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Journal of Electrical and Computer Engineering
Volume 2012 (2012), Article ID 290390, 9 pages
http://dx.doi.org/10.1155/2012/290390
Research Article

Minimum Symbol Error Probability MIMO Design under the Per-Antenna Power Constraint

Department of Electrical and Computer Engineering, Polytechnic Institute of NYU, 6 MetroTech Center, Brooklyn, NY 11201, USA

Received 21 October 2011; Revised 26 January 2012; Accepted 31 January 2012

Academic Editor: D. Laurenson

Copyright © 2012 Enoch Lu and I.-Tai Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Approximate minimum symbol error probability transceiver design of single user MIMO systems under the practical per-antenna power constraint is considered. The upper bound of a lower bound on the minimum distance between the symbol hypotheses is established. Necessary conditions and structures of the transmit covariance matrix for reaching the upper bound are discussed. Three numerical approaches (rank zero, rank one, and permutation) for obtaining the optimum precoder are proposed. When the upper bound is reached, the resulting design is optimum. When the upper bound is not reached, a numerical fix is used. The approach is very simple and can be of practical use.

1. Introduction

Since multiple-input and multiple-output (MIMO) is a very promising technology for mitigating the spectrum scarcity problem, many MIMO transceiver designs have been published recently. The designs have been based on a variety of criteria, for example, maximum capacity, minimum mean square error (MMSE) and minimum bit error rate (Min BER). Considered in this paper is the minimum symbol error probability transceiver design subject to the per-antenna power constraint. This is due to a number of reasons. Firstly, the minimum symbol error probability criterion is directly related to the system performance. Secondly, the per-antenna power constraint is more practical than the commonly used total power constraint in MIMO systems (because each antenna has its own power amplifier and each power amplifier has a limited dynamic range). Lastly, both criterion and metric are difficult to tackle.

Minimum symbol error probability and the related Min BER design problems have been formulated in various different ways (e.g., [17]). For example, [1] performs its minimum symbol error probability design by maximizing a lower bound for the minimum distance of the symbol hypotheses. As the metric is a lower bound, the design is approximate in nature. However, it is also thus independent of the symbol alphabet. Their lower bound is the minimum eigenvalue of a positive definite system matrix thus making the problem a max min eigenvalue one. For the sum power constraint, their optimum design is neatly obtained by making all eigenvalues equal.

In this paper, we follow the formulation in [1] because it involves interesting and challenging signal processing issues. As already mentioned, the constraint here however is the per-antenna one. The upper bound of the cost function is established using the per-antenna power constraint and the special structure of a system matrix (which involves the precoder matrix, noise covariance matrix, and channel matrix). The necessary conditions and structures of the transmit covariance matrix for reaching the upper bound are discussed in detail. Three numerical approaches (rank zero, rank one, and permutation) for obtaining the optimum precoder are proposed. When the upper bound is reached, the resulting design is optimum. When the upper bound is not reached, a numerical fix is used. Extensive numerical studies have been performed to assess the performance of the proposed methodology. Although the upper bound is not reached in most cases, good performances in mutual information and signal to interference plus noise ratio (SINR) are achieved. Moreover, the approach is very simple and can be of practical use.

Notations: boldface letters denote either vectors (lower case) or matrices (upper case). 𝐀𝑇, 𝐀, 𝐀1, tr(A), E(A), 𝜆max(A), and 𝜆min(A) stand for the transpose, conjugate transpose, inverse, trace, expectation, maximum eigenvalue, and minimum eigenvalue of A, respectively. 𝜆𝑖(A) stands for the 𝑖th eigenvalue of A when its eigenvalues are arranged in increasing order. Ir is the 𝑟×𝑟 identity matrix. 0 is the zero matrix of appropriate dimension. ei denotes the 𝑖th column of the identity matrix (the size of which will be clear from the context). diag(a) denotes the diagonal matrix with a on the main diagonal. On the other hand, if A is a square matrix, diag(A) is the main diagonal of A. 𝐀>𝐁 and 𝐀𝐁 mean that 𝐀𝐁 is positive definite and positive semidefinite, respectively. A°B denotes the elementwise product of A and B. max(𝑎,𝑏) means the maximum of real numbers 𝑎 and b. CN(0,𝐑) denotes a zero-mean circularly symmetric complex normal random vector with covariance matrix R.

2. Background, Problem, and Overview of Design

The received signal of the considered MIMO system is 𝐲=𝐇𝐅𝐬+𝐧 where H is the full rank 𝑛×𝑛 channel matrix, F is the full rank 𝑛×𝑛 precoder, s is the 𝑛×1 data vector, and n is the 𝑛×1 received noise vector. The full rank 𝑛×𝑛 decoder G is applied to y to yield Gy. For convenience and without loss of generality, let the source covariance matrix 𝐸(𝐬𝐬)=𝐈 and the noise covariance matrix 𝐸(𝐧𝐧)=𝐑𝐧𝐧>𝟎. As the transmitter’s power must be constrained, the precoder F is required to satisfy the per-antenna power constraint𝐞𝑖𝐅𝐅𝐞𝑖𝑑𝑖,𝑖.(1) Here, 𝑑𝑖>0,𝑖. Note that (1) also results in the average total power being upper bounded by P =𝑛𝑖=1𝑑𝑖. Let 𝑆 denote the set of all feasible F’s.

