The Capacity and Optimal Signaling for Gaussian MIMO Channels Under Interference Constraints (full version)

02/18/2020 ∙ by Sergey Loyka, et al. ∙ uOttawa 0

Gaussian MIMO channel under total transmit and interference power constraints (TPC and IPC) is considered. A closed-form solution for the optimal transmit covariance matrix in the general case is obtained using the KKT-based approach (up to dual variables). While closed-from solutions for optimal dual variables are possible in special cases, an iterative bisection algorithm (IBA) is proposed to find the optimal dual variables in the general case and its convergence is proved for some special cases. Numerical experiments illustrate its efficient performance. Bounds for the optimal dual variables are given, which facilitate numerical solutions. An interplay between the TPC and IPC is studied, including the transition from power-limited to interference-limited regimes as the total transmit power increases. Sufficient and necessary conditions for each constraint to be redundant are given. A number of explicit closed-form solutions are obtained, including full-rank and rank-1 (beamforming) cases as well as the case of identical eigenvectors (typical for massive MIMO settings). A bound on the rank of optimal covariance is established. A number of unusual properties of optimal covariance matrix are pointed out.

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I Introduction

Growing volume of high-rate mobile wireless traffic stimulated active development of 5G standards and systems. Due to very high expectations, several new key technologies have been identified to meet those demands, including massive MIMO, millimeter waves (mmWave) and non-orthogonal multiple-access [1]. The ultimate goal is to increase significantly the available bandwidth as well as spectral efficiency to meet the growing traffic demands. However, aggressive frequency re-use and non-orthogonal access schemes can potentially generate significant amount of inter-user interference, which thus has to be carefully managed [2]-[5]. This is somewhat similar to cognitive radio (CR) systems, which also emerged as a powerful approach to exploit underutilized spectrum and hence possibly resolve the spectrum scarcity problem [6]. In both settings, allowing spectrum re-use calls for a careful management of possible interference. In this respect, multi-antenna (MIMO) systems have significant potential due to their significant signal processing capabilities, including interference cancellation and precoding [7], which can also be done in an adaptive and distributed manner [13]. A promising approach is to limit interference to primary receivers (PR) by properly designing secondary transmitters (Tx) while exploiting their multi-antenna capabilities.

The capacity and optimal signalling for the Gaussian MIMO channel under the total power constraints (TPC) is well-known: the optimal (capacity-achieving) signaling is Gaussian and, under the TPC, is on the eigenvectors of the channel with power allocation to the eigenmodes given by the water-filling (WF) [7]-[9]. Under per-antenna power constraints (PAC), in addition or instead of the TPC, Gaussian signalling is still optimal but not on the channel eigenvectors anymore so that the standard water-filling solution over the channel eigenmodes does not apply [10][11]. Much less is known under the added interference power constraint (IPC), which limits the power of interference induced by the secondary transmitter to a primary receiver. A game-theoretic approach to this problem was proposed in [13], where a fixed-point equation was formulated from which the optimal covariance matrix can in principle be determined. Unfortunately, no closed-form solution is known for this equation. In addition, this approach is limited in the following respects: the channel to the primary receiver is required to be full-rank (hence excluding the important case of single-antenna devices communicating to a multi-antenna base station or, in general, the cases where the number of Rx antennas is less than the number of Tx antennas - typical for massive MIMO downlink); the TPC is not included explicitly (rather, being ”absorbed” into the IPC), hence eliminating the important case of inactive IPC (since this is the only explicit constraint); consequently, no interplay between the TPC and the IPC can be studied.

Earlier studies on cognitive radio MIMO system optimization under interference constraint using game-theoretic approach are extended to the case of channel uncertainty in [14] by developing a number of numerical algorithms for Tx optimization under global interference constraints. Due to the non-convex nature of the original problem, a number of approximate and sub-optimal approaches are adopted, for which provable convergence to a global optimum is out of reach. No closed-form solutions are known for this setting either. Weighted sum-rate maximization in multiuser MISO channel is considered in [15] under interference constraints and numerically-efficient algorithms for Tx optimization based on zero-forcing beamforming are developed. No closed-form solutions are obtained for this problem. Gaussian MIMO broadcast (BC) and multiple-access channels (MAC) are considered in [16] under general linear Tx covariance constraint, which can also be interpreted as interference constraint, and the earlier BC-MAC duality result is extended to this more general setting. However, no closed-form solutions are obtained for an optimal Tx covariance matrix. In the related context of physical-layer security, the secrecy capacity of the Gaussian MIMO wiretap channel under interference constraints has been characterized in [12] as a non-convex maximization problem (over feasible Tx covariance matrices under the TPC and IPC) or as a convex-concave max-min problem (where ”min” is over noise covariance matrices of a genie-aided channel), for which no closed-form solution is known in general.

