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

Authors: Enoch Lu; I.-Tai Lu
Journal: Journal of Electrical and Computer Engineering
 (2012)

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## 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.


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## Body

## 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).AT, A*, A-1, 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 A≥B mean that A-B is positive definite and positive semidefinite, respectively. A°B denotes the elementwise product of Aand 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 isy=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 constraint(1)ei*(FF*)ei≤di,∀i.
Here, di>0,∀i. Note that (1) also results in the average total power being upper bounded by P =∑i=1ndi. Let S denote the set of all feasible F’s.


### 2.1. Problem Formulation

The goal of this paper is to designF 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*Rnn-1H and(2)Ψ(F,G)≜F*H*G*(GRnnG*)-1GHF.λmin(Ψ(F,G)) actually lower bounds the minimum distance between the symbol hypotheses. In addition,(3)Ψ(F,G)≤F*AF
with equality when G=F*H*(HFF*H*+Rnn)-1, that is, the MMSE decoder is used. Thus, we will choose G=F*H*(HFF*H*+Rnn)-1 and design the precoder F according to the optimization problem:(4)max⁡F∈Sμ(F),μ(F)≜λmin⁡(F*AF).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,(5)VΛ-1/2(Ptr⁡(Λ-1))1/2=argmaxtr⁡(FF*)=P,F∈Cn×nμ(F),
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 ofF

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,(6)μ(F)=z*FF*zz*A-1z≤x*FF*xx*A-1x,
where xis any nonzero n×1 vector. Define B=[bij]≜A-1 for convenience. Plugging in ei for x in (6) and using the constraint (1), it is then clear that, for all F∈S,(7)μ(F)≤min⁡idibii=min⁡i{d1b11,d2b22,…,dnbnn}≜dkbkk≜ρ.
That is, the cost function in (4) is upper bounded by ρ.The upper boundρ 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:(8)μ(Fρ)=ρ.
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*.


## 2.1. Problem Formulation

The goal of this paper is to designF 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*Rnn-1H and(2)Ψ(F,G)≜F*H*G*(GRnnG*)-1GHF.λmin(Ψ(F,G)) actually lower bounds the minimum distance between the symbol hypotheses. In addition,(3)Ψ(F,G)≤F*AF
with equality when G=F*H*(HFF*H*+Rnn)-1, that is, the MMSE decoder is used. Thus, we will choose G=F*H*(HFF*H*+Rnn)-1 and design the precoder F according to the optimization problem:(4)max⁡F∈Sμ(F),μ(F)≜λmin⁡(F*AF).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,(5)VΛ-1/2(Ptr⁡(Λ-1))1/2=argmaxtr⁡(FF*)=P,F∈Cn×nμ(F),
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 ofF

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,(6)μ(F)=z*FF*zz*A-1z≤x*FF*xx*A-1x,
where xis any nonzero n×1 vector. Define B=[bij]≜A-1 for convenience. Plugging in ei for x in (6) and using the constraint (1), it is then clear that, for all F∈S,(7)μ(F)≤min⁡idibii=min⁡i{d1b11,d2b22,…,dnbnn}≜dkbkk≜ρ.
That is, the cost function in (4) is upper bounded by ρ.The upper boundρ 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:(8)μ(Fρ)=ρ.
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

IfFρ 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,(9)AUρek=ρek⟺Uρek=ρA-1ek=ρBek.
In particular, the kth column of Uρ
must equal thekth-column of B multiplied by ρ. Consequently, if 1<k<n and  [b1Tbkkb2T]T denotes the kth column of B, then, necessarily,(10)Uρ=FρFρ*=[𝚺1ρb1Lρb1*dkρb2*L*ρb2𝚺2]>0,(11)Ik-1°𝚺1=diag⁡([d1⋯dk-1]),In-k°𝚺2=diag⁡([dk+1⋯dn]).
For the sake of clarity, the cases when k=1 or n are omitted in this paper.