2.1. Problem Formulation

The goal of this paper is to design F and G to approximately minimize the probability of error in an alphabet independent manner. To accomplish this, we will begin as in [1]. Define 𝐀𝐇𝐑1𝐧𝐧𝐇 and𝚿(𝐅,𝐆)𝐅𝐇𝐆𝐆𝐑𝐧𝐧𝐆1𝐆𝐇𝐅.(2)𝜆min(Ψ(𝐅,𝐆)) actually lower bounds the minimum distance between the symbol hypotheses. In addition,𝚿(𝐅,𝐆)𝐅𝐀𝐅(3) with equality when 𝐆=𝐅𝐇(𝐇𝐅𝐅𝐇+𝐑𝐧𝐧)1, that is, the MMSE decoder is used. Thus, we will choose 𝐆=𝐅𝐇(𝐇𝐅𝐅𝐇+𝐑𝐧𝐧)1 and design the precoder F according to the optimization problem:max𝐅𝑆𝜇(𝐅),𝜇(𝐅)𝜆min𝐅𝐀𝐅.(4)

Note that if the equality average total power constraint tr(𝐅𝐅)=𝑃 was used instead, the solution to (4) would be given by [1]. That is,𝐕𝚲1/2𝑃𝚲tr11/2=argmaxtr𝐅𝐅=𝑃,𝐅𝑛×𝑛𝜇(𝐅),(5) where the unitary matrix V and the diagonal matrix Λ are obtained from the eigenvalue decomposition of 𝐀=𝐕Λ𝐕 (eigenvalues in descending order).

2.2. Overview of Design of F

The optimization problem, (4), is very complicated. Though not mentioned in Section 2.1, we desire a low complexity algorithm to design F. We will thus take some simplifications.

Before detailing the simplifications and the algorithm, we will first need some analysis on the cost function of (4): since 𝐅𝐀𝐅 and 𝐀𝐅𝐅 have the same eigenvalues, 𝐀𝐅𝐅𝐳=𝜇(𝐅)𝐳 where z is an eigenvector of 𝐀𝐅𝐅 generated by the minimum eigenvalue 𝜇(𝐅). Noting that 𝐀>𝟎,𝐳𝜇(𝐅)=𝐅𝐅𝐳𝐳𝐀1𝐳𝐱𝐅𝐅𝐱𝐱𝐀1𝐱,(6) where x is any nonzero 𝑛×1 vector. Define 𝐁=[𝑏𝑖𝑗]𝐀1 for convenience. Plugging in ei for x in (6) and using the constraint (1), it is then clear that, for all 𝐅𝑆,𝜇(𝐅)min𝑖𝑑𝑖𝑏𝑖𝑖=min𝑖𝑑1𝑏11,𝑑2𝑏22𝑑,,𝑛𝑏𝑛𝑛𝑑𝑘𝑏𝑘𝑘𝜌.(7) That is, the cost function in (4) is upper bounded by ρ.

The upper bound ρ may be reachable. Consequently, the algorithm to design F is as follows. We first try to find a precoder Fρ∈S with maximum power (i.e., satisfies (1) with equality) and which reaches the upper bound ρ:𝜇𝐅𝜌=𝜌.(8) Any of the three approaches in Section 4 can be used for this search. If successful, we have found an optimal solution to (4) and are done; set F equal to Fρ. If unsuccessful, we get F by implementing a fix for the approach used to search for Fρ. This fix is simply lowering the power of the kth antenna and is explained in Section 5. The low complexity of the three approaches in Section 4 is due to Section 3 which reveals necessary structure and conditions for 𝐔𝜌=𝐅𝜌𝐅𝜌.

3. Necessary Structure and Conditions

If Fρ exists, the transmit covariance matrix Uρ would have a special structure and certain conditions would be true. Assume Fρ exists. Then, observing from (6), ek must be an eigenvector of AUρ paired with the eigenvalue ρ, that is,𝐀𝐔𝜌𝐞𝑘=𝜌𝐞𝑘𝐔𝜌𝐞𝑘=𝜌𝐀1𝐞𝑘=𝜌𝐁𝐞𝑘.(9) In particular, the kth column of Uρ must equal the kth-column of B multiplied by ρ. Consequently, if 1<𝑘<𝑛 and [𝐛𝑇1𝑏𝑘𝑘𝐛𝑇2]𝑇 denotes the kth column of B, then, necessarily,𝐔𝜌=𝐅𝜌𝐅𝜌=𝚺1𝜌𝐛1𝐋𝜌𝐛1𝑑𝑘𝜌𝐛2𝐋𝜌𝐛2𝚺2𝐈>𝟎,(10)𝑘1𝚺1𝑑=diag1𝑑𝑘1,𝐈𝑛𝑘𝚺2𝑑=diag𝑘+1𝑑𝑛.(11) For the sake of clarity, the cases when k=1 or 𝑛 are omitted in this paper.

