Design of Spherical Lattice Space–Time Codes - Semantic Scholar

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 11, NOVEMBER 2008

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Design of Spherical Lattice Space–Time Codes Narayan Prasad, Inaki Berenguer, and Xiaodong Wang, Fellow, IEEE

Abstract—In this paper, we propose a systematic procedure for designing spherical lattice (space–time) codes. By employing stochastic optimization techniques we design lattice codes which are well matched to the fading statistics as well as to the decoder used at the receiver. The decoders we consider here include the optimal albeit of highest decoding complexity maximum-likelihood (ML) decoder, the suboptimal lattice decoders, as well as the suboptimal lattice-reduction-aided (LRA) decoders having the lowest decoding complexity. For each decoder, our design methodology can be tailored to obtain low error-rate lattice codes for arbitrary fading statistics and signal-to-noise ratios (SNRs) of interest. Further, we obtain fundamental lower bounds on the error probabilities yielded by lattice and LRA decoders and characterize their asymptotic behavior. Index Terms—Lattice coder, lattice decoder, multiple-input multiple-output (MIMO) fading channels, stochastic optimization.

I. INTRODUCTION

S

PACE–TIME block code (STBC) design for wireless fading channels has been the focus of intensive research for the past several years. As a result, several powerful STBCs such as orthogonal designs [1] and linear dispersion (LD) codes [2] have been discovered. Algebraic number-theoretic tools for code design have also been employed for the independent and identically distributed (i.i.d.) Rayleigh-fading model [3], [20]. Recently, it has been shown in [4] that all STBCs proposed in the literature are, in fact, lattice codes. This reveals that the traditional STBC design where input information symbols are drawn from quadrature amplitude modulation (QAM) constellations (or equivalently pulse amplitude modulation (PAM) in the real representation) results in lattice codes with suboptimum (in terms of energy efficiency) shaping regions. Thus, there is a possibility to further improve performance by designing lattice codes with optimized shaping regions. On the other hand, a benefit of fixing input information symbols to be QAM symbols is efficient maximum-likelihood (ML) decoding via the sphere decoder [6], [7]. Unfortunately, the complexity of ML decoding can significantly increase for lattice codes with optimized shaping due to the problem of boundary control [4]. In this paper, we investigate the design of spherical lattice codes Manuscript received October 6, 2006; revised September 13, 2007. Current version published October 22, 2008. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Seattle, WA, July 2006. N. Prasad is with the NEC Labs America, Princeton, NJ 08540 USA (e-mail: [email protected]). I. Berenguer is with the Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02142 USA (e-mail: [email protected]). X. Wang is with the Electrical Engineering Department, Columbia University, New York, NY 10027 (e-mail: [email protected]). Communicated by A. J. Goldsmith, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2008.929916

for the ML decoder. Although spherical lattice codes have the highest energy efficiency, they are not provably optimal for the ML decoder. Nevertheless, one of the advantages is that the spherical shaping region is relatively easy to accommodate in the existing efficient tree search algorithms so the complexity is not adversely increased. One way to accrue the benefits of optimized shaping without the complexity increase is to employ suboptimal lattice and lattice-reduction-aided (LRA) decoders, which avoid boundary control and hence the increase in complexity. Thus, another interesting and important problem that we address here is the design of locally optimal (in terms of error rate) lattice codes for multiple-input multiple-output (MIMO) systems, where the receiver employs these suboptimal decoders. We note that no such systematic lattice code design procedure has been proposed previously and the examples given in [4] were obtained through random search. Further, even the linear dispertion (LD) codes of [5] were obtained via gradient-descent-based optimization as well as a brute-force search in a large set of realizations of an isotropic semi-unitary random matrix. One of the main problems in our quest for optimal lattice codes is that obtaining closed-form objective functions needed for deterministic optimization or other analytical techniques seem intractable, even for the simple albeit impractical i.i.d. Rayleigh-fading model. For such class of problems, a promising approach is to use the technique of stochastic approximation with gradient estimation [8]. We adopt this technique and propose several formulations that can be used to obtain low error-rate lattice codes for arbitrary fading statistics and any signal-to-noise ratio (SNR) of interest. In particular, we propose a method to obtain low error-rate spherical lattice codes for the ML decoder. We note that a gradient-descent based optimization for obtaining LD codes with ML decoding was also suggested in [5]. However, as opposed to [5] which proposed to maximize the coding gain (under a rank constraint) and relied on perturbations of the objective function to estimate the gradient, here we use the exact error probability for designing spherical lattice codes and provide an expression for the gradient which can be estimated via simulations. In our experience, the perturbation technique to estimate the gradient does not perform well and in most cases yields codes which are no better than the initial point. Also, note that unlike [5] we do not always restrict the generator matrices to be semi-unitary, which while allowing efficient parametrization, is not optimal for correlated environments. In addition, we also propose methods to obtain low error-rate spherical lattice codes for the lattice as well as LRA decoders, satisfying average or peak energy constraints, respectively. Furthermore, fundamental lower bounds on the error probabilities achievable via lattice coding and lattice or LRA decoding are derived and also analyzed in the limit of large code lengths.

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The remainder of the paper is organized as follows. Section II provides a description of the spherical lattice space–time codes along with the various decoders. Section III describes the general design algorithm and the different constraint sets considered in this paper. Section IV derives the objective functions and associated gradients for each one of the decoders considered here. Section V derives fundamental performance limits. Section VI provides code design examples and demonstrates their performance. Finall,y Section VII concludes the paper. II. SYSTEM DESCRIPTIONS Consider an -transmit -receive MIMO channel with no channel state information (CSI) at the transmitter and perfect CSI at the receiver. The wireless channel is assumed to be quasimastatic and flat fading and can be represented by an trix , which is assumed to remain fixed for . The complex-baseband model of the received signal can be expressed as (1) where denotes the average transmit power and is the transmitted signal at time , , is the denotes the i.i.d. circularly symreceived signal, and metric Gaussian noise, . The random variables in are assumed to be drawn from some continuous joint distribution. The equivalent real-valued channel model corresponding to (1) can be written as (2) where to a codebook