### 3.1. Necessary Conditions forUρ (1st Representation)

As to be expected, the remaining unspecified elements ofUρ (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 if(12)𝚺1>0,𝚺2>0,𝚺2-L*Σ1-1L>0,(13)bkk2dk>b1*𝚺1-1b1+(b2-L*𝚺1-1b1)*(𝚺2-L*𝚺1-1L)-1(b2-L*𝚺1-1b1).
Since Σ2-L*Σ1-1L>0 (see (12)), the following necessary condition can be derived from (13): (14)bkk2dk>b1*𝚺1-1b1.
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:(15)bkk2dk>b1*b1λmax⁡(𝚺1)≥b1*b1tr⁡(𝚺1)=b1*b1d1+d2+⋯+dk-1.An alternate necessary and sufficient condition forUρ > 0 can be given by simply interchanging the subscripts 1 and 2, and L* and L in (12), (13). Thus, one can easily show that(16)bkk2dk>b2*𝚺2-1b2,bkk2dk>b2*b2dk+1+dk+2+⋯+dn
are also necessary.


### 3.2. Necessary Conditions forUρ (2nd Representation)

Use an appropriate invertible, symmetric, realn×n permutation matrix P to permute dk to the upper left corner:(17)P*UρP=[dkρb̃*ρb̃Q].
(The structures of P, Q, and b̃, 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))(18)Q-dkbkk2b̃b̃*>0.
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,…,n-1, are just the diagonal elements of Σ1, Σ2 permutated. Consequently, requiring that the qii, 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 bkk:(19)qii-dkbkk2|bĩ|2>0,(i=1,…,n-1).


## 3.1. Necessary Conditions forUρ (1st Representation)

As to be expected, the remaining unspecified elements ofUρ (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 if(12)𝚺1>0,𝚺2>0,𝚺2-L*Σ1-1L>0,(13)bkk2dk>b1*𝚺1-1b1+(b2-L*𝚺1-1b1)*(𝚺2-L*𝚺1-1L)-1(b2-L*𝚺1-1b1).
Since Σ2-L*Σ1-1L>0 (see (12)), the following necessary condition can be derived from (13): (14)bkk2dk>b1*𝚺1-1b1.
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:(15)bkk2dk>b1*b1λmax⁡(𝚺1)≥b1*b1tr⁡(𝚺1)=b1*b1d1+d2+⋯+dk-1.An alternate necessary and sufficient condition forUρ > 0 can be given by simply interchanging the subscripts 1 and 2, and L* and L in (12), (13). Thus, one can easily show that(16)bkk2dk>b2*𝚺2-1b2,bkk2dk>b2*b2dk+1+dk+2+⋯+dn
are also necessary.


## 3.2. Necessary Conditions forUρ (2nd Representation)

Use an appropriate invertible, symmetric, realn×n permutation matrix P to permute dk to the upper left corner:(17)P*UρP=[dkρb̃*ρb̃Q].
(The structures of P, Q, and b̃, 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))(18)Q-dkbkk2b̃b̃*>0.
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,…,n-1, are just the diagonal elements of Σ1, Σ2 permutated. Consequently, requiring that the qii, 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 bkk:(19)qii-dkbkk2|bĩ|2>0,(i=1,…,n-1).


## 4. Three Approaches for Obtaining Uρ

Each of the following three approaches seeks to find anUρ (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 matrixL 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, …, dk-1) 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 correspondingG) 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)

ChooseΣ1 and Σ2 as in R0A. If b1*Σ1-1b1=b2*Σ2-1b2, use another approach. If b1*Σ1-1b1>b2*Σ2-1b2, choose a rank one choice of L,(20)L=b1b2*b1*Σ1-1b1.
This L makes the right-hand side of (13) as small as possible and independent of Σ2. If b1*Σ1-1b1<b2*Σ2-1b2, choose an alternative rank one choice,(21)L=(b1b2*)(b2*𝚺2-1b2).
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 b1*Σ1-1b1>b2*Σ2-1b2 when (20) is used and that b1*Σ1-1b1<b2*Σ2-1b2 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 correspondingG) 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 forUρ using the 2nd representation of necessary conditions (Section 3.2). It is based on two facts. The first is that Q = Q* and(22)λmin⁡(Q)>ηmin⁡≜dkbkk-2b̃*b̃
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 alli (just apply the Schur-Horn Theorem [8]).Granted thatηmin<di, for all i, the approach is as follows. First choose some ε>0 such that ηmin+ε≡γ≤di, for alli. 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))(23)Uρ=(P*)-1[dkρb̃*ρb̃Q]P-1.
If ρ is the minimum eigenvalue of AUρ, obtain Fρ and a correspondingG fromUρ 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 HermitianQwith 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.