3.1. Necessary Conditions for 𝐔𝜌 (1st Representation)

As to be expected, the remaining unspecified elements of Uρ (i.e., L and the off diagonal elements of 𝚺1 and 𝚺2) are not arbitrary. By reducing Uρ to direct sum form using elementary block row and column operations, it can be seen that Uρ > 0 if and only if𝚺1>𝟎,𝚺2>𝟎,𝚺2𝐋Σ11𝐋>𝟎,(12)𝑏2𝑘𝑘𝑑𝑘>𝐛1𝚺11𝐛1+𝐛2𝐋𝚺11𝐛1𝚺2𝐋𝚺11𝐋1𝐛2𝐋𝚺11𝐛1.(13) Since 𝚺2𝐋𝚺11𝐋>𝟎 (see (12)), the following necessary condition can be derived from (13): 𝑏2𝑘𝑘𝑑𝑘>𝐛1𝚺11𝐛1.(14) Furthermore, since 𝚺1>𝟎, a redundant though useful necessary condition can be derived for the antenna powers {𝑑1,,𝑑𝑘}, which depends only on the known parameters b1, and 𝑏𝑘𝑘:𝑏2𝑘𝑘𝑑𝑘>𝐛1𝐛1𝜆max𝚺1𝐛1𝐛1𝚺tr1=𝐛1𝐛1𝑑1+𝑑2++𝑑𝑘1.(15)

An alternate necessary and sufficient condition for Uρ > 0 can be given by simply interchanging the subscripts 1 and 2, and L* and L in (12), (13). Thus, one can easily show that𝑏2𝑘𝑘𝑑𝑘>𝐛2𝚺21𝐛2,𝑏2𝑘𝑘𝑑𝑘>𝐛2𝐛2𝑑𝑘+1+𝑑𝑘+2++𝑑𝑛(16) are also necessary.

3.2. Necessary Conditions for 𝐔𝜌 (2nd Representation)

Use an appropriate invertible, symmetric, real 𝑛×𝑛 permutation matrix P to permute 𝑑𝑘 to the upper left corner:𝐏𝐔𝜌𝑑𝐏=𝑘𝜌̃𝐛𝜌̃𝐛𝐐.(17) (The structures of P, Q, and ̃𝐛, are omitted here). It turns out that 𝐔𝜌 > 0 if and only if P*UρP > 0. Thus, Uρ > 0 if and only if (noting that 𝑑𝑘>0 given in (1))𝑑𝐐𝑘𝑏2𝑘𝑘̃𝐛̃𝐛>𝟎.(18) Since both (12)-(13), and (18) are necessary and sufficient for Uρ > 0, (12)-(13) and (18) are equivalent to each other. Conveniently, the diagonal elements of 𝐐,𝑞𝑖𝑖, 𝑖=1,,𝑛1, are just the diagonal elements of 𝚺1, 𝚺2 permutated. Consequently, requiring that the 𝑞𝑖𝑖, 𝑖=1,,𝑛1, be equal to the correct antenna powers is equivalent to requiring conditions (11). Moreover, a redundant though useful necessary condition can easily be derived from (18) which only uses the antenna powers, ̃𝐛, and 𝑏𝑘𝑘:𝑞𝑖𝑖𝑑𝑘𝑏2𝑘𝑘||𝑏𝑖||2>0,(𝑖=1,,𝑛1).(19)

4. Three Approaches for Obtaining Uρ

Each of the following three approaches seeks to find an Uρ (i.e., L, 𝚺1, and 𝚺2 in (10) or equivalently Q in (17)) which satisfies the necessary structure and conditions in Section 3.

4.1. Rank 0 Approach (R0A)

In this simple approach, the matrix L is chosen to have rank 0 (hence the name of the approach), that is, equal to all zeros. 𝚺1 and 𝚺2 are chosen to be diagonal matrices with the diagonal entries (𝑑1, …, 𝑑𝑘1) and (𝑑𝑘+1, …, 𝑑𝑛), respectively. Such a choice for L, 𝚺1, and 𝚺2 automatically satisfies (11), (12). If the last remaining necessary condition (13) is satisfied, construct Uρ using (10) and check whether ρ is the minimum eigenvalue of AUρ. If both of these conditions are satisfied, decompose Uρ to get a Fρ (and a corresponding G) and an optimum solution has been found. If either condition fails, use the fix in Section 5. It is interesting to note that the decomposition from Uρ to Fρ is not unique; indeed, using Fρ right multiplied by a unitary matrix is also a valid decomposition.