is a codeword belonging

. In the Euclidean space, the closest lattice associated with is defined by

point quantizer

if

(6)

where ties are broken arbitrarily. The Voronoi cell of is the set of points in closest to the zero codeword, i.e., . The Voronoi cell associated with each is a shift of by and is denoted by . The ( -dimensional) volume of the Voronoi cell is given by [9]. . A finite set of points in the -dimensional Let translated lattice can be used as codewords of a codebook and for a rate bits per channel use, the codesuch points. Here, we specify a book will contain , with lattice code using the three-tuple , , and . is referred to as are called the coordinate vectors. the translation vector and For a given and , the code will be referred to as a spherical lattice code if the coordinate vectors are a solution to (7) For ease in exposition, in Sections III–VI, we first consider . The changes needed for the general case are the case listed in Section IV-D. Moreover, in this work we will consider has only nondegenerate lattices, i.e., we will assume that full column rank with probability one. B. Lattice Decoders In lattice decoding, the receiver assumes that any point in the infinite lattice could have been transmitted. For a given lattice, the naive lattice decoder determines

with

(8)

(3) The goal of this paper is to design a lattice codebook satisfying either the average energy constraint (4) or the peak energy constraint (5) such that the code yields a low error probability. Note that the bit per channel use. rate of the code is A. Lattice Space–Time Codes is defined by a set of basis An -dimensional lattice (column) vectors in . The lattice is composed of all integral combinations of the basis vectors, i.e., , where and

An interesting property of lattice codes with naive lattice decoder is that owing to the lattice symmetry (geometric uniformity), the error probability is invariant to conditioning on a particular transmitted lattice codeword and only depends on the lattice generator. However, from the energy efficiency point of view, selecting the codewords with minimum norm minimizes the average transmit power and hence spherical lattice codes are optimal for the naive lattice decoder. It has been shown in [4] that a minimum mean-square error (MMSE) front-end can dramatically improve the performance of the lattice decoding algorithms in MIMO systems. This minimum mean-square error (MMSE) lattice decoder determines an upper triangular matrix from the Cholesky decomposition of and an matrix and rethe matrix turns (9) Consider the statistics of the equivalent system model for the , we have that MMSE lattice decoder. Defining

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Assuming to be zero mean with and is independent of , it can be using the fact that . Although contains a signal shown that [4] dependent term, assuming is very effective.1 Henceforth, we will make this assumption and then the error probability yielded by the MMSE lattice decoder is identical to that of a naive lattice decoder operating on (10) with being an independent additive white Gaussian noise (AWGN) vector so that spherical lattice codes are optimal for the MMSE lattice decoder as well. In the sequel, we will use the phrase “assumed . model in (10)” to mean the model in (10) where C. Lattice-Reduction-Aided (LRA) Decoders The LRA-linear receiver2 described in [11] is a low-complexity detector that offers good performance. The idea behind LRA-linear receivers is to assume that the signal was trans, and mitted in a reduced basis, i.e., to equalize in the new basis, which is more robust against noise enhancement, and then to return the decoded symbol to the original basis. That is (11) quantizes its input vector componentwhere the quantizer wise to the nearest integer and is a unimodular matrix3 which is obtained via the Lenstra–Lenstra–Lovász (LLL) reduction . Recently, it has also been observed in [14] that [12] of yields better performance. In particular this reducing decoder which we refer to as type-2 LRA-linear decoder works be the reduced version of as follows. Let , typically obtained through LLL reduction. Then the decision vector is obtained as (12) It is shown in Section IV-B that spherical lattice codes are optimal for LRA-linear decoders. The performance of both LRAlinear decoders can also be further improved by MMSE preproand assume cessing [10]. As before, we obtain . the resulting model is of the form in (10) where The LRA-linear decoders can then be applied to (10), where is the effective generator matrix. In the sequel, we will design spherical lattice codes for MMSE-LRA-linear decoders using the assumed model (10). Finally, note that the LRA-linear decoders described above are extensions of the well-known ZF and LMMSE detectors. To further improve performance at the expense of a slight increase in complexity, the LRA-successive interference cancellation (LRA-SIC) decoders, which are extensions of the ZF-SIC and MMSE-SIC detectors, can also be derived. It can be shown that spherical lattice codes are optimal for LRA-SIC decoders as well.

D. Maximum-Likelihood (ML) Decoder The ML decoder is the optimal receiver in terms of error probability. The ML decoding rule is given by (13) Note that the decoding regions are not identical due to the boundary of the codebook and in fact some are not bounded. As a consequence, for ML decoding we cannot claim that spherical lattice codes are optimal over arbitrary fading channels and other shaping regions may in fact be better suited. However, when the shaping region of a lattice code is a sphere—as considered in this paper—boundary control can be (relatively) efficiently incorporated. A low-complexity implementation of ML decoding for spherical lattice (space–time) codes that efficiently incorporates boundary control is proposed in [15]. The design of spherical lattice codes (albeit with the translation vector set to zero) over interleaved single-input singleoutput (SISO) Rayleigh-fading channels has been treated in [16] using tools from algebraic number theory. However, the main focus of the research on lattice code design has been the design of LD codes, see for instance [17] for the interleaved SISO case and [2] for the block-fading MIMO case. LD codes are lattice codes with suboptimum (in terms of energy efficiency) shaping regions, which permit an efficient ML decoding via the sphere decoder. In particular, LD codes are lattice codes where a set of integer coordinate vectors is fixed a priori (typically each element of belongs to a PAM constellation) and the optimization is conducted over only .4 III. DESIGN ALGORITHM We propose to use a stochastic gradient descent algorithm to minimize the probability of error (or its bounds) over a feasible set of generator matrices. We provide a brief description of the algorithm in its general form and elaborate on particular designs in Section IV. Let denote a random vector defined over some sample space. Also let denote the vector of parameters lying in a feasible set . Our objective is to minover using “noisy” but unbiased imize estimates of .5 Then the stochastic gradient descent algorithm works as follows. Let deth note the vector of parameters at the th step. Then the iteration proceeds as follows. 1. Draw samples . 2. Obtain unbiased gradient estimate:

3. Update:

.

