connection between outage probability and the achievable FEP for long framelengths When no outer codes are employed, the FEP of the optimized stacked orthogonal block code is shown, at a moderate spectral efficiency, to improve upon the FEP of uncoded V- BLAST system when the latter is detected using even the maximum likelihood rule This improvement is obtained with a much lower detection complexity In the case of the optimized stacked space-time systems with SISO outer coding, we show that much better diversity orders and frame error probabilities of about 4 db in one example are obtained relative to the codes of [1] and these improvements are also obtained with a lower decoding complexity I INTRODUCTION In order to achieve high spectral efficiencies over the frequency non-selective, quasi-static MIMO Rayleigh fading channel at a reasonable decoding complexity, [1] proposed a multi-layered space-time architecture In this scheme the transmit antennas were partitioned into two-antenna groups and the hand-crafted space-time trellis codes of [] were used as component codes to transmit information from each group The decoder was based on the group interference suppression idea originally proposed in [3] Such a combination of coding and signal processing techniques could potentially result in high spectral efficiency designs as well as savings in complexity because the original decoding problem is partitioned into several -antenna decoding problems at the receiver However, spectral efficiencies are limited by the available rates of the component codes and there is also a significant loss of diversity order, and hence the error rate performance is not satisfactory In [4] we proposed stacked orthogonal space-time block codes where codewords are constructed by vertically stacking component block codes which are taken to be orthogonal designs [5] The objective was to obtain space-time codes having This work was supported in part by NSF grant CCR and ARO grant DADD a decoding complexity no greater than that of the uncoded V- BLAST decoded using the decorrelating decision feedback detector D-DFD cf [6] and simultaneously achieve large gains at moderate to high spectral efficiencies The low decoding complexity requirement precluded the use of outer codes and we derived the exact FEP which was then optimized over rates and powers to obtain significant gains of upto 6-9 db over the uncoded V-BLAST system with the D-DFD In this paper, we consider the stacked orthogonal space-time architecture for the case where the component orthogonal designs are taken to be the -antenna Alamouti designs Our objective here is to exactly analyze and optimize the outage probability of this architecture with the receiver this time employing a much improved successive group interference suppression strategy that specifically exploits the structure of the component designs The optimizations are over rate-power allocations across groups as well as via channel dependent ordering rules that determine the order in which antenna groups are decoded The improved interference suppression strategy not only results in much higher diversity orders than are obtained in [1] but it also preserves the outer code pair-wise error probability design criterion to be a Euclidean distance measure for each group so that independent SISO trellis coded modulation TCM codes can be used as outer codes Consequently, the overall coding scheme provides significant performance improvements, lower decoding complexity and a greater flexibility in the design of space-time trellis coding schemes in that off-the-shelf SISO TCM codes and their decoding algorithms can be leveraged For very high speed applications with stringent decoding delay constraints, it may be appropriate to not use outer codes, and in this case, the problem of interest is the minimization of the FEP of the stacked orthogonal block code Such an optimization is also considered in this paper The improved successive group interference suppression idea was proposed in [7] in the context of a two-user twotransmit antenna Rayleigh fading channel where each user employs the Alamouti design as an inner code While [7] recognizes the improved diversity order, no analytical justification is provided The so-called threaded space-time architecture was proposed in [8] whose salient features are the iterative MMSE receiver based on the Turbo principle and a much improved performance over the codes of [1] However, no performance analysis is provided in [8] Moreover, as opposed to [8], the improvements obtained in this work do not involve interleavers and iterative decoding and are in fact obtained with a lower decoding complexity GLOBECOM /03/$ IEEE