4.2. Rank 1 Approach (R1A)

Choose 𝚺1 and 𝚺2 as in R0A. If 𝐛1𝚺11𝐛1=𝐛2𝚺21𝐛2, use another approach. If 𝐛1𝚺11𝐛1>𝐛2𝚺21𝐛2, choose a rank one choice of L,𝐛𝐋=1𝐛2𝐛1Σ11𝐛1.(20) This L makes the right-hand side of (13) as small as possible and independent of 𝚺2. If 𝐛1𝚺11𝐛1<𝐛2𝚺21𝐛2, choose an alternative rank one choice,𝐛𝐋=1𝐛2𝐛2𝚺21𝐛2.(21) With these 𝚺1, 𝚺2, and L, the power constraints in (11) and the first two conditions of (12) are automatically satisfied. Furthermore, the condition 𝚺2𝐋𝚺11𝐋>𝟎 in (12) is also satisfied—use the fact that 𝐛1𝚺11𝐛1>𝐛2𝚺21𝐛2 when (20) is used and that 𝐛1𝚺11𝐛1<𝐛2𝚺21𝐛2 when (21) is used. However, (13) needs to be checked. In addition, whether ρ is the minimum eigenvalue of AUρ also needs to be checked. Same as in R0A, if both conditions are satisfied, decompose Uρ to get a Fρ (and a corresponding G) and an optimum solution has been found. If either condition fails, use the fix in Section 5.

4.3. Permutation Approach (PA)

This third approach, unlike the previous two, searches for Uρ using the 2nd representation of necessary conditions (Section 3.2). It is based on two facts. The first is that Q = Q* and𝜆min(𝐐)>𝜂min𝑑𝑘𝑏2𝑘𝑘̃𝐛̃𝐛(22) together imply (18). The second is that a Hermitian Q satisfying (22) and having the correct diagonal entries (the antenna powers permutated) exists if and only if 𝜂min<𝑑𝑖, for all i (just apply the Schur-Horn Theorem [8]).

Granted that 𝜂min<𝑑𝑖, for all i, the approach is as follows. First choose some 𝜀>0 such that 𝜂min+𝜀𝛾𝑑𝑖, for all i. Next, find a Hermitian Q whose diagonal entries are the correct antenna powers, {𝑞𝑖𝑖}, and whose eigenvalues are lower bounded by γ. Once a Q is found, construct (in light of (17))𝐔𝜌=𝐏1𝑑𝑘𝜌̃𝐛𝜌̃𝐏𝐛𝐐1.(23) If ρ is the minimum eigenvalue of AUρ, obtain Fρ and a corresponding G from Uρ as in R0A and R1A; an optimum solution has been found. If there is a smaller eigenvalue than ρ, use the fix in Section 5.

There are various ways to find a Hermitian Q with diagonal entries {𝑞𝑖𝑖} and eigenvalues lower bounded by γ. A closed form solution is shown in Appendix A and a projection approach is shown in Appendix B.

5. A Fix

Consider a L, 𝚺1, and 𝚺2 which satisfies (11) and (12). If (13) does not hold and/or if ρ = 𝑑𝑘/𝑏𝑘𝑘 is greater than the smallest eigenvalue of AUρ, proceed as following. First, temporarily change the per-antenna power constraint for the kth antenna by replacing 𝑑𝑘 by 𝑑𝑘. This makes ρ become 𝑑𝑘/𝑏𝑘𝑘, Uρ become 𝚺1𝜌𝐛1𝐋𝜌𝐛1𝑑𝑘𝜌𝐛2𝐋𝜌𝐛2𝚺2,(24) and so forth. Second, lower 𝑑𝑘 (maintaining 𝑑𝑘>0) until (13) holds. Then, continue lowering 𝑑𝑘 (maintaining 𝑑𝑘 > 0) and thus ρ = 𝑑𝑘/𝑏𝑘𝑘 until ρ is the smallest eigenvalue of AUρ. As Appendix C shows, one can always lower 𝑑𝑘 until this happens. Using this fix, (8) is thus obtained—granted, for a lower power constraint.

Lastly, now that 𝑑𝑘 is low enough, decompose Uρ to get Fρ (and a corresponding G). This Fρ is full rank since (12-13) hold. In addition, it satisfies the inequality per-antenna power constraint (1) with the true 𝑑𝑘 and the 𝑑𝑖, for all 𝑖𝑘. Indeed,𝐞𝑘𝐅𝜌𝐅𝜌𝐞𝑘=𝑑𝑘<𝑑𝑘.(25) Thus, set F equal to Fρ.

Lowering 𝑑𝑘 until (13) holds is understandable—F needs to be full rank. But, why lower it to satisfy (8) for a lower power constraint? Instead of lowering 𝑑𝑘 further, decomposing Uρ to get F at that point would yield a legitimate precoder. So, why continue to lower 𝑑𝑘? The reason is that it is observed numerically that continuing to lower 𝑑𝑘 actually raises the minimum eigenvalue of AUρ. See Figure 6 and the discussion for it.

6. Numerical Results

The numerical results are divided into two parts. In the first part, two examples are given to demonstrate the proposed approaches of Section 4 and the fix of Section 5. In the second part, Monte Carlo simulations are used to investigate how suboptimal, if at all, is the proposed design methodology for F. It also investigates how often the fix is needed.

6.1. Demonstration of the Proposed Design Methodology for F

Two examples are given here, each of which corresponds to one H and 𝐑𝐧𝐧 (i.e., one A). Without loss of generality, consider 10 antennas (𝑛=10) with identical power constraints (𝑑𝑖 = 10, 𝑖=1,,𝑛). Thus, maximum allowable total power 𝑃=𝑛𝑖=1𝑑𝑖=100.