1It will be demonstrated subsequently via simulation results that this approximation is particularly effective for spherical lattice codes.

is (usually) chosen as the harmonic The step-size sequence , where is a positive scalar. resembles series

2As observed in [10], the receiver structure for this decoder is not linear but since it is the extension of the common zero-forcing (ZF) linear receiver, it is referred to as LRA-linear receiver. 3A matrix P is known as a unimodular matrix if P and P have integer 6 . entries. Such a matrix satisfies jP j

set

= 1

4Since

Gz

f

u = 0. 5The

=

the commonly assumed set of fz g is simplex i.e., z 0, the is also simplex for any G so that the optimal translation vector is

g

exchange of derivative and the expectation is required for this method.

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a projection operator6 in that it finds a point in the feasible set close to the input argument when the latter falls outside the feasible set. For the problem at hand, several objective functions which are defined in the sequel can be used since their gradients are derived in the required form, i.e., their unbiased estimates can be obtained via simulations. In the subsequent subsections, we will consider various choices for the feasible set and the associated . A. Average Energy Constraint In order to satisfy the average energy constraint in (4), the feasible set of generator matrices is

obtained after taking a step along the negative gradient direction. For that , we recompute a good spherical code using the iterative method7 and determine using (15) and set . As will be revealed in the simulation results, all of our suboptimal methods (for the lattice, LRA and ML decoders, respectively) work well in practice. A special case where an unconstrained optimization is possible for all three types of decoders is when we restrict to be a scaled real-orthogonal (real-unitary) matrix. In are the optimal (energy this case, we see that if are opminimizing) set for generator , then timal for any real-orthogonal . Thus, the optimal scaling is and8 the optimization

(14) We first consider the lattice and LRA decoders. The error probabilities obtained with these decoders for a given generator are invariant to the choice of coordinate vectors as well as the translation vector. As a result, we can minimize the error . Unfortunately, it seems intractable probability over the set . In fact, for a given an efficient to parameterize the set way of obtaining an optimal spherical code, i.e., an optimal set of codewords (which minimize the average energy) is also not known. As a consequence, we adopt the following suboptimum and and define as approach. We set follows. For a given input , we determine a “good” spherical code (having low average energy) using an iterative technique provided in Appendix B, which converges to a fixed point very fast in a few iterations. With the codewords so determined we compute a scaling factor

is over the set . Note that in this case, the optimization even for the ML decoder is over only the generator matrix since the set of coordinate vectors remains fixed. Then, since the orthogonal group is a differentiable manifold, we can express as a differentiable function parameters.9 The main benefit is that we can now implement an unconstrained stochastic gradient-descent and the iterative method to determine a “good” spherical code needs to be implemented only once. Moreover, the best known generators that have been obtained for ML decoder and i.i.d. Rayleigh-fading channels using analytical techniques [3], [23] are all unitary so at least for this channel model, restricting the set of generator matrices to be unitary is reasonable. For the next two constraint sets, we only consider the lattice and LRA decoders. The reason again is that unlike the ML decoder, the error probabilities yielded by these two decoders for a given generator matrix are invariant to the choice of codewords, which allows us to obtain unconstrained gradient descent algorithms. B. Peak Energy Constraint

(15) If then and we let ; otherwise ), we set . (i.e., if On the other hand, for the ML decoder the error probability in addition to also depends on the choice of the coordinate vectors. The role of the translation vector is limited to transmit enany translation vector ergy reduction and for a given results in the same error probability. Consequently, without loss of optimality, we can take the (negative) centroid to be the translation vector and define the feasible set of spherical lattice codes be the curas shown in (16) at the bottom of the page. Let rent set of coordinate vectors and let be the generator matrix 6Usually this method is employed to solve unconstrained problems. In the constrained version there is no universal rule or method to enforce the constraints.

We now consider the design under a peak energy constraint, where all codewords must satisfy . denote an -dimensional sphere centered at Letting , we can leverage a result stated in the origin and of radius [4], which says that if , then there exists a translation vector and coordinate vectors such that , . Moreover, since the error probability and its bounds for a given generator monotonically decrease in , without loss of optimality we consider the set of generators denoted by (17)

0

~ z~ . G fz g; u) for I via the

~= 2 that the iterative method always yields u our implementation, we first obtained a good set ( iterative method. 7Note 8In

9For brevity, we do not provide the parametrization here but refer the reader to [22].

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The nice feature of the set is that any can be expressed as a differentiable function of a parameter vector . To see this, let be the QR decomposition of , where is unitary and is lower triangular with positive diagonal elements. As mentioned in the previous section, is a differentiable function of parameters. Setting and

we see that is a differentiable function of paand all its strictly lower triangular elements). rameters (i.e., paCollecting all the is a differenrameters into a vector , we have that tiable function of . Thus, we can implement an unconstrained stochastic gradient-descent. A spherical code for the optimized generator matrix is determined by generating a large number of translation vectors using the method outlined in Appendix B and codewords satisfy the peak enpicking any one for which ergy constraint.

Hence, at sufficiently high rates, using the fact that the error monotoniprobability and its bounds for a given generator cally decrease in , we consider the constraint set defined as

(19) and the advantage again is that Clearly, we have an unconstrained gradient descent algorithm. A “good” spherical code is determined only for the final (optimized) generator and scaled to satisfy the energy constraint in (4). The following table provides the objective functions and the gradients that are derived in Section IV. Only the expressions for the naive versions (i.e., without MMSE processing) of the lattice and LRA decoders are listed. As a consequence of our earlier assumption discussed in Section II-B [cf. (10)], code designs for the MMSE lattice and LRA decoders are obtained by designing spherical lattice codes for their naive counterparts over the assumed model in (10) and the corresponding bounds and gradients employed are obtained after replacing in the expressions given in the following table by .