2 II SYSTEM MODEL AND CODE DESCRIPTION Consider the discrete-time, block fading, complex baseband model of a wireless communication system in a flat fading environment with N receive, K transmit antennas and a coherence interval of T symbol periods given as, Y = XH + V 1 Y is the T N received matrix and X is the T K stacked Alamouti design The fading is described by the K N matrix H having independent, identically distributed iid, zeromean, unit variance, complex normal CN0, 1 elements The random matrix H stays constant for T symbol periods after which it jumps to an independent value The T N matrix V represents the additive noise at the receiver and has iid, CN0,σ elements The code matrix X, henceforth referred to as the stacked Alamouti design, has the partition into p groups or layers X = [ ] X G1,, X Gp, where the groups {G 1,G,,G p },p 1, represent a partition of the K transmitters and comprise of two transmit antennas each so that K =p The code blocks code layers {X }, 1 q p are generated by p space-time encoders operating in parallel on independent data streams The q th code block X of size T is transmitted by the transmitters in group in T symbol periods X can be expanded as X =[X T G,, q,1 XT,l ]T where l = T/ and each {X,k} 1 k l is the complex Alamouti design given by, [ ] x,k,1 x,k, X,k = x,k, x,k,1 All the indeterminates of the group, { x,k,r}, 1 r, 1 k l satisfy a power constraint, l k=1 r=1 E[ x,k,r ] w q T, where E[] denotes the expectation operator and w q > 0 Also, since by construction, the code blocks across different groups are independent, we have, E[ x,k,r x G m,s,t ]=0,m q The average received SNR per receive antenna per symbol duration is given by ρ = EtrXX Tσ, where tr denotes the trace of its matrix argument Next, starting from the model in 1 we will obtain an equivalent real representation To do so, we first assume T = or l = 1 and let Y k and V k denote the k th, 1 k N, columns of the matrices Y and V with Yk R, YI k and VR k, VI k denoting their real and imaginary parts, respectively and define the 4N 1 vectors ỹ =[Y 1 R T, Y1 I T,,YN R T, YN I T ] T and ṽ = [V1 R T, V1 I T,,VN R T, VN I T ] T Further we partition H as, H =[H T G 1,, H T G p ] T, where H, 1 q p contains the rows of H corresponding to the group and define the 4 N matrix h =[H R T, H I T ] T Then letting ĥ = vech, 1 and x = [ x T G 1,, x T G p ] T, where, x =[ x R G, q,1,1 xi G, q,1,1 xr G, q,1, xi,1, ]T, ỹ can be expanded as, ỹ = H x + ṽ, 1 The vec operator returns a column vector obtained by placing the columns of the matrix argument on top of each other where, H = [I N D 1 ĥ,, I N D 4 ĥ ], 1 q p H = [ H G1,, H Gp ], 3 and where the matrices D 1 to D 4 are given by, D 1 = D = D 3 = D 4 = For the case T =l, l > 1, in a similar manner we obtain l such linear models, ỹ k = H x k + ṽ k, 1 k l, 4 where x k =[ x k G 1,, x k G p ] and x k =[ x R,k,1,, xi,k, ]T In the following analysis we take N p III OPTIMUM DECODING In this section we derive the diversity order of the FEP achieved by the stacked Alamouti design with the maximum likelihood ML decoder In order to derive the diversity order of the FEP we consider the model in 1 and using the rank criterion developed for the optimum decoder over quasi-static MIMO channels in [], the diversity { order, denoted by D ML,is given by, D ML = N min X ˆX RankX ˆX X ˆX } Then using the structure of the code matrices it is readily verified that, min X ˆX Rank X ˆX X ˆX =and thus D ML =N We next consider the following theorem whose proof can be found in [9] Theorem 1: The matrix H in 3 has rank 4p with probability 1 when N p Now, an important consequence of Theorem 1 is that when N p, and when the indeterminates of each group are uncoded symbols from some QAM constellation, the ML decoder can be implemented using the sphere decoder [10] In the following section we will consider decoding strategies that have a lower and deterministic decoding complexity as opposed to the SD IV GROUP DECISION FEEDBACK DECODERS In this section we work with the vector model in We first consider the group decorrelating decision feedback decoder GD-DFD that is based on the group interference suppression idea proposed in [3] We then consider the group MMSE decision feedback decoder GM-DFD, where the decorrelating operation in the GD-DFD is replaced by MMSE filtering Before describing the decoders we introduce some notation We let C =[ C G1,, C Gp ]= 1 [ w 1 HG1,, w p HGp ] Further, we define the matrices, C Ḡj =[ CGj+1,, C Gp ], 1 j p 1, and ĈG =[ CGj j,, C Gp ], 1 j p GLOBECOM /03/$ IEEE

3 A Group Decorrelating Decision Feedback Decoder We assume that the groups are decoded in the increasing order of group indices ie group G 1 is decoded first and G p is the group to be decoded last We then obtain the decomposition H = Ũ L, where Ũ is a 4N 4p matrix such that ŨT Ũ = I L is a 4p 4p lower triangular matrix with positive diagonal elements and is partitioned as, L G1,G L G,G 1 L G,G 0 L = 5 L Gp,G 1 L Gp,G L Gp,G p Using Ũ we obtain zk = ŨT ỹ k = L x k + q k for 1 k l The soft statistics for the first group, denoted by { z k G 1 } l k=1,are obtained as, z k G 1 = z k G 1 = L G1,G 1 x k G 1 + q k G 1, 1 k l 6 Note that the noise vectors { q k G 1 } l k=1 have iid N 0,σ / elements The decoder for group G 1 is restricted to be a function of { z k G 1 } l k=1 The decision vectors {ˆxk G 1 } l k=1 are fed back before decoding the subsequent groups In particular, the soft statistics for the q th group are obtained as z k = z k q 1 m=1 L,G m ˆx k G m, where 1 k l and can be expanded as, z k = L, x k + q 1 m=1 L,G m x k G m ˆx k G m + q k, 7 The decoder for the group is restricted to be a function of { z k } l k=1 and obtains the decisions {ˆxk } assuming perfect feedback B Group MMSE Decision Feedback Decoder The GM-DFD operates in a manner similar to the GD-DFD except that the soft statistics for the q th group denoted by { r k } l k=1 are obtained as, q 1 r k = R 1/ F T ỹ k H Gm ˆx k G m, 8 m=1 where F denotes the group MMSE filter for and is given 1 σ by, F = I + Ĉ Ĉ T G C q R which models the covariance matrix of the effective noise ie the additive noise plus the multi antenna interference MAI, is given by, R = F T CḠq CTḠq F + σ FT F Inspecting the models in 7 and 8, we see that in the scenario when no outer codes are employed, the ML rule for each group can be trivially implemented through simple quantizers provided the matrices L, and R 1/ F T H for each group are diagonal Further with outer codes, this property enables the use of SISO encoders and decoders We will henceforth refer to this as the decoupling property In the following section we list several important properties of the stacked Alamouti design V STACKED ALAMOUTI DESIGN-PROPERTIES We now list the important properties of the stacked Alamouti design in the following theorems Using Theorem 1 along with the properties of the matrices D 1 to D 4 we can prove the following theorem which establishes the decoupling property for the GD-DFD Theorem : In the decomposition H = Ũ L, the4n 4p matrix Ũ can be expanded as, Ũ = [Ũ1,, Ũ4,, Ũ4p] 9 = [I N D 1 û 1,,I N D 4 û 1,,I N D 4 û p ] Further, the sub-matrices of L for 1 m<q p, can be expanded as, L, = L I 4,L > 0, 10 δ q,m λ q,m β q,m η q,m L,G m = λ q,m δ q,m η q,m β q,m β q,m η q,m δ q,m λ q,m η q,m β q,m λ q,m δ q,m Thus the decoupling property for the stacked Alamouti designs with the GD-DFD is now established In order to conduct any performance analysis for the stacked Alamouti design with the GD-DFD, the joint distribution of the distinct non-zero elements of the matrix L needs to be determined The following theorem proved in [9], provides the joint distribution Theorem 3: The distinct non-zero off diagonal elements of L ie the distinct elements of the matrices L,G m, 1 m<q p are iid N 0, 1/ and the squares of the distinct diagonal elements, say L, 1 q p have a Chi-squared distribution with D q =4N p + q degrees of freedom respectively and are independent of each other and of the off diagonal elements In particular the density function of L, 1 q p is given by, f L y = exp yyn p+q 1 11 N p + q 1! The following theorem proved in [9] shows that the decoupling property holds with the GM-DFD also Theorem 4: For each group where 1 q p, wehave that F T H = θ q I and R = ζ q I, for some non-zero constants θ q and ζ q VI FEP AND CODE DESIGN CRITERION