For the first numerical example, (8) is achieved by the R0A, R1A, and PA. The resulting eigenvalues of AU for each of them are plotted in Figure 1. The upper bound ρ and the optimum result under the total power constraint (see (5)) are also plotted as benchmarks. Several interesting observations can be made. Firstly, since 𝑃=𝑛𝑖=1𝑑𝑖, the optimum solution for the total power case always is at least as good as that of the per-antenna case. Indeed, in Figure 1, the 𝜆1(𝐀𝐔) (i.e.,𝜆min(𝐀𝐔)) for the total power case is greater than ρ, an upper bound for the 𝜆1(𝐀𝐔)’s of the per-antenna case. Secondly, the R0A, R1A, and PA all result in optimum F’s here; 𝜆1(𝐀𝐔) for each of the three approaches is numerically equal to ρ. Thirdly, the eigenvalues are all equal for the total power case (as is always the case. See [1]). However, the eigenvalues for any of the approaches subject to the per-antenna constraint are clearly not all equal.

290390.fig.001
Figure 1: 𝜆𝑗(𝐀𝐔) as a function of j (the index of eigenvalues) when the upper bound in (8) is reached.

In the second numerical example, the R0A’s solution does not satisfy (13). Thus, the fix in Section 5 is applied to it. The eigenvalues and the ρ resulting from the fix (i.e., 𝑑𝑘/𝑏𝑘𝑘) are plotted in Figure 2. The original ρ and the optimum result under the total power constraint 𝑃=100 are also plotted for reference. As it is supposed to be, 𝜆1(𝐀𝐔) for the fixed solution is numerically equal to the lowered ρ. The 𝑑𝑘 is approximately equal to 4.4138.

290390.fig.002
Figure 2: 𝜆𝑗(𝐀𝐔) as a function of j (the index of eigenvalues) when the fix is used.
6.2. Investigation into the Effectiveness of the Proposed Design Methodology for F

In this subsection, we use Monte Carlo simulation to assess the effectiveness of the proposed approach and to show how far our suboptimum solution is from the optimum solution. We did not prove that the upper bound in (8) is always achievable. How then do we get the optimum solution needed for this comparison? For a 𝐑𝐧𝐧, P, and H, we obtain it by the following methodology. We do not specify the per-antenna power constraints {𝑑𝑖} at the beginning. Instead, we calculate the closed-form precoder in (5) and set its antenna powers as the {𝑑𝑖}. In other words, if Fo denotes the precoder from (5), we set 𝐞𝑖(𝐅𝑜𝐅𝑜)𝐞𝑖 as 𝑑𝑖, for all i. For this resulting problem, Appendix D proves that the precoder from (5) is an optimum solution to (4). Moreover, it achieves the upper bound in (8). With a 𝐑𝐧𝐧, H, and {𝑑𝑖} in hand where we know the upper bound ρ is achievable, we can run our algorithm to get F and analyze its performance.

More specifically, this simulation is run as follows. The noise covariance matrix 𝐑𝐧𝐧 is set equal to 𝜎2I. The transmit signal-to-noise ratio (SNR) is defined as =10log10(𝑃/𝜎2). Transmit SNRs of 0, 6, 12, and 18 dB are run. Both 4 and 8 antenna scenarios (𝑛=4,8) are run. For each transmit SNR and n, 1000 H’s are randomly generated; elements of H are independent identically distributed CN(0,1) random variables. After the {𝑑𝑖} are determined by the total power closed-form solution (5), the R0A and, if necessary, the fix are run.

First, consider the 8 antennas case (𝑛=8). For all the transmit SNRs and all the randomly generated H, the R0A does not achieve the upper bound in (8) and the fix in Section 5 is employed. The top figure in Figure 3 shows the histogram of the ratio between the new 𝑑𝑘 (after the fix) and 𝑑𝑘 for the transmit SNR 6 dB case. Since 𝑑𝑘 is smaller than 𝑑𝑘, the total power of the proposed F is smaller than the maximum total power allowed, P. The bottom figure in Figure 3 thus shows the histogram of the ratio between the total power of the proposed F and 𝑃. The histograms for the other transmit SNRs are not shown since they are so similar to Figure 3. Recall that if the fix is used, the obtained cost function value is 𝑑𝑘/𝑏𝑘𝑘. Since the optimum solution obtains the cost function value of ρ = 𝑑𝑘/𝑏𝑘𝑘, the top histogram of Figure 3 also shows how suboptimal the proposed algorithm is.

290390.fig.003
Figure 3: Eight-antenna example. Top: Histogram of the ratio between 𝑑𝑘 (after the fix) and 𝑑𝑘. Bottom: histogram of the ratio between the total power of the proposed F and 𝑃.