C. Average Energy Constraint: Continuous Approximation We invoke a continuous approximation from [24] (also used in [25]) to obtain a parameterization for the set in (14) which is accurate for high rates. The main idea is that at high rates the random vector uniformly distributed over the set of codewords of a spherical lattice code can be (approximately) considered as a spherically uniform random vector. In particular, consider the high-rate regime and any and let denote, respectively, its optimal (energy minimizing) coordinate vectors and translation vector. Note that the codewords must lie within or on a sphere cenand denoted by for some radius . tered at Then at high rates, using the continuous approximation, we and we can get that approximate the probability mass function (over the set of codewords) by the probability density function of a spherically uniform random vector, which results in the Riemann integral approximation

(18) term vanishes as . Using a result on the where the average energy of a spherically uniform random vector [24], we have that the right-hand side (RHS) of (18) equals , so that the average power constraint (4) implies that .10 Thus, invoking the continuous approximation, we see that any should satisfy

IV. OBJECTIVE FUNCTIONS AND GRADIENTS In the sequel, we consider each decoder and derive its error probability (or bounds on its error probability) and obtain the corresponding gradient in the form required by the design algorithm. A. Lattice Decoders In Sections V and VI, we derive three objective functions along with their gradients for the lattice decoders, which can be employed in the gradient-descent-based design algorithm. The first one is the exact error probability whereas the second and third ones are an upper and a lower bound, respectively. The exact error probability is computationally most intensive whereas the lower bound has the least computational complexity. The upper bound (in particular, its improved version) provides a good balance between computational complexity and accuracy with respect to the actual error probability. 1) Exact Error Probability: Let us start with the exact error probability of the naive lattice decoder. As mentioned earlier, for the naive lattice decoder, without loss of generality we can is the transmitted coordinate vector. Then assume that denote the error probability (averaged over the letting channel realizations) of a spherical lattice code with generator , from the decision rule in (8) we see that

which leads to 10Recall

that n

= 2MT .

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Unfortunately, as noted in [25], the integral in (20) is in general impossible to obtain in a closed form. However, its derivative can be estimated. To show this, first we invoke a result from [26] (also used in [27]), which says that for a fixed generator and elements, the a random vector having i.i.d. uniform is uniformly distributed over , vector is defined in (6). As a result where

design algorithm we would like to be able to estimate the gradient of the upper bound. Our next result allows us to do just that. Lemma 2: Let be a relevant coordinate vector for the lattice . Then small enough such that generated by , remains a relevant coordinate vector for the . lattice generated by Proof: Using Proposition 2 from [29] we note that if and only if

(21) (25)

where we have used the fact that . Comparing (21) and (20), we see that

Further, since the lattice is a sphere packing we must have that for each

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(22) Let , denote the maximum and minimum sinis a difgular values of the matrix argument and suppose ferentiable function of some vector parameter . Henceforth, with some abuse of notation, while computing the partial derivato tive with respect to any element of , we will use denote the matrix-valued function of with the remaining elements of held fixed. We offer the following lemma. The proof is given in Appendix A. Lemma 1: For any element

for some . Next, to check if is relevant for , using (25) we see that it is sufficient to check if no coordinate vector lies in the set

Next, it can be seen that

of , we have

(23) denotes the indicator function and where set of coordinate vectors such that (bounded) fundamental parallelotope

is any finite covers the .

The derivative of (22) with respect to can now be computed by first exchanging it with expectation over (which is justified by the bounded convergence theorem [28]) and then using (23). 2) Union Upper Bound: Next we consider the union upper bound on the conditional error probability, given by [9]

(24) where denotes the standard Q-function and is the set such that of all relevant coordinate vectors for a given determine all the facets of . An algorithm to determine all such coordinate vectors is given in [29]. Now, for an -dimensional lattice generator, it is known that the max[29]. For our imum number of relevant vectors is

Using the continuity of in , we can define a small interval , , . where this ratio is bounded above by a positive constant , Letting . Thus, we have obtained a large enough we set of coordinate vectors and small enough , if is greater than , then remains relevant for . Next, using the continuity of in we can show that for any but finite set such that

where , are positive terms that go to zero as . and are finite, we can now conclude that Since the set , where is small enough, all the as well. coordinate vectors in remain relevant for Now, suppose for a given the number of relevant cois equal to the upper bound . ordinate vectors in Then, as a consequence of the Lemma 2, we can conclude that , the set contains all the relevant coordinate . This observation allows us to take the vectors of

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derivative of the upper bound (conditioned on to be a fixed set and we have that

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) assuming

(27) In our case, the fading matrix is drawn from a continuous distrialways bution and the generator has no structure so that has the maximum number of relevant vectors. Thus, the unconditional upper bound and its derivative can be obtained after averaging (24) and (27) over , respectively.11 A better upper bound, referred to as the improved upper bound, along with its derivative can be obtained as

(28) where is called the outage set and is its complement. Equation (28) is obtained by . The improved taking the upper bound to be one when bound is dramatically tighter for the naive decoder since the conditional upper bound in (24) almost always exceeds one when . Further, since the set is independent of the derivative of (28) is readily obtained using (27). The intuition behind defining the outage set in this manner rises from the asymptotic analysis of a fundamental performance limit which will be presented in Section V. 3) Lower Bound: Here we derive a lower bound and obtain its derivative in the desired form. Suppose the kissing number is , i.e., there are exactly two of lattice generated by shortest (nonzero) vectors in the lattice. Letting and denote these vectors, we must have that and that and are relevant.12 Then since the half-spaces and do not overlap, we can obtain a conditional lower bound given by

Again, since the fading matrix is drawn from a continuous distrialways has the bution and the generator has no structure, minimum kissing number. Thus, the unconditional lower bound and its derivative can be obtained after averaging (29) and (30) over , respectively. The corresponding bounds and gradients employed in the code design over the assumed model in (10) are obtained after replacing by . Thus, for the improved upper bound we take . As an aside, the continuous approximation made in Section III-C can be used to motivate our assumption in (10) (albeit in a nonrigorous fashion) as follows. From the decision rule in (9), we can see that the (true) error probability yielded by the MMSE lattice decoder conditioned on the channel realization and the transmitted codeword is equal to