Unlike most of the studies above, we concentrate here on analysis of the problem and obtain a number of closed-form solutions, which are validated via numerical experiments. This provides deeper understanding of the problem and a number of insights unavailable from numerical algorithms alone. Specifically, we obtain closed-form solutions for an optimal covariance matrix of the Gaussian MIMO channel under the TPC and the IPC using the KKT conditions. Both constraints are included explicitly and hence anyone is allowed to be inactive. This allows us to study the interplay between the power and interference constraints and, in particular, the transition from power-limited to interference limited regimes as the Tx power increases. As an added benefit, no limitations is placed on the rank of the channel to the PR, so that the number of antennas of the PR can be any. Under the added IPC, independent signaling is shown to be sub-optimal for parallel channels to the intended receiver (Rx), unless the PR channels are also parallel or if the IPC is inactive. These results are also extended to multiple IPCs.

Optimal signaling for the Gaussian MIMO channel under the TPC and the IPC has been also studied in [17][18] using the dual problem approach, and was later extended to multi-user settings in [19]. However, constraint matrices are required to be full-rank and no closed-form solution was obtained for optimal dual variables. Hence, various numerical algorithms or sub-optimal solutions were proposed (e.g. partial channel projection). Similar problem has been also considered in [20] under multiple linear constraints at the transmitter and an iterative numerical algorithm was developed via a min-max reformulation of the problem. However, no closed-form solution was obtained and the properties of optimal signaling as well as those of the capacity were not studied. Here, a closed-form solution for an optimal covariance matrix is obtained in the general case (up to dual variables), its properties are studied and a number of more explicit solutions are obtained in some special cases (including explicit solutions for dual variables). Our KKT-based approach does not require full-rank constraint matrices and includes explicit equations for the optimal dual variables, which can be solved efficiently. To this end, we propose an iterative (gradient-free) bisection algorithm (IBA) and prove its convergence. Numerical experiments demonstrate its efficient performance. In some cases, our KKT-based approach leads to closed-form solutions for the optimal dual variables, including full-rank and rank-1 (beamforming) solutions. Bounds to the optimal dual variables are derived, which facilitate numerical solutions. Properties of the optimal Tx covariance as a function of dual variables are explored: the total Tx power as well as interference power are shown to be decreasing functions of dual variables, which is an important part in the proof of the IBA convergence.

The above solutions for optimal covariance posses a number of unusual properties (not found in the standard WF procedure), namely: an optimal covariance is not necessarily unique; its rank can exceed the main channel rank; the TPC can be inactive; signaling on the main channel eigenmodes is not optimal anymore. Ultimately, these are due to an interplay between the TPC and the IPC.

A simple rank condition is given to characterize the cases where spectrum sharing is possible for any interference power constraint. In general, the primary user has a major impact on the capacity at high SNR while being negligible at low SNR. The high-SNR behaviour of the capacity is qualitatively determined by the null space of the PR’s channel matrix.

The presented closed-form solutions of optimal signaling can be used directly in massive MIMO settings. Since numerical complexity of generic convex solvers can be prohibitively large for massive MIMO (in general, it scales as with the number of antennas), the above analytical solutions are a valuable low-complexity alternative.

In all considered cases, optimal Tx covariance matrices are significantly different from those of the standard Gaussian MIMO channel under the TPC and/or the PAC [10][11], and from those of the wiretap channel in [12]. In the latter two cases, optimal Tx covariance matrix remains unknown in the general case while some special cases have been solved.

Finally, it should be pointed out that the channel model we consider here (the standard point-to-point Gaussian MIMO channel under interference constraints at the transmitter) is different from the Gaussian interference channel (G-IC) where multi-user interference is present at each receiver and there is no interference constraint at the transmitters, as in e.g. [21]-[29]. In the latter case, the capacity and optimal signaling are not known in general, even for the 2-user SISO channel [21] (it is not even known whether Gaussian signaling is optimal in general), except for some special cases, such as strong and weak interference regimes [22][23], so that various bounds [24][25] and ad-hoc signaling techniques [26]-[29] are used instead. On the contrary, optimal signaling is known to be Guassian for the channel model considered here and the capacity can be expressed as an optimization problem over all feasible transmit covariance matrices. An analytical solution of this problem in the general case and its properties are the main contributions of the present paper.

Notations: bold capitals () denote matrices while bold lower-case letters (

) denote column vectors;

is the Hermitian conjugation of ; means that is positive semi-definