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

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., [1–7]). 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). 𝐀^𝑇, 𝐀^∗, 𝐀⁻¹, tr(A), E(A), 𝜆_max(A), and 𝜆_min(A) stand for the transpose, conjugate transpose, inverse, trace, expectation, maximum eigenvalue, and minimum eigenvalue of A, respectively. 𝜆_𝑖(A) stands for the 𝑖th eigenvalue of A when its eigenvalues are arranged in increasing order. I_r is the 𝑟×𝑟 identity matrix. 0 is the zero matrix of appropriate dimension. e_i 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 𝐀≥𝐁 mean that 𝐀−𝐁 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^∗)=𝐑_𝐧𝐧>𝟎. As the transmitter’s power must be constrained, the precoder F is required to satisfy the per-antenna power constraint𝐞^∗𝑖𝐅𝐅^∗𝐞_𝑖≤𝑑_𝑖,∀𝑖.(1) Here, 𝑑_𝑖>0,∀𝑖. Note that (1) also results in the average total power being upper bounded by P =∑^𝑛𝑖=1𝑑_𝑖. 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 𝐀≜𝐇^∗𝐑^−1𝐧𝐧𝐇 and𝚿(𝐅,𝐆)≜𝐅^∗𝐇^∗𝐆^∗𝐆𝐑_𝐧𝐧𝐆^∗⁻¹𝐆𝐇𝐅.(2)𝜆_min(Ψ(𝐅,𝐆)) actually lower bounds the minimum distance between the symbol hypotheses. In addition,𝚿(𝐅,𝐆)≤𝐅^∗𝐀𝐅(3) with equality when 𝐆=𝐅^∗𝐇^∗(𝐇𝐅𝐅^∗𝐇^∗+𝐑_𝐧𝐧)⁻¹, that is, the MMSE decoder is used. Thus, we will choose 𝐆=𝐅^∗𝐇^∗(𝐇𝐅𝐅^∗𝐇^∗+𝐑_𝐧𝐧)⁻¹ and design the precoder F according to the optimization problem:max_𝐅∈𝑆𝜇(𝐅),𝜇(𝐅)≜𝜆_min𝐅^∗𝐀𝐅.(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,𝐕𝚲^−1/2tr𝑃𝚲⁻¹^1/2=argmax_{tr𝐅𝐅∗=𝑃,𝐅∈ℂ𝑛×𝑛}𝜇(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 𝐅^∗𝐀𝐅 and 𝐀𝐅𝐅^∗ have the same eigenvalues, 𝐀𝐅𝐅^∗𝐳=𝜇(𝐅)𝐳 where z is an eigenvector of 𝐀𝐅𝐅^∗ generated by the minimum eigenvalue 𝜇(𝐅). Noting that 𝐀>𝟎,𝜇(𝐅)=𝐳^∗𝐅𝐅^∗𝐳𝐳^∗𝐀⁻¹𝐳≤𝐱^∗𝐅𝐅^∗𝐱𝐱^∗𝐀⁻¹𝐱,(6) where x is any nonzero 𝑛×1 vector. Define 𝐁=[𝑏_𝑖𝑗]≜𝐀⁻¹ for convenience. Plugging in e_i for x in (6) and using the constraint (1), it is then clear that, for all 𝐅∈𝑆,𝜇(𝐅)≤min_𝑖𝑑_𝑖𝑏_𝑖𝑖=min_𝑖𝑏𝑑¹₁₁,𝑑²𝑏₂₂,…,𝑑^𝑛𝑏_𝑛𝑛≜𝑏𝑑^𝑘_𝑘𝑘≜𝜌.(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 F_ρ∈S with maximum power (i.e., satisfies (1) with equality) and which reaches the upper bound ρ:𝜇𝐅_𝜌=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 F_ρ. If unsuccessful, we get F by implementing a fix for the approach used to search for F_ρ. 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 𝐔_𝜌=𝐅_𝜌𝐅^∗𝜌.