FOR THE GD-DFD We define the FEP, denoted by PrE, as the probability that not all symbols transmitted in a frame are decoded correctly We first consider the case when the indeterminates of the group are uncoded symbols from a scaled square QAM constellation of size M q, where M q = rq for some positive integer r q, with average energy w q and with the minimum distance being equal to 6wq M q 1 We use the fact that the true FEP conditional FEP yielded by the GD-DFD is identical to the FEP conditional FEP of a genie aided GD-DFD which enjoys perfect feedback Thus, we consider the genie aided counterpart where, z kg = L xk + q k, 1 q p, 1 k l 1 GLOBECOM /03/$ IEEE

4 Then letting E g denote the error event for the q th group under perfect feedback and using the independence of the coefficients {L }, the FEP can be computed as [9], PrE = 1 p q=1 1 PrE g, where, PrE g =1 E L [1 P e q L T ] with, P e q L =4A q QB q 4A qq B q, 13 where B q = α q L 3w, α q = q M q 1σ and A q =1 1 Mq Although a closed for expression for PrE seems intractable, it can be numerically evaluated using the density in 11 Also it can be verified that the diversity order of the FEP, D = lim σ 0 logpre logσ equals N p +1 Similar results can be obtained for non-square QAM constellations as well but since the details are tedious we omit the derivations here In order to obtain design criteria for constructing good outer codes, note that the outer code for should be chosen to minimize PrEq g We focus on the q th group and letting x q and s q be two sequences of indeterminates of length T, we conduct a pairwise error probability PWEP analysis Using the density function of L we obtain the Chernoff bound, Prx q s q T 4σ k=1 xq k Dq 14 sq k From 14 using the worst case PWEP as the design metric, the code design criterion is to maximize the minimum Euclidean distance Hence TCM [11] designed to optimize the free Euclidean distance over the SISO AWGN channel are eminently suitable In the following section we develop the outage formulation which allows us to benchmark the performance yielded by the stacked Alamouti designs employing particular outer codes and decoded through the group decoders VII OUTAGE PROBABILITY AND OPTIMAL ORDER Outage Probability is usually discussed for the non-ergodic infinite frame length fading channels but as noted in [1] for single transmit antenna systems, it is effective in determining the performance of practical coded systems even for moderate frame lengths T 100 Here we assume that the transmission duration is long enough for the information theoretic coding arguments to apply but is sufficiently short so that the fading remains unchanged during a transmission In [1 14], the outage formulation was developed to predict the behavior of the codeword or frame error probability We now evaluate the outage probability for the specified rate and power tuples R = [R 1,,R p ] T, w = [w 1,,w p ] T and first consider the GD-DFD Now, as stated in Section VI, the true FEP conditional FEP yielded by the GD-DFD is identical to the FEP conditional FEP obtained using a genie-aided GD- DFD Hence, without loss of generality, we focus on the genie aided GD-DFD Further, we take all source messages to be equi-probable and assume that all SISO decoders employ maximum likelihood decoding Then under perfect feedback, the soft statistics for the q th group are given by 1 Recall that each x k contains the real and imaginary parts of two indeterminates { x,k,r} r=1 Further, note that for any positive L and σ, the channel in 1 is Gaussian and memoryless, so that under the imposed power constraints the instantaneous mutual information is maximized by setting the indeterminates { x,k,r} to be iid CN0,w q Denoting the resulting instantaneous mutual information in bits per channel use PCU by Iq g, we have that Iq g = log1 + wq σ L We next define the outage event, O = p q=1 O, where O = Iq g <R q, denotes the outage event for the q th group Letting O c and OG c q denote the complements of the events O and O, respectively and using the independence of {L } p k=1, we can expand the outage probability PrO as, PrO =1 PrO c =1 p 1 PrO 15 q=1 with, PrO =F Rq 1σ L and where F L represents the cumulative density function of the chi squared random variable L, 1 q p with D q degrees of freedom and is given by F L x =1 exp x D q 1 x k k=0 k!,x 0 To