Letting , we see that the error probability conditioned on the channel realization equals

For a spherical lattice code at high rates, invoking the continuous approximation can be approximated by a spherically uniform ), random vector. Then for large dimensions (i.e., as indeed behaves like a white Gaussian vector [24]. Consequently, behaves like a white Gaussian vector which suffices to justify our assumption in (10). B. LRA Decoder Here, we consider the LRA decoders and obtain the gradients of their exact error probabilities in a form that can be estimated via simulations. We first consider the type-2 LRA-linear be the decoder without the MMSE preprocessing. Let transmitted codeword. From the decision rule in (12) it is seen . Since that an error event occurs if , equivalently, we have that an error occurs if which itself is identical to the event where . Thus the error probability is given by

(29) Also, using arguments similar to those used before we can show that the derivative of (29) is equal to

(30) 11The exchange of derivative and the expectation over justified.

H can be rigorously

kissing number can be no less than 2. Further, the fact that both are relevant follows from the necessary and sufficient condition given in (25). 12The

z ;zz

z

=0

(31) Clearly, the error probability depends only on the generator maand for a given generator , it is monotonically detrix . Thus, spherical lattice codes are optimal for creasing in LRA-linear decoders. We now obtain the gradient of the error probability in the desired form and toward this end we expand the error probability in (31) as shown in (32) at the top of the following page, where follows after noting that since is unimodular. Under some technical conditions which are satisfied with probability one, when the random variables in are drawn from a continuous distribution and the matrix has no structure, we can show that the gradient of is then given by (33) at

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(32)

(33)

the top of the page. Clearly, (33) is in a form that can be estimated via simulations. The derivatives for the other LRA-linear decoders as well as the LRA-SIC decoders can be similarly derived. For the latter decoders, we need to invoke a key property of the SIC detector which is that the (joint) error event of the detector is identical to that of its genie-aided (perfect-feedback) counterpart [30]. C. ML Decoder We consider the exact error probability of the ML decoder for a specified spherical lattice code , where and write the channel model as

such that , remains the enough unique solution to , with . The iterative method of Appendix B, which is employed in this paper to compute a “good” spherical code for a given generator , almost always yields a pair , where , such that possesses the aforementioned uniqueness property. As a to be fixed or invariant to result, we can consider the set while computing the derivative and we have that

(34)

(37)

Then the exact (block) error probability is given by (35) is the indicator function of the block error event. We where can expand (35) as

which using the chain rule yields (38) at the bottom of the page, where the fact that is proved in [31] in the context of ML decoding of LD codes. Hence, is given by the second term on the RHS of (38), which can be estimated via simulations. Also note that when is restricted to be a scaled unitary matrix, the set of coordinate vectors remains fixed at its initial value across all iterations and the condition needed for the derivative in (38) to be valid is clearly satisfied.

(36) D. Asymmetric MIMO Systems The error probability clearly depends on the generator matrix as well as the (discrete) integer-valued coordinate vectors so obtaining a derivative seems impossible. However, an important exception is when the set is the unique solution to . In this case, using standard continuity arguments it is easy to verify the existence of a small

Recall that for convenience so far we assumed . Next we consider the general MIMO systems. Note that in order to enable naive lattice decoding, LRA decoding (without MMSE processing) and efficient ML decoding, we need for the effective to be one-to-one. On the other hand, to generator matrix obtain high spectral efficiencies with these decoders we need

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to keep the symbol rate13 as high as possible. To balance these where . requirements, we assume In the following, we enumerate the changes that have to be made in the preceding sections to accommodate the general case. We first consider the naive lattice decoder. , where 1. We first obtain the QR decomposition and is lower triangular, and transform the to obtain an equivalent model model in (2) by . Naive lattice decoding is performed on . All the objective this model and note that functions and their gradients can be obtained by using this model where is the effective generator matrix such that . , , 2. With regards to the constraints sets defines the -dimensional volume we note that of the Voronoi region and we have to use , as the constraining spheres. Further, to implement the LRA decoders in the general case, we just have to replace the inverses involved by the corresponding pseudo-inverses and proceed. The ML decoder requires no changes. For the MMSE lattice and LRA decoders we simply use the changes suggested above over the assumed model (10), i.e., we by and choose . Interestingly, replace assuming (10) to be the equivalent model after MMSE proand nonsingular, we can cessing, we see that since is generator and still enjoy efficient decoding. In fact, use an this observation was used in [32] to obtain an efficient lattice decoder with near-optimal performance for under-determined systems. Moreover, the Gaussian approximation involved in obtaining (10) always seems to be the most accurate . However, we have observed that when when we choose optimizing a generator that is , for the naive decoder over the assumed model yields the best results over the actual model with the MMSE lattice or LRA decoders.

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Theorem 1: A lower bound on the error probabilities yielded by the naive lattice decoder and the LRA decoders for any genand is given by erator (39)

(40) follows after replacing The lower bound for any in the numerator of (40) by . In the case , the is given by (41) at the bottom lower bound for any of the page, with (42) The corresponding bound for any follows after in the numerator of by . replacing Proof: We first consider the naive lattice decoder. For a be any matrix in with given rate , let being its QR decomposition. For a given channel realization , we note that its error probability satisfies (43) which follows after using a change of variable . We next consider the RHS of (43). Note that generates a -dimensional lattice with

To obtain the lower bound, we note that and use Shannon’s classical idea developed for lattices in [25], which yields

V. FUNDAMENTAL PERFORMANCE LIMITS In this section, we consider a general MIMO system and let and . A drawback of our gradient descent approach is that it can only provide us with locally optimal solutions. A natural question then is to determine the scope for further improvement or in other words, the gap in performance with respect to the globally optimal solution. To address this question, we consider lattice as well as LRA decoders and obtain fundamental performance limits in the form of firm lower bounds on the error probability achieved by any lattice code (not necessarily spherical) whose generator lies or in the set . We offer the foleither in the set lowing theorem.