3. Necessary Structure and Conditions

If F_ρ exists, the transmit covariance matrix U_ρ would have a special structure and certain conditions would be true. Assume F_ρ exists. Then, observing from (6), e_k must be an eigenvector of AU_ρ paired with the eigenvalue ρ, that is,𝐀𝐔_𝜌𝐞_𝑘=𝜌𝐞_𝑘⟺𝐔_𝜌𝐞_𝑘=𝜌𝐀⁻¹𝐞_𝑘=𝜌𝐁𝐞_𝑘.(9) In particular, the kth column of U_ρ must equal the kth-column of B multiplied by ρ. Consequently, if 1<𝑘<𝑛 and [𝐛^𝑇1𝑏_𝑘𝑘𝐛^𝑇2]^𝑇 denotes the kth column of B, then, necessarily,𝐔_𝜌=𝐅_𝜌𝐅^∗𝜌=⎡⎢⎢⎢⎢⎣𝚺¹𝜌𝐛_∗1𝜌𝐛₁𝐋𝑑_𝑘𝜌𝐛^∗2𝐋^∗𝜌𝐛₂𝚺₂𝐈⎤⎥⎥⎥⎥⎦>𝟎,(10)_𝑘−1𝐈∘𝚺₁=diag𝑑₁⋯𝑑_𝑘−1_𝑛−𝑘,∘𝚺₂=diag𝑑_𝑘+1⋯𝑑_𝑛.(11) For the sake of clarity, the cases when k=1 or 𝑛 are omitted in this paper.

3.1. Necessary Conditions for 𝐔_𝜌 (1st Representation)

As to be expected, the remaining unspecified elements of U_ρ (i.e., L and the off diagonal elements of 𝚺₁ and 𝚺₂) are not arbitrary. By reducing U_ρ to direct sum form using elementary block row and column operations, it can be seen that U_ρ > 0 if and only if𝚺₁>𝟎,𝚺₂>𝟎,𝚺₂−𝐋^∗Σ¹⁻¹𝐋>𝟎,(12)𝑏^2𝑘𝑘𝑑_𝑘>𝐛^∗1𝚺¹⁻¹𝐛₁+𝐛₂−𝐋^∗𝚺¹⁻¹𝐛₁^∗𝚺₂−𝐋^∗𝚺¹⁻¹𝐋⁻¹𝐛₂−𝐋^∗𝚺¹⁻¹𝐛₁.(13) Since 𝚺₂−𝐋^∗𝚺¹⁻¹𝐋>𝟎 (see (12)), the following necessary condition can be derived from (13): 𝑏^2𝑘𝑘𝑑_𝑘>𝐛^∗1𝚺¹⁻¹𝐛₁.(14) Furthermore, since 𝚺₁>𝟎, a redundant though useful necessary condition can be derived for the antenna powers {𝑑₁,…,𝑑_𝑘}, which depends only on the known parameters b₁, and 𝑏_𝑘𝑘:𝑏^2𝑘𝑘𝑑_𝑘>𝐛^∗1𝐛₁𝜆_max𝚺₁≥𝐛^∗1𝐛₁tr𝚺₁=𝐛^∗1𝐛₁𝑑₁+𝑑₂+⋯+𝑑_𝑘−1.(15)

An alternate necessary and sufficient condition for U_ρ > 0 can be given by simply interchanging the subscripts 1 and 2, and L* and L in (12), (13). Thus, one can easily show that𝑏^2𝑘𝑘𝑑_𝑘𝑏_2𝑘𝑘>𝐛^∗2𝚺²⁻¹𝐛₂,𝑑_𝑘>𝐛^∗2𝐛₂𝑑_𝑘+1+𝑑_𝑘+2+⋯+𝑑_𝑛(16) are also necessary.