connect the FEP obtained with the GD-DFD to the computed outage probability, we consider any number ɛ>0and any rate tuple 0 r R, where for two p 1 vectors a and b we say a b a b when a q <b q a q b q for all 1 q p Weletγ =[L 1,1,,L p,p] T so that γ q = L, 1 q p and let X O γ, R, w, X O cγ, R, w and X O c γ q,r q,w q denote the indicator functions for the events O, O c and OG c q, respectively Then we define the sets W = {γ IR p + : X O γ, R, w =1}, W c = {γ IR p + : X O cγ, R, w =1} and WG c q = {γ IR p + : X O c γ q,r q,w q =1}, 1 q p and note that W c WG c q for all 1 q p Then the FEP achieved by the GD-DFD for the stacked Alamouti design with any p independent outer codes, can be upper bounded as, w q PrE = E γ [PrE γx O cγ, R, w] 16 + E γ [PrE γx O γ, R, w] p E γq [PrE g γ q X O c γ q,r q,w q ] + PrO q=1 For each 1 q p let q = Rq r q > 0 Now since for all γ q WG c q, Iq g r q q, we consider the channel in 1 and using the random coding upper bound computed for iid CN0,w q indeterminates, which is then averaged over the fading γ q, we can find a ˆT q > 0 such that for each T = l ˆT q, there exists a outer code X with J q = r qt codewords of length T, denoted by {x m } Jq m=1, which satisfies 1 Jq the power constraint J q m=1 xm w q T and yields E γq [PrE g γ q X O c q γ q,r q,w q ] ɛ/p Then we set T = max ˆT 1, ˆT p Note that with this choice of T,wehave l = T/ Alamouti designs in each group so that the length of the q th outer code, l = T, is greater than ˆT q Then using 16 we can guarantee the existence of outer codes satisfying the A stronger constraint x m w qt, 1 m J q can also be guaranteed GLOBECOM /03/$ IEEE

5 power constraints and yielding a FEP through the GD-DFD no greater than PrO+ɛ In order to establish the outage probability as a tight lower bound to the FEP for large framelengths, we consider a rate tuple r R and lower bound the FEP as, PrE E γ [PrE g γx O γ, R, w] 17 Then to show that the lower bound to the FEP in 17 approaches PrO as the framelength goes to, we need a strong converse This fact was assumed but not proved in [1] On the other hand [14] considers only the weak converse which says that conditioned on the outage event, the FEP is bounded away from zero However a strong converse can be obtained under certain assumptions In [9, 15], it is shown that when each group, 1 q p, employs an outer code X satisfying the power constraint, x m w q T, 1 m J q = rqt, we have lim l E γ [PrE g γx O γ, R, w] = PrO Now, the outage probability in 15, for specified values of N, p, total rate R and the operating SNR, can be optimized over the rates and powers by keeping the total rate fixed and under the constraint imposed on the powers by the specified SNR value Also, it can be verified that the diversity order of PrO equals N p+1and is identical to the diversity order of the FEP obtained with uncoded QAM symbols as indeterminates Hence, the benefits of employing outer codes appears not as an improvement in diversity order of the resulting FEP but rather as an increased energy gain The outage probability achieved through symbols from finite alphabets can also be similarly defined Letting F q denote the constellation for the q th, 1 q p, group with M q signal points enumerated as {s q,1,,s q,mq }, the outage probability can be defined as before, but where, PrO = inf p P q Pr I g q <R q, 18 where P q is the set of all M q 1 probability vectors and where Iq g is evaluated for the probability assignment p = [p 1,,p Mq ] T and the scaled constellation c q F q, where w c q = q Mq, using the formula in [16] In general j=1 pj sq,j a closed form solution to 18 seems intractable However, in the special case when each F q is a PSK constellation with s q,j =exp iπj 1 M q, 1 j M q, it can be verified using the Karush-Kuhn-Tucker KKT conditions that the solution to 18 is obtained using p = 1 M q [1,, 1] T with c q = w q With this result in hand, using an argument similar to the preceding discussion, employing the random coding upper bound, now evaluated for equi probable iid scaled PSK symbols, for any ɛ>0and any rate tuple 0 r R, we can guarantee the existence of p independent outer codes, with code symbols drawn from scaled PSK constellations, which yield a FEP through the GD-DFD no greater than PrO +ɛ For the converse, we consider any rate tuple r R and again lower bound the FEP as in 17 Note that