(44) is a -dimensional sphere, centered at the where origin and having the same volume as . Then, letting , we get that

13To conform to the usual meaning, we define symbol rate as the ratio of the size of the coordinate vector and T .

(45)

(41)

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Next consider the type-2 LRA-linear decoder (without MMSE processing) and note that the error probability satisfies

(46) with

(47)

The volume of

is equal to the volume of the

fundamental parallelotope of

which, in turn, is equal to

. Further, as is unimodular, so that the lower bound in the RHS of (44) is also a lower bound for the type-2 LRA-linear decoder. Similarly, it can be shown that the type-2 LRA-SIC decoder as well as the original LRA-linear and LRA-SIC decoders (each without MMSE processing) also permit the same lower bound. Clearly, . Using the fact the lower bound in (44) is decreasing in we see that when (48) so that a lower bound valid is given by (39). , we first parse as Next, when , where each and note that is nonsingular. Using Hadamard’s inequality [33] we get that

The following theorem determines the fundamental lower bound on the performance of the naive lattice decoder as well in the limit of large as the LRA decoders, over the set code lengths, i.e., . The fundamental lower bound also represents the best achievable performance in the limit of large code length for the naive lattice decoder. For convenience, we and let denote the generator matrix of a code assume of length and define to be the set of such codes satisfying (19). Theorem 2: The error probabilities of any sequence of lat, tice codes of unbounded length with generators obtained via either the naive lattice decoder , must satisfy or the LRA decoders, denoted by

Further, for the naive lattice decoder there exists a sequence for which

Proof: It is proved in [4] the existence of a sequence of lat, such that tice codes with generators their error probabilities obtained with the naive lattice decoder . satisfy Thus, to prove the theorem we prove the converse for the larger . First note that sets

(49) where is given by (3). Using the concavity of set of positive definite matrices, we note that

over the

(53) where are i.i.d (independent of ). Then using the weak law of large numbers in (53) along with (40) (with replaced by ), it is seen that

(50)

, we recall that since

Letting

, (54)

, and we have

(51) is given by (52) at the so that a lower bound valid bottom of the page. Then, using the structure of and the fact that circularly symmetric complex Gaussian vectors are entropy maximizers [34], we can show that the lower bound in (52) is equal to (41).

Using (54) with the dominated convergence theorem [28] and is a set of measure the fact that zero, we can infer that the lower bound in the limit becomes . In the case

, the limiting value is equal to (55)

(52)

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PRASAD et al.: DESIGN OF SPHERICAL LATTICE SPACE–TIME CODES

Fig. 1. Fundamental lower bound and outage probability. T

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= M = N = 2, R = 4.

On the other hand, the lower bounds for the sets , , and the (assumed) model in (10) with can be derived by simply replacing with in the corresponding bounds derived above. The limiting value of both the lower bounds can be readily verified to be . In Fig. 1, we consider a two transmit, two receive antenna MIMO system experiencing i.i.d. Rayleigh fading and transmitbits per channel use. In the figure, we plot the ting at rate and the naive lattice or LRA lower bound for the set and , redecoders, computed using Theorem 2 for spectively, along with the corresponding large code-length limit (i.e., the outage probability). Also plotted are the lower bounds and the outage probability for the naive decoder over the assumed model in (10). As seen from the plots, the lower bounds nearly coincide with their respective outage curves. for Moreover, since the lower bounds are fairly close (within 0.7 dB ) to the outage curves, we can also infer that the even for error probability of the optimal code can at best improve marginally upon the outage probability. Thus, in the event of an outage, a block error is very likely for any spherical code whose generor . This observation allowed ator lies in the set us to obtain the improved upper bounds given in Section IV-A2. We also note that although a fundamental lower bound (and hence its large code-length limit) has not been derived for the in (14), the lower bound derived for and the set outage probability are effective indicators of the best achievable error probability over this set, particularly at high rates.

VI. NUMERICAL RESULTS In this section, we provide several examples to show the performance of the new lattice space–time codes obtained by the design procedure described in the preceding section. We will see that the codes optimized for a particular SNR work well over a wide range of SNR values of interest. 1) Lattice Space–Time Code Design With ML Decoding: with i.i.d. Rayleigh fading Consider first bits per channel use (i.e., a codebook with 256 and ) with ML decoding. The codewords, and dimension block error rate (BLER) performance of the spherical lattice 15 dB) is code (optimized for ML decoding at an SNR of plotted in Fig. 2. The optimized generator matrix is given by the matrix (56) at the bottom of the following page. For comparison, we also show the block error rate performance using the spherical lattice space–time codebook obtained with the eight-dimensional generator matrix given in [4], denoted as GCD. We also compare the optimized code with the performance of the spherical lattice codes which are obtained from the generator matrices given in [3], [37] and are denoted by Golden and LG in the legend, respectively. The 2X2 codes in [3], [37] are isomorphic to the Golden code presented in [19], which was shown in [18] to be a perfect code having several desirable properties. The plotted results are obtained after averaging as many channel realizations as to obtain 150 errors. The outage probability curve obtained with i.i.d. Gaussian inputs for 4 bits/s/Hz and is also plotted in the same

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Fig. 2. Block error rate (BLER) performance of lattice codes under ML decoding. T

figure. It is seen that our optimized code offers similar performance compared to the best known lattice space–time codes, and the performance is close to the outage probability curve. Note that our technique, which is a gradient-descent-based optimization of the exact error probability (in the case of ML decoder), yields locally optimal solutions and the code employed here was obtained after using several initial points.14 Consequently, this example strongly suggests that the codes in [3], [37] are very close to optimal with respect to the error probability over the i.i.d. Rayleigh fading channel. Next, we compare the peak-to-mean envelope power ratio (PMEPR), which is a practically important factor for selection of space–time block codes. We use two definitions: the one in [20] as well as the per-antenna PMEPR definition from [21]. Reference [20] defines PMEPR as the ratio of the maximum 14Surprisingly, although different initial points yield different spherical codes, the obtained codes have very similar performance.