3.2. Necessary Conditions for 𝐔_𝜌 (2nd Representation)

Use an appropriate invertible, symmetric, real 𝑛×𝑛 permutation matrix P to permute 𝑑_𝑘 to the upper left corner:𝐏^∗𝐔_𝜌𝐏=⎡⎢⎢⎣𝑑^𝑘𝜌𝐛̃^∗𝜌𝐛̃𝐐⎤⎥⎥⎦.(17) (The structures of P, Q, and 𝐛̃, are omitted here). It turns out that 𝐔_𝜌 > 0 if and only if P*U_ρP > 0. Thus, U_ρ > 0 if and only if (noting that 𝑑_𝑘>0 given in (1))𝐐−𝑑^𝑘𝑏^2𝑘𝑘𝐛̃𝐛̃^∗>𝟎.(18) Since both (12)-(13), and (18) are necessary and sufficient for U_ρ > 0, (12)-(13) and (18) are equivalent to each other. Conveniently, the diagonal elements of 𝐐,𝑞_𝑖𝑖, 𝑖=1,…,𝑛−1, are just the diagonal elements of 𝚺₁, 𝚺₂ permutated. Consequently, requiring that the 𝑞_𝑖𝑖, 𝑖=1,…,𝑛−1, 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, 𝐛̃, and 𝑏_𝑘𝑘:𝑞_𝑖𝑖−𝑑^𝑘𝑏^2𝑘𝑘||𝑏_𝑖||²>0,(𝑖=1,…,𝑛−1).(19)

4. Three Approaches for Obtaining U_ρ

Each of the following three approaches seeks to find an U_ρ (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, …, 𝑑_𝑘−1) and (𝑑_𝑘+1, …, 𝑑_𝑛), respectively. Such a choice for L, 𝚺₁, and 𝚺₂ automatically satisfies (11), (12). If the last remaining necessary condition (13) is satisfied, construct U_ρ using (10) and check whether ρ is the minimum eigenvalue of AU_ρ. If both of these conditions are satisfied, decompose U_ρ to get a F_ρ (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 U_ρ to F_ρ is not unique; indeed, using F_ρ 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 𝐛^∗1𝚺¹⁻¹𝐛₁=𝐛^∗2𝚺²⁻¹𝐛₂, use another approach. If 𝐛^∗1𝚺¹⁻¹𝐛₁>𝐛^∗2𝚺²⁻¹𝐛₂, choose a rank one choice of L,𝐋=𝐛¹𝐛^∗2𝐛^∗1Σ¹⁻¹𝐛₁.(20) This L makes the right-hand side of (13) as small as possible and independent of 𝚺₂. If 𝐛^∗1𝚺¹⁻¹𝐛₁<𝐛^∗2𝚺²⁻¹𝐛₂, choose an alternative rank one choice,𝐋=𝐛¹𝐛^∗2𝐛^∗2𝚺²⁻¹𝐛₂.(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 𝚺₂−𝐋^∗𝚺¹⁻¹𝐋>𝟎 in (12) is also satisfied—use the fact that 𝐛^∗1𝚺¹⁻¹𝐛₁>𝐛^∗2𝚺²⁻¹𝐛₂ when (20) is used and that 𝐛^∗1𝚺¹⁻¹𝐛₁<𝐛^∗2𝚺²⁻¹𝐛₂ when (21) is used. However, (13) needs to be checked. In addition, whether ρ is the minimum eigenvalue of AU_ρ also needs to be checked. Same as in R0A, if both conditions are satisfied, decompose U_ρ to get a F_ρ (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 U_ρ using the 2nd representation of necessary conditions (Section 3.2). It is based on two facts. The first is that Q = Q* and𝜆_min(𝐐)>𝜂_min≜𝑑_𝑘𝑏^−2𝑘𝑘𝐛̃^∗𝐛̃(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 𝜂_min<𝑑_𝑖, for all i (just apply the Schur-Horn Theorem [8]).