for any positive γ q,σ, Iq g which is evaluated for the scaled PSK constellation w q F q, using the uniform probability assignment is equal to the capacity of the channel in 1 for the finite input alphabet w q F q Thus as in [13], we can invoke the strong converse for block codes which is applicable to memoryless channels with discrete inputs, and have that lim l E γ [PrE g γx O γ, R, w] = PrO For the GM-DFD we proceed as before and consider the soft statistics for the genie-aided GM-DFD given by, r kg = R 1/ F T H x k +R 1/ F T p s=q+1 H Gs x k G s +ṽ k, 19 where 1 q p and 1 k l We again take the indeterminates of of the q th group to be iid CN0,w q Although we cannot prove that this choice is optimal, nevertheless it yields a useful upper bound to the outage probability Then the resulting instantaneous mutual information in bits PCU, is given by Iq g = log1 + wqθ q ζ q and the upper bound to the outage probability is given by PrO UB =Pr p q=1 O A closed form expression for this upper bound seems intractable However it can be easily determined by Monte Carlo simulations For the special case when all transmitters transmit at the same rate with equal average powers, [3,6] proposed a channel dependent ordering rule for the D-DFD which maximized the the worst case post detection SNR under the perfect feedback assumption We formulate the optimal channel dependent ordering rule as the one which minimizes the outage probability This approach allows us to consider systems with group sizes greater than one and with possibly different rates and powers In particular we consider the GD-DFD 3 and we let π denote a permutation on the set S = {1,,,p} Then defining H π =[ H Gπ1,, H Gπp ],welet H π = Ũπ Lπ be the corresponding decomposition with L π =[L π G k,g j ] having the expansion in 10 We let L π, = L π I, 1 q p Then the outage event corresponding to the order of decoding specified by π, denoted by O π is the event, min {1 + w πq 1 q p σ Lπ R πq } < 0 A sufficient condition for a channel dependent ordering rule to yield the minimum outage probability is that for each realization H, it selects the order maximizing the left hand side of 0 among all p! orders Thus the optimal channel dependent ordering rule denoted by ˆπ 4, can be written as, ˆπ = arg max π min {1 + w πq 1 q p σ Lπ R πq } 1 Further for a given channel realization the following greedy algorithm determines ˆπ, hence avoiding an exhaustive search The proof of optimality is provided in [9] and is a simple extension of the proof in [6] 1 Initialize: a G = H =[ H G1,, H Gp ] b S = {1,,,p}, k =1 Recursion: a ˆπk = arg max { I + w q q S σ H T P H q 4Rq } 3 The ordering rule for the GM-DFD is a simple variation 4 Note that although not explicitly indicated ˆπ is a function of H and the rates and powers GLOBECOM /03/$ IEEE

6 where P q = I J q J T q J q 1 J T q with J q = G/ H 5 b G = G/ H Gˆπk, S = S/ˆπk, k = k +1 3 Stop if k>p Also the outage probability with the optimal ordering rule can be determined through Monte Carlo simulations by simply replacing H by Hˆπ the channel matrix corresponding to the optimal order and proceeding as before In figure 1 we plot the FEP achieved by an uncoded V- BLAST system using the SD, where all transmitters employ a BPSK constellation and transmit with equal average powers Also plotted is the FEP achieved by the stacked Alamouti design p = without outer codes, through the GD-DFD with optimized rates and powers, as described in Section VI Also plotted is the FEP achieved using the GD-DFD for equal rates and powers but with ordering based on the algorithm in Section VII Note that at the FEP of 10, the GD-DFD with ordering gains about 8 db on the SD even with a lower and deterministic implementation complexity In the same plot we then consider the same stacked Alamouti design but now with outer codes and plot the FEP achieved by the GD-DFD with and without ordering referred to as FEP GD-DFD 3 and FEP GD-DFD 3ord, respectively Both the groups employ the 3 state rate /3 trellis code for 8 PSK [11] and transmits with equal average powers Further, for the rate tuple R 1 = R = and equal average powers along with ordering, the outage probability achieved by the GD-DFD using iid equi-probable 8 PSK symbols and the upper bound to the outage probability achieved by the GM-DFD using iid complex normal symbols, are also plotted