= M = N = 2, R = 4.

codeword energy and the average codeword energy. Using this definition, we obtain that our optimized spherical code as well as the two spherical codes from [3], [37] have identical , whereas the GCD spherical code has a PMEPR of . On the other hand, using the slightly higher PMEPR of per-antenna PMEPR definition of [21], we see that our code as the PMEPRs of transmitters 1 and 2, yields respectively. The corresponding pairs for the LG code and the and , respecGolden code are tively, whereas the GCD spherical code yields . From these numbers, we see that our code enjoys a small advantage over the competing codes. and . We select the Now consider rate as bits per channel use (i.e., a codebook with 4096 ). Fig. 3 illustrates the block codewords, and dimension error rate performance of the optimized code. For comparison we show the performance of a lattice space–time code obtained

(56)

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Fig. 3. BLER performance of lattice codes under ML decoding.

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M = N = 2, T = 3, R = 4.

using Construction A15 [9], [4], as well as the outage probability curve. It is seen that the new lattice code exhibits slightly better performance than the code obtained via Construction A. Moreover, there is a gap of less than 1.5 dB between the performance of our code and the outage probability at error rate. 2) Lattice Space–Time Code Design With Lattice Decoders: In Fig. 4, we consider a MIMO system with and , experiencing i.i.d. Rayleigh fading. The spherical lattice code used was obtained after optimizing the improved upper bound for the naive decoder and the assumed model (10) using the method in Section III-A, at an SNR of 18.5 dB. We plot the BLER, the upper bound (24) (with replaced by ) as well as the improved upper bound obtained for the naive lattice decoder and the assumed model (10). Also plotted are the BLER of this code obtained with the MMSE lattice decoder (over the original or actual model in (2)) along with the outage probability. The outage probability is computed for i.i.d. Gaussian inputs and as shown in Section V, it is also the large code-length limit of the fundamental lower bound on the error probability of any code—yielded by the naive decoder over the assumed model or in in (10)—whose generator lies either in the set . From the figure we see that our assumption in (10) is indeed very accurate since the BLER yielded by the MMSE lattice decoder over the actual model and the naive decoder over the as15The authors would like to thank M. O. Damen for providing the lattice generator of Construction A.

sumed model are close. Thus, using the improved upper bound derived under that assumption for optimization is well justified. Also, using the unconstrained formulation in Section III-C with the same objective function yielded a code having similar performance. On the other hand, using the exact error probability as the objective function proved computationally too intensive. Finally, we note that the improved upper bound for the naive decoder over the actual model (2) was found to be dramatically tighter than the original one in this example. In Figs. 5 and 6, we again consider a system with and . We plot the BLER of the optimized code over correlated Rayleigh fading and the BLER of the optimized code over correlated Rician fading, respectively. For each case, the spherical lattice code was obtained after optimizing the improved upper bound computed for the naive decoder over the assumed model in (10). The plotted BLERs are however the true BLERs obtained for the optimized codes with the MMSE lattice decoder. The correlated Rayleigh-fading channel matrix is , where has i.i.d. elmodeled as ements and . For the correlated Rician case we used the model in [36] with the parameters , , , , and . To demonstrate the gains achieved by the optimization procedure, in each case we plot the BLERs yielded by the MMSE lattice decoder for spherical lattice codes constructed from the generator matrices given in [3], [37], respectively. Note that by matching the code to the channel statistics, we get a gain of about 1 dB

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Fig. 4. BLER performance of lattice codes under MMSE lattice decoding.

M = N = T = 2, R = 4.

Fig. 5. BLER performance of lattice codes under MMSE lattice decoding in correlated Rayleigh-fading channels.

M = N = T = 2, R = 4.

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Fig. 6. BLER performance of lattice codes under MMSE lattice decoding in correlated Rician-fading channels.

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M = N = T = 2, R = 4.

Fig. 7. BLER performance of lattice codes under MMSE lattice decoding in i.i.d. and correlated Rayleigh-fading channels.

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M = 4, N = T = 2, R = 4.

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2, R = 4.

Fig. 8. BLER performance of lattice codes under type-2 LRA-linear decoding with MMSE processing, in correlated Rician-fading channels.

over the spherical codes from [3], [37]. Moreover, the codes optimized for correlated scenarios also have PMEPRs which are at least as good as the competing codes. In particular, the spherical code optimized for the correlated Rayleigh fading yields as per the definition in [20] and per-ana PMEPR of . Similarly, the spherical code tenna PMEPRs of optimized for the correlated Rician fading yields a PMEPR of and per-antenna PMEPRs of . In Fig. 7, we consider a MIMO system with transmit receive antennas, and bits per channel and use, experiencing correlated Rayleigh fading. The correlation parameters are set to model the Scenario C (urban). For comparison, along with the BLER yielded by our optimized code, we plot the BLER yielded by the spherical lattice code whose generator matrix (of the same dimensions) was optimized for ML decoding and i.i.d. Rayleigh fading using analytical tools in [37]. We have also included the BLER yielded by the spherical code from [37] over i.i.d. Rayleigh fading as well as that of our spherical code which was optimized for i.i.d. Rayleigh fading. As done in the previous example, in each case our spherical lattice code was obtained after optimizing the improved upper bound for the naive decoder over the assumed model in (10). The plotted BLERs are however the true BLERs obtained with the MMSE lattice decoder. Note that while the spherical code from [37] is superior over i.i.d. Rayleigh fading, we get a huge gain over the considered correlated scenario. 3) Lattice Space–Time Code Design With LRA Decoders: Fi, , nally, in Fig. 8 we reconsider the case of