Granted that 𝜂_min<𝑑_𝑖, for all i, the approach is as follows. First choose some 𝜀>0 such that 𝜂_min+𝜀≡𝛾≤𝑑_𝑖, for all i. Next, find a Hermitian Q whose diagonal entries are the correct antenna powers, {𝑞_𝑖𝑖}, and whose eigenvalues are lower bounded by γ. Once a Q is found, construct (in light of (17))𝐔_𝜌=𝐏^∗⁻¹⎡⎢⎢⎣𝑑_𝑘𝜌𝐛̃^∗𝜌𝐛̃𝐐⎤⎥⎥⎦𝐏⁻¹.(23) If ρ is the minimum eigenvalue of AU_ρ, obtain F_ρ and a corresponding G from U_ρ 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 {𝑞_𝑖𝑖} 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 = 𝑑_𝑘/𝑏_𝑘𝑘 is greater than the smallest eigenvalue of AU_ρ, proceed as following. First, temporarily change the per-antenna power constraint for the kth antenna by replacing 𝑑_𝑘 by 𝑑^𝑘. This makes ρ become 𝑑^𝑘/𝑏_𝑘𝑘, U_ρ become ⎡⎢⎢⎢⎢⎣𝜌𝚺𝐛_1∗1𝜌𝐛₁𝐋𝑑^𝑘𝜌𝐛^∗2𝐋^∗𝜌𝐛₂𝚺₂⎤⎥⎥⎥⎥⎦,(24) and so forth. Second, lower 𝑑^𝑘 (maintaining 𝑑^𝑘>0) until (13) holds. Then, continue lowering 𝑑^𝑘 (maintaining 𝑑^𝑘 > 0) and thus ρ = 𝑑^𝑘/𝑏_𝑘𝑘 until ρ is the smallest eigenvalue of AU_ρ. As Appendix C shows, one can always lower 𝑑^𝑘 until this happens. Using this fix, (8) is thus obtained—granted, for a lower power constraint.

Lastly, now that 𝑑^𝑘 is low enough, decompose U_ρ to get F_ρ (and a corresponding G). This F_ρ is full rank since (12-13) hold. In addition, it satisfies the inequality per-antenna power constraint (1) with the true 𝑑_𝑘 and the 𝑑_𝑖, for all 𝑖≠𝑘. Indeed,𝐞^∗𝑘𝐅_𝜌𝐅^∗𝜌𝐞_𝑘=𝑑_𝑘<𝑑_𝑘.(25) Thus, set F equal to F_ρ.

Lowering 𝑑^𝑘 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 𝑑^𝑘 further, decomposing U_ρ to get F at that point would yield a legitimate precoder. So, why continue to lower 𝑑^𝑘? The reason is that it is observed numerically that continuing to lower 𝑑^𝑘 actually raises the minimum eigenvalue of AU_ρ. 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 𝐑_𝐧𝐧 (i.e., one A). Without loss of generality, consider 10 antennas (𝑛=10) with identical power constraints (𝑑_𝑖 = 10, 𝑖=1,…,𝑛). Thus, maximum allowable total power 𝑃=∑^𝑛𝑖=1𝑑_𝑖=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=1𝑑_𝑖, 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.,𝜆_min(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.

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., 𝑑^𝑘/𝑏_𝑘𝑘) 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 𝑑_𝑘 is approximately equal to 4.4138.

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 𝐑_𝐧𝐧, P, and H, we obtain it by the following methodology. We do not specify the per-antenna power constraints {𝑑_𝑖} at the beginning. Instead, we calculate the closed-form precoder in (5) and set its antenna powers as the {𝑑_𝑖}. In other words, if F_o denotes the precoder from (5), we set 𝐞^∗𝑖(𝐅_𝑜𝐅^∗𝑜)𝐞_𝑖 as 𝑑_𝑖, 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 𝐑_𝐧𝐧, H, and {𝑑_𝑖} 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 𝐑_𝐧𝐧 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 {𝑑_𝑖} 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 𝑑^𝑘 (after the fix) and 𝑑_𝑘 for the transmit SNR 6 dB case. Since 𝑑^𝑘 is smaller than 𝑑_𝑘, 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 𝑑^𝑘/𝑏_𝑘𝑘. Since the optimum solution obtains the cost function value of ρ = 𝑑_𝑘/𝑏_𝑘𝑘, the top histogram of Figure 3 also shows how suboptimal the proposed algorithm is.

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+𝐇𝐅𝐅^∗𝐇^∗𝐑^−1𝐧𝐧||,(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=[𝐟₁⋯𝐟_𝑛] and 𝐆=[𝐠^𝑇1⋯𝐠^𝑇𝑛]^𝑇 are used is simply||𝐠_𝑖𝐇𝐟_𝑖||²𝐠_𝑖𝐑_𝐧𝐧𝐠^∗𝑖+∑_𝑗≠𝑖||𝐠_𝑖𝐇𝐟_𝑗||².(27)

Figure 4 

Eight-antenna example. Mutual information for the optimum precoder and the proposed precoder.