Finally the minimum achievable outage probability computed as per Telatar s conjecture [17], is plotted and is referred to as Out ML Note that at the outage probability of 10, the FEP of the ordered GD-DFD GD-DFD 3ord is about 45 and 1 db away from OutML and its outage probability lower bound OutGD-DFD PSKord, respectively Comparing with the code for the same parameters in [1], we see that at the FEP of 10, the ordered GD-DFD with 3 state outer codes, yields a 4 db gain Further the use SISO outer codes results in a lower decoding complexity compared to the code in [1] VIII CONCLUSIONS Stacked orthogonal space-time architecture using Alamouti designs as component inner codes was proposed The architecture along with the proposed low complexity decoders permit an extensive performance analysis in that the exact FEP and the outage probability can be computed These parameters can then be optimized over rates and powers to obtain further performance improvements Moreover, SISO TCM codes were found to be suitable outer codes and the outage probability can be used to benchmark the performance of the outer codes The optimal channel dependent ordering rule was determined Finally through simulation results it was demonstrated that significant gains are achievable through the proposed system with a low decoding complexity Outage Prob and FEP Rate 4, 4 transmit, 4 receive antennas, Framelength FEP SD FEP GD DFDopt FEP GD DFDord FEP GD DFD 3 FEP GD DFD 3ord OutGD DFD PSKord OutGM DFDUBord OutML SNR db Fig 1 FEP and Outage Probability Comparison REFERENCES [1] V Tarokh, A F Naguib, N Seshadri, and A R Calderbank, Combined array processing and space time coding, IEEE Trans Inform Theory, vol 45, no 4, pp , May 1999 [] V Tarokh, N Seshadri, and A R Calderbank, Space time codes for high data rate wireless communications: Performance criterion and code construction, IEEE Trans Inform Theory, vol 44, no, pp , Mar 1998 [3] M K Varanasi, Group detection for synchronous gaussian code-division multiple-access channels, IEEE Trans Inform Theory, vol 41, no 4, pp , July 1995 [4] N Prasad and M K Varanasi, Optimum efficiently decodable layered space time block codes, in Proc Asilomar Conf on Signals, Systems, and Computers, Monterey, CA, Nov 001 [5] V Tarokh, H Jafarkhani, and A R Calderbank, Space time block codes from orthogonal designs, IEEE Trans Inform Theory, vol 45, no 5, pp , July 1999 [6] P W Wolniansky, G J Foschini, G D Golden, and R A Valenzuela, V- BLAST: An architecture for realizing very high data rates over the richscattering wireless channel, in Proc of the ISSSE, Pisa, Italy, Sept 1998, invited [7] A F Naguib, N Seshadri, and A R Calderbank, Applications of spacetime block codes and interference suppression for high data rate wireless systems, in Proc Asilomar Conference on Signals, Systems and Computers, 1998, pp [8] H El Gamal and A R J Hammons, A new approach to layered space time coding and signal processing, IEEE Trans Inform Theory, vol 47, no 6, pp , Sept 001 [9] N Prasad and M K Varanasi, Outage analysis and optimization of stacked orthogonal space-time architecture and near-outage codes, under preparation [10] M O Damen, A Chkeif, and J-C Belfiore, Lattice code decoder for space time codes, IEEE Commun Letters, vol 4, no 5, pp , May 000 [11] G Ungerboeck, Channel coding with amplitude/phase signals, IEEE Trans Inform Theory, vol 8, no 1, pp 55 65, Jan 198 [1] R Knopp and P A Humblet, On coding for block fading channels, IEEE Trans Inform Theory, vol 46, no 1, pp , Jan 000 [13] E Malkamaki and H Leib, Coded diversity on block fading channels, IEEE Trans Inform Theory, vol 45, no, pp , Mar 1999 [14] L Zheng and D N C Tse, Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels, IEEE Trans Inform Theory, vol 49, pp , May 003 [15] N Prasad and M K Varanasi, Outage based analysis for Multiaccess/V- BLAST architecture over MIMO block Rayleigh fading channels, in Proc Allerton Conf on Comm, Control, and Comput, Monticello, IL, Oct 003, University of Illinois [16] R E Blahut, Principles and Practice of Information Theory, Addison- Wesley, Reading, MA, 1988 [17] İ E Telatar, Capacity of multi-antenna Gaussian channels, European Trans on Telecommun, vol 10, no 6, pp , Nov 1999, Originally a Bell Laboratories, Lucent Technologies, Technical Report, Oct Note that / denotes set subtraction GLOBECOM /03/$ IEEE