The optimal solution in (5) is better than the proposed solution with respect to the cost function in (4). However, the proposed solution has a much larger mutual information than the optimal solution (see Figure 4). According to [1], the mutual information for a F islog2||𝐈+𝐇𝐅𝐅𝐇𝐑1𝐧𝐧||,(26) when the MMSE decoder is used—as is done here. The reason for the observed larger mutual information is as follows. The optimum solution in (5) diagonalizes the equivalent channel matrix GHF and equalizes all eigen-channels so that the resulting SINRs for all data streams are the same and equal to ρ (see Appendix D and [1]). But, for the proposed solution, the SINRs of all the data streams are not the same and, moreover, most of them are larger than ρ(see Figure 5). The SINR for the 𝑖th stream when 𝐅=[𝐟1𝐟𝑛] and 𝐆=[𝐠𝑇1𝐠𝑇𝑛]𝑇 are used is simply||𝐠𝑖𝐇𝐟𝑖||2𝐠𝑖𝐑𝐧𝐧𝐠𝑖+𝑗𝑖||𝐠𝑖𝐇𝐟𝑗||2.(27)

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Figure 4: Eight-antenna example. Mutual information for the optimum precoder and the proposed precoder.
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Figure 5: Eight-antenna example. Normalized SINRs for the 8 data streams for some channel realizations when the proposed algorithm is employed. The normalization factor is ρ, the SINR of every data stream when the optimum solution is used.
290390.fig.006
Figure 6: Eight-antenna example. Typical example of how the fix works.

Figure 5 shows the normalized SINRs for the 8 data streams of the proposed solution for a sampling of the channel realizations. The transmit SNR is 6 dB. The figures for the other channel realizations and transmit SNRs are not shown since they are so similar to Figure 5. For a given channel realization, the normalization factor is the same for all 8 data streams. It is ρ, the SINR of every data stream when the optimum solution is used (see Appendix D). For the channel realizations shown in Figure 5, only one normalized SINR (out of 8) is less than 1. Thus, only one data stream for the proposed solution has a lower SINR than the SINR of the 8 data streams for the optimal solution. This is also roughly the case for the other channel realizations as well.

Note that the optimal solution may have a smaller symbol error rate than the suboptimum solution if a ML receiver is used (according to the logic in Section 2.1). If the ML receiver is not employed, the optimal solution may not have any advantage over the proposed solution. For practical implementations, appropriate modulation and coding schemes can be selected to maximize the throughput when the precoder derived from the low complexity proposed algorithm is employed.

Section 5 said that numerical results showed that continuing to lower 𝑑𝑘 actually raises the minimum eigenvalue of AUρ. Figure 6 shows a typical plot of the results of continuing to lower 𝑑𝑘 for a channel realization. At each iteration, 𝑑𝑘 is lowered by one percent. In the figure, ρ = 𝑑𝑘/𝑏𝑘𝑘 and Uρ is given by (23) in accordance with the notation in Section 5. Indeed, the minimum eigenvalue of AUρ increases as 𝑑𝑘, and thus ρ = 𝑑𝑘/𝑏𝑘𝑘, is lowered. In fact, the increase is always monotonic (the value for an iteration is greater or equal to that of the previous iteration) in all the simulations save for one exception. In some channels (e.g., the one for Figure 6), 𝜆1(𝐀𝐔𝜌) for the last iteration may be smaller than 𝜆1(𝐀𝐔𝜌) for the second to last iteration. This may be due to overshooting since the step size for 𝑑𝑘 is fixed.

Next, consider the 4 antennas case (𝑛=4). Figure 7 shows the histogram of the ratio between 𝑑𝑘 (after the fix) and 𝑑𝑘. It also shows the histogram of the ratio between the total power of the proposed F and 𝑃. Figure 8 shows the mutual information for the optimum precoder and the proposed precoder. Figure 9 shows the proposed solution’s normalized SINRs for the 4 data streams for some channel realizations. For a given channel realization, the normalization factor is ρ, the SINR of the 4 data streams when the optimum precoder in (5) is used.

290390.fig.007
Figure 7: Four-antenna example. Top: histogram of the ratio between 𝑑𝑘 (after the fix) and 𝑑𝑘. Bottom: histogram of the ratio between the total power of the proposed F and P.
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Figure 8: Four-antenna example. Mutual information for the optimum precoder and the proposed precoder.
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Figure 9: Four-antenna example. Normalized SINRs for the 4 data streams for some channel realizations when the proposed algorithm is employed. The normalization factor is ρ, the SINR of every data stream when the optimum solution is used.

Basically, the observations made for the eight-antenna example are still applicable here for the four-antenna example. Some notable differences and points are as follows. Firstly, the histogram of 𝑑𝑘/𝑑𝑘 shifts to the right when going from 𝑛=8 to 𝑛=4. Secondly, the mutual information is smaller in the 𝑛=4 case. The transmit SNR gap between the two approaches is also smaller here as well. Lastly, for the 𝑛=4 case, the normalized SINRs suffer some degradation as transmit SNR increases. For the lower transmit SNRs, there are roughly 3 data streams with normalized SINRs above 1 and 1 data stream with normalized SINR below 1. For the higher transmit SNRs, there are roughly 2 data streams with normalized SINRs above 1 and 2 data streams with normalized SINRs below 1.