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.

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 𝑑^𝑘 actually raises the minimum eigenvalue of AU_ρ. Figure 6 shows a typical plot of the results of continuing to lower 𝑑^𝑘 for a channel realization. At each iteration, 𝑑^𝑘 is lowered by one percent. In the figure, ρ = 𝑑^𝑘/𝑏_𝑘𝑘 and U_ρ is given by (23) in accordance with the notation in Section 5. Indeed, the minimum eigenvalue of AU_ρ increases as 𝑑^𝑘, and thus ρ = 𝑑^𝑘/𝑏_𝑘𝑘, 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(𝐀𝐔_𝜌) for the last iteration may be smaller than 𝜆₁(𝐀𝐔_𝜌) for the second to last iteration. This may be due to overshooting since the step size for 𝑑^𝑘 is fixed.

Next, consider the 4 antennas case (𝑛=4). Figure 7 shows the histogram of the ratio between 𝑑^𝑘 (after the fix) and 𝑑_𝑘. 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.

Figure 7 

Four-antenna example. Top: histogram of the ratio between 𝑑𝑘 (after the fix) and 𝑑𝑘. Bottom: histogram of the ratio between the total power of the proposed F and P.

Figure 8 

Four-antenna example. Mutual information for the optimum precoder and the proposed precoder.

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 𝑑^𝑘/𝑑_𝑘 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 (𝑛−1)×(𝑛−1) matrix Y= γI_n-1 + X where X ≥ 0. For any X ≥ 0, Y is Hermitian and has eigenvalues lower bounded by γ. So, as long asdiag(𝐗)=𝑞₁₁−𝛾,𝑞₂₂−𝛾,…,𝑞_{𝑛−1,𝑛−1}−𝛾,(A.1)Y is a valid choice of Q.