## 4.1. Rank 0 Approach (R0A)

In this simple approach, the matrixL 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, …, dk-1) 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 correspondingG) 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)

ChooseΣ1 and Σ2 as in R0A. If b1*Σ1-1b1=b2*Σ2-1b2, use another approach. If b1*Σ1-1b1>b2*Σ2-1b2, choose a rank one choice of L,(20)L=b1b2*b1*Σ1-1b1.
This L makes the right-hand side of (13) as small as possible and independent of Σ2. If b1*Σ1-1b1<b2*Σ2-1b2, choose an alternative rank one choice,(21)L=(b1b2*)(b2*𝚺2-1b2).
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 b1*Σ1-1b1>b2*Σ2-1b2 when (20) is used and that b1*Σ1-1b1<b2*Σ2-1b2 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 correspondingG) 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 forUρ using the 2nd representation of necessary conditions (Section 3.2). It is based on two facts. The first is that Q = Q* and(22)λmin⁡(Q)>ηmin⁡≜dkbkk-2b̃*b̃
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 alli (just apply the Schur-Horn Theorem [8]).Granted thatηmin<di, for all i, the approach is as follows. First choose some ε>0 such that ηmin+ε≡γ≤di, for alli. 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))(23)Uρ=(P*)-1[dkρb̃*ρb̃Q]P-1.
If ρ is the minimum eigenvalue of AUρ, obtain Fρ and a correspondingG fromUρ 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 HermitianQwith 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 aL, 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
(24)[𝚺1ρb1Lρb1*dk′ρb2*L*ρb2𝚺2],
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 thatdk′ is low enough, decompose Uρ to get Fρ (and a correspondingG). 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 i≠k. Indeed,(25)ek*(FρFρ*)ek=dk′<dk.
Thus, set F equal to Fρ.Loweringdk′ 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 Section4 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 forF

Two examples are given here, each of which corresponds to oneH 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=∑i=1ndi=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=∑i=1ndi, 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, andPA 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, theR0A’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.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 forF

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 matrixRnn 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 Fis 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.Figure 3
Eight-antenna example. Top: Histogram of the ratio betweendk′ (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 is(26)log2|I+HFF*H*Rnn-1|,
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⋯fn] and G=[g1T⋯gnT]T  are used is simply(27)|giHfi|2giRnngi*+∑j≠i|giHfj|2.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 isρ, the SINR of every data stream when the optimum solution is used.Figure 6
Eight-antenna example. Typical example of how the fix works.Figure5 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 Section2.1). 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.Section5 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.Figure 7
Four-antenna example. Top: histogram of the ratio betweendk′ (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 isρ, 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 ofdk′/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.


## 6.1. Demonstration of the Proposed Design Methodology forF

Two examples are given here, each of which corresponds to oneH 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=∑i=1ndi=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=∑i=1ndi, 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, andPA 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, theR0A’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.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 forF

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 matrixRnn 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 Fis 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.Figure 3
Eight-antenna example. Top: Histogram of the ratio betweendk′ (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 is(26)log2|I+HFF*H*Rnn-1|,
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⋯fn] and G=[g1T⋯gnT]T  are used is simply(27)|giHfi|2giRnngi*+∑j≠i|giHfj|2.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 isρ, the SINR of every data stream when the optimum solution is used.Figure 6
Eight-antenna example. Typical example of how the fix works.Figure5 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 Section2.1). 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.Section5 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.Figure 7
Four-antenna example. Top: histogram of the ratio betweendk′ (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 isρ, 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 ofdk′/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.



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*Source: 290390-2012-05-16.xml*