M = 4, N = T =

, with correlated Rician fading, where the fading reand alizations were generated using the model in [36] with the same parameters as in Fig. 6. The receiver employs the type-2 LRAlinear decoder with MMSE processing. We plot the BLERs of a spherical lattice code with a randomly chosen isotropic semiunitary generator matrix along with that of a spherical lattice code obtained using the generator matrix given in [37]. Also plotted are the BLERs yielded by spherical lattice codes which were optimized for the lattice decoder and the LRA decoder, respectively. It is seen that the code optimized for i.i.d. Rayleigh fading and ML decoding results in a severe performance degradation for this correlated Rician fading model and is matched by any randomly picked code. On the other hand, the code optimized for the lattice decoder offers significant performance improvement of over 3 dB since it is tuned or matched to the channel statistics. Further improvement of about 0.5 dB is obtained by the code optimized for the channel statistics as well as the employed decoder. VII. CONCLUSION We have proposed systematic procedures for designing spherical lattice space–time codes for the ML decoder, the lattice decoders, as well as the LRA decoders. The design methods are universal in the sense that they can be applied to optimize the lattice codes for arbitrary channel statistics and SNR. Simulation results have shown that our optimization methods yield low error rate lattice codes that outperform other lattice (or linear dispersion) codes proposed in the literature.

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APPENDIX

. Thus, ratio is bounded above by a positive constant and allow us to , we have that conclude (57). As a result, for any

A. Proof of Lemma 1 and , we note that lies in For a given . First, we will the (bounded) fundamental parallelotope of show that we can define a large enough albeit finite set and small enough, such that for all and

(57) Toward this end, we note that for all

and any (58)

where denotes the covering radius of the lattice generated by [9]. A convenient upper bound on covering . For a given vector , radius is be a lattice point closest to it. Then, we have let the inequalities in (59), shown at the bottom of the page. Using in we the continuity of , , where this can define a small interval

(60) which implies that for obtaining the derivative in (23) we can take to be a fixed set. Then the relation in the second equation shown at the bottom of the page, using the chain rule and the observation in (60), can be written in (61), also shown at the bottom of the page. Comparing (61) and (23) we see that the theorem will be proved if we show that the first term in the RHS of (61) is zero. To do so, . Using we first note the relation given in (62), where (62) we can expand the first term in the RHS of (61) as shown in denotes the equation at the top of the following page, where and , the delta function. Since the above term becomes zero by summing the corresponding and terms.

(59)

(61)

(62)

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B. Generating a “Good” Spherical Lattice Code Given a lattice generator matrix , we describe a simple iterative method to select a translation vector and coordinate vectors to reduce the average transmit energy. Define the as centroid [39] for a pair (63) Set an initial (random) translation vector lying in the Voronoi lattice points that are closest region and find a set of , as described in [13]. Then, replace by the negative of to the centroid of this set of lattice points, and repeat. It is observed that the procedure converges after a few iterations. Take negative as of the final centroid as the translation vector, and the set the coordinate vectors. The next method yields a spherical code satisfying the peak energy constraint when the given generator matrix lies in the set . Obtain (where is a large number) random translation vectors , uniformly distributed over the Voronoi relattice points closest to . gion. For each , obtain a set of Among all the , choose the one such that the corresponding spherical code satisfies the peak energy constraint. In order to uniformly distributed over the Voronoi obtain vectors region, we proceed as follows [26]. Generate a random vector , with elements of distributed as i.i.d. . Find such that is closest to , i.e., the coordinate vector . Then, it can be proved that the difference vector (64) is uniformly distributed over the Voronoi region. REFERENCES [1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 4, pp. 1456–1467, Jul. 1999. [2] B. Hassibi and B. M. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inf. Theory, vol. 48, no. 7, pp. 1804–1824, Jul. 2002. [3] P. Dayal and M. K. Varanasi, “An optimal two transmit antenna spacetime code and its stacked extensions,” in Proc. Asilomar Conf. Signals, Systems, and Computers, Thousand Oaks, CA, Nov. 2003. [4] H. El Gamal, G. Caire, and M. O. Damen, “Lattice coding and decoding achieve the optimal diversity–multiplexing tradeoff of MIMO channels,” IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 968–985, Jun. 2004.

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[27] E. Agrell and T. Eriksson, “Optimization of lattices for quantization,” IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 1814–1828, Sep. 1998. [28] H. Royden, Real Analysis, 3rd ed. Englewood Cliffs, NJ: PrenticeHall, 1988. [29] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory, vol. 48, no. 8, pp. 2201–2214, Aug. 2002. [30] M. K. Varanasi, “Decision feedback multiuser detection: A systematic approach,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 219–240, Jan. 1999. [31] X. Wang, V. Krishnamurthy, and J. Wang, “Stochastic gradient algorithms for minimum error-rate linear dispersion codes in MIMO wireless systems,” IEEE Trans. Signal Process., vol. 54, no. 4, pp. 1242–1255, Apr. 2006. [32] M. O. Damen, H. El Gamal, and G. Caire, “MMSE-GDFE lattice decoding for solving under-determined linear systems with integer unknowns,” in Proc. IEEE Int. Symp. Information Theory, Chicago, IL, Jun. 2004, p. 539. [33] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1993.

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˙ E. Telatar, “Capacity of multi-antenna Gaussian channels,” Europ. [34] I. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov. 1999. [35] H. Yao and G. W. Wornell, “Achieving the full MIMO diversity-multiplexing frontier with rotation-based space-time codes,” in Proc. Allerton Conf. Communications, Control, and Computing, Monticello, IL, Oct. 2003. [36] M. R. Mckay and I. B. Collings, “General capacity bounds for spatially correlated rician MIMO channels,” IEEE Trans. Inf. Theory, vol. 51, no. 9, pp. 3121–3145, Sep. 2005. [37] L. G. Electronics, “Open loop MIMO scheme for EUTRA,” in 3GPP R1-051338, Seoul, Korea, Nov. 2005. [38] W. H. Mow, “Universal lattice decoding: A review and some recent results,” in Proc. IEEE Intl. Conf. Communications, Paris, France, Jun. 2004. [39] J. H. Conway and N. J. A. Sloane, “A fast encoding method for lattice codes and quantizers,” IEEE Trans. Inf. Theory, vol. IT-29, no. 6, pp. 820–824, Nov. 1983.

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