To find a 𝐗≥𝟎 that satisfies (A.1) is not difficult. First, choose any (𝑛−1)×(𝑛−1) matrix 𝐙≥𝟎 with positive diagonal entries {𝑧_𝑖𝑖}. Then, form𝐑≡diag𝑞₁₁−𝛾𝑧₁₁,𝑞₂₂−𝛾𝑧₂₂,…,𝑞_{𝑛−1,𝑛−1}−𝛾𝑧_{𝑛−1,𝑛−1}.(A.2) The product RZR is a valid choice of X. Consequently, Y= γI_n-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 (𝑛−1)×(𝑛−1) unitary matrix V⁽⁰⁾ and 𝑛−1 real numbers, 𝜂₁≥⋯≥𝜂_𝑛−1≥𝛾. It then creates the initial guess at Q:𝐐⁽⁰⁾=𝐕⁽⁰⁾⋅diag𝜂₁,…,𝜂_𝑛−1⋅𝐕⁽⁰⁾^∗.(B.1) At the 𝑗th iteration step (𝑗>1), it proceeds as follows.(i)Force the diagonal elements of 𝐐^(𝑗−1) to be equal to {𝑞_𝑖𝑖}.(ii)Then, decompose 𝐐^(𝑗−1) as 𝐐^(𝑗−1)=𝐕^(𝑗−1)⋅diag𝜉₁,…,𝜉_𝑛−1⋅𝐕^(𝑗−1)^∗(B.2) to get the unitary matrix 𝐕^(𝑗−1) and the real numbers 𝜉₁≥⋯≥𝜉_𝑛−1.(iii)If 𝜉_𝑛−1 ≥γ, set Q equal to 𝐐^(𝑗−1) and stop iterating. If not, let 𝜉_𝑖 = max(𝜉_𝑖, γ), for all i, and create𝐐^(𝑗)=𝐕^(𝑗−1)⋅diag𝜉₁,…,𝜉_𝑛−1⋅𝐕^(𝑗−1)^∗.(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 ℂ^𝑛×𝑛:𝐓∈ℂ^𝑛×𝑛∣diag(𝐓)=𝑞₁₁⋯𝑞_{𝑛−1,𝑛−1},𝐓=𝐓^∗,(B.4)𝐕Θ𝐕Θ^∗∣=𝐕di⁻¹ag=𝐕𝜃^∗₁∈ℂ^𝑛×𝑛,,…,𝜃_𝑛∈ℝ^𝑛×𝑛,𝜃_𝑖≥𝛾(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 𝑑_𝑘 and thus ρ(maintaining 𝑑_𝑘, 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 AU_ρ (use (9)). The second part is to find a (𝑛−1)×(𝑛−1) matrix whose eigenvalues are precisely AU_ρ’s other 𝑛−1 eigenvalues. To this end, introduce the partition𝐀=⎡⎢⎢⎢⎢⎣𝐂¹¹𝐜₁₂𝐂₁₃𝐜^∗12𝑐₂₂𝐜₂₃𝐂^∗13𝐜^∗23𝐂₃₃⎤⎥⎥⎥⎥⎦,(C.1) where 𝑐₂₂ is the kkth element of A. In addition, note that A times the kth column of U_ρ is equal to ρe_k due to 𝐁=𝐀⁻¹. With some straightforward steps, it can thus be seen that𝐌𝑑_𝑘=⎡⎢⎢⎣𝐂¹¹𝐂₁₃𝐂^∗13𝐂₃₃⎤⎥⎥⎦⎡⎢⎢⎣𝐋𝚺_1∗𝐋𝚺₂⎤⎥⎥⎦+𝑑_𝑘𝑏_𝑘𝑘⎡⎢⎢⎣𝐜₁₂𝐛^∗1𝐜₁₂𝐛^∗2𝐜^∗23𝐛^∗1𝐜^∗23𝐛^∗2⎤⎥⎥⎦(C.2) is a suitable candidate matrix. The third part is to realize that the eigenvalues of⎡⎢⎢⎣𝐂₁₁𝐂₁₃𝐂^∗13𝐂₃₃⎤⎥⎥⎦⎡⎢⎢⎣𝐋𝚺_1∗𝐋𝚺₂⎤⎥⎥⎦,(C.3) the first term in M(𝑑_𝑘), are positive and independent of 𝑑_𝑘. 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 𝑑_𝑘 and thus ρ (maintaining 𝑑_𝑘,ρ > 0) until all the eigenvalues of M(𝑑_𝑘) are ≥ ρ.

D.

Let F_o denote the precoder from (5). As it is an optimal solution to the optimization problem in (5), tr{𝐅_𝑜𝐅^∗𝑜}=𝑃. But, what are its antenna powers? It turns out that the 𝑖th antenna has power 𝑏_𝑖𝑖𝑃/tr(Λ⁻¹) because𝐅_𝑜𝐅^∗𝑜=𝐕Λ^−𝟏/𝟐𝚲^−𝟏/𝟐𝐕^∗tr𝑃𝚲^−𝟏=𝐁tr𝑃𝚲^−𝟏.(D.1) Here, we used the fact that 𝐁=𝐀⁻¹ and 𝐀=𝐕Λ𝐕^∗.

Let us say, for all i, we set 𝑑_𝑖 equal to F_o’s 𝑖th antenna power:𝑑_𝑖=𝑏_𝑖𝑖𝑃tr𝚲^−𝟏.(D.2) Then from (7) and (D.2), we have the upper bound𝜌=𝑃tr𝚲^−𝟏.(D.3) It turns out that F_o reaches this upper bound. Direct computation shows this. From (5), we have 𝐅^∗𝑜𝐀𝐅_𝑜=Λ^−1/2𝐕^∗𝐕𝚲𝐕^∗𝐕𝚲^−1/2=𝐈𝑃tr𝑃𝚲^−𝟏tr𝚲^−𝟏.(D.4) From (D.3) and (D.4), we have𝜆_min𝐅^∗𝑜𝐀𝐅_𝑜=𝑃tr𝚲^−𝟏=𝜌.(D.5) In summary, if the 𝑑₁,…,𝑑_𝑛 of the per-antenna power constraint (1) are defined using the antenna powers of F_o, F_o 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.