7. Conclusion

Considered here is the approximate minimum symbol error probability transceiver design subject to the practical per-antenna power constraint. The metric to be maximized is a lower bound for the minimum distance of the symbol hypotheses. As in [1], the bound used is the minimum eigenvalue of a positive definite system matrix involving the precoder matrix, noise covariance matrix, and channel matrix. This max min problem is both interesting and challenging because the differentiation of the minimum eigenvalue cannot be performed explicitly. Remarkably, we are able to develop approaches to solve the design problem without using differentiation of eigenvalues or the popular method of Lagrange multipliers.

First, the upper bound for the cost function is established using the special structure of the system matrix and the power constraint. Then, necessary conditions and structures of the transmit covariance matrix for reaching the upper bound are obtained. Based on these necessary conditions and structures, three numerical approaches (rank zero, rank one and permutation) for obtaining the optimum precoder are developed. Since the upper bound is not always achieved, a possibly suboptimum fix is also given to be used, when necessary, after the proposed approaches.

In the total power constraint case, the eigenvalues of the optimum solution in [1] were always equal. Interestingly, the numerical results here show that this is not always the case for the per-antenna power constraint. Extensive numerical studies have been performed to assess the performances of the proposed methodology. Although the upper bound is not reached in most cases, good performances in mutual information and SINR are achieved. Moreover, the approach is very simple and can be of practical use.

Appendices

A.

Consider the (𝑛1)×(𝑛1) matrix Y= γIn-1 + X where X 0. For any X 0, Y is Hermitian and has eigenvalues lower bounded by γ. So, as long as𝑞diag(𝐗)=11𝛾,𝑞22𝛾,,𝑞𝑛1,𝑛1𝛾,(A.1)Y is a valid choice of Q.

To find a 𝐗𝟎 that satisfies (A.1) is not difficult. First, choose any (𝑛1)×(𝑛1) matrix 𝐙𝟎 with positive diagonal entries {𝑧𝑖𝑖}. Then, form𝐑diag𝑞11𝛾𝑧11,𝑞22𝛾𝑧22,,𝑞𝑛1,𝑛1𝛾𝑧𝑛1,𝑛1.(A.2) The product RZR is a valid choice of X. Consequently, Y= γIn-1 + RZR is a valid choice of Q and can be used in the PA.

B.

The projection approach finds a Q as follows. It begins by randomly generating a (𝑛1)×(𝑛1) unitary matrix V(0) and 𝑛1 real numbers, 𝜂1𝜂𝑛1𝛾. It then creates the initial guess at Q:𝐐(0)=𝐕(0)𝜂diag1,,𝜂𝑛1𝐕(0).(B.1) At the 𝑗th iteration step (𝑗>1), it proceeds as follows.(i)Force the diagonal elements of 𝐐(𝑗1) to be equal to {𝑞𝑖𝑖}.(ii)Then, decompose 𝐐(𝑗1) as 𝐐(𝑗1)=𝐕(𝑗1)𝜉diag1,,𝜉𝑛1𝐕(𝑗1)(B.2) to get the unitary matrix 𝐕(𝑗1) and the real numbers 𝜉1𝜉𝑛1.(iii)If 𝜉𝑛1γ, set Q equal to 𝐐(𝑗1) and stop iterating. If not, let 𝜉𝑖 = max(𝜉𝑖, γ), for all i, and create𝐐(𝑗)=𝐕(𝑗1)𝜉diag1,,𝜉𝑛1𝐕(𝑗1).(B.3)(iv)Move onto the (𝑗+1)th iteration step.

A Q is always found. That is, the above iteration always converges. The reason is as follows. The iteration simply projects between two closed, convex subsets of the Hilbert space 𝑛×𝑛:𝐓𝑛×𝑛𝑞diag(𝐓)=11𝑞𝑛1,𝑛1,𝐓=𝐓,(B.4)𝐕Θ𝐕𝐕1=𝐕𝑛×𝑛,𝜃Θ=diag1,,𝜃𝑛𝑛×𝑛,𝜃𝑖𝛾(B.5) As the intersection of the two subsets is nonempty (by the Schur-Horn Theorem [8]), the iteration will converge [9].

C.

Assume that (10)–(13) hold but not (8). It will be proved here that (8) can be satisfied as well by simply lowering 𝑑𝑘 and thus ρ(maintaining 𝑑𝑘, 𝜌>0). The proof can be split into four parts. The first part is to realize that (10) implies that ρ is an eigenvalue of AUρ (use (9)). The second part is to find a (𝑛1)×(𝑛1) matrix whose eigenvalues are precisely AUρ’s other 𝑛1 eigenvalues. To this end, introduce the partition𝐂𝐀=11𝐜12𝐂13𝐜12𝑐22𝐜23𝐂13𝐜23𝐂33,(C.1) where 𝑐22 is the kkth element of A. In addition, note that A times the kth column of Uρ is equal to ρek due to 𝐁=𝐀1. With some straightforward steps, it can thus be seen that𝐌𝑑𝑘=𝐂11𝐂13𝐂13𝐂33𝚺1𝐋𝐋𝚺2+𝑑𝑘𝑏𝑘𝑘𝐜12𝐛1𝐜12𝐛2𝐜23𝐛1𝐜23𝐛2(C.2) is a suitable candidate matrix. The third part is to realize that the eigenvalues of𝐂11𝐂13𝐂13𝐂33𝚺1𝐋𝐋𝚺2,(C.3) the first term in M(𝑑𝑘), are positive and independent of 𝑑𝑘. To this end, note that (C.3) is the product of two positive definite matrices. Then, note that the eigenvalues of such a product must be positive [10]. Finally, using the fact that “the eigenvalues of a square…complex matrix depend continuously upon its entries” ([11]: Appendix D) and a limiting argument, the last part is to realize that one can lower 𝑑𝑘 and thus ρ (maintaining 𝑑𝑘,ρ > 0) until all the eigenvalues of M(𝑑𝑘) are ≥ ρ.

D.

Let Fo denote the precoder from (5). As it is an optimal solution to the optimization problem in (5), tr{𝐅𝑜𝐅𝑜}=𝑃. But, what are its antenna powers? It turns out that the 𝑖th antenna has power 𝑏𝑖𝑖𝑃/tr(Λ1) because𝐅𝑜𝐅𝑜=𝐕Λ𝟏/𝟐𝚲𝟏/𝟐𝐕𝑃𝚲tr𝟏𝑃=𝐁𝚲tr𝟏.(D.1) Here, we used the fact that 𝐁=𝐀1 and 𝐀=𝐕Λ𝐕.

Let us say, for all i, we set 𝑑𝑖 equal to Fo’s 𝑖th antenna power:𝑑𝑖=𝑏𝑖𝑖𝑃𝚲tr𝟏.(D.2) Then from (7) and (D.2), we have the upper bound𝑃𝜌=𝚲tr𝟏.(D.3) It turns out that Fo reaches this upper bound. Direct computation shows this. From (5), we have 𝐅𝑜𝐀𝐅𝑜=Λ1/2𝐕𝐕𝚲𝐕𝐕𝚲1/2𝑃𝚲tr𝟏𝑃=𝐈𝚲tr𝟏.(D.4) From (D.3) and (D.4), we have𝜆min𝐅𝑜𝐀𝐅𝑜=𝑃𝚲tr𝟏=𝜌.(D.5) In summary, if the 𝑑1,,𝑑𝑛 of the per-antenna power constraint (1) are defined using the antenna powers of Fo, Fo reaches the upper bound ρ and is thus an optimum solution to (4).

Disclosure

A part of this manuscript appears in Enoch Lu’s dissertation, submitted to the Faculty of the Polytechnic Institute of New York University in partial fulfillment of the requirements for the degree Doctor of Philosophy (Electrical Engineering) January 2012.

Acknowledgment

The authors would like to express their gratitude to Professor Dante Youla for his comments.

References

  1. A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, and H. Sampath, “Optimal designs for space-time linear precoders and decoders,” IEEE Transactions on Signal Processing, vol. 50, no. 5, pp. 1051–1064, 2002. View at Publisher · View at Google Scholar · View at Scopus
  2. D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization,” IEEE Transactions on Signal Processing, vol. 51, no. 9, pp. 2381–2401, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. D. P. Palomar, “Unified framework for linear MIMO transceivers with shaping constraints,” IEEE Communications Letters, vol. 8, no. 12, pp. 697–699, 2004. View at Publisher · View at Google Scholar · View at Scopus
  4. F. Rey, M. Lamarca, and G. Vazquez, “Robust power allocation algorithms for MIMO OFDM systems with imperfect CSI,” IEEE Transactions on Signal Processing, vol. 53, no. 3, pp. 1070–1085, 2005. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. K. Yazarel and D. Aktas, “Downlink beamforming under individual SINR and per antenna power constraints,” in Proceedings of the IEEE Pacific Rim Conference on Communications, Computers and Signal Processing (PACRIM '07), pp. 422–425, August 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. C. C. Weng and P. P. Vaidyanathan, “Per-antenna power constrained MIMO transceivers optimized for BER,” in Proceedings of the 42nd Asilomar Conference on Signals, Systems and Computers (ASILOMAR '08), pp. 1300–1304, October 2008. View at Publisher · View at Google Scholar · View at Scopus
  7. M. C. H. Lim, M. Ghogho, and D. C. McLernon, “Reduced complexity design for weighted harmonic mean SINR maximization in the multiuser MIMO downlink,” in Proceedings of the IEEE 10th Workshop on Signal Processing Advances in Wireless Communications (SPAWC '09), pp. 206–210, June 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. M. T. Chu, “Constructing a Hermitian Matrix from Its Diagonal Entries and Eigenvalues,” SIAM Journal on Matrix Analysis and Applications, vol. 16, pp. 207–217, 1995.
  9. H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics, John Wiley & Sons, 1998.
  10. S. Hu-yun, “Estimation of the eigenvalues of AB for A>0, B>0,” Linear Algebra and Its Applications, vol. 73, pp. 147–150, 1986. View at Scopus
  11. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, NY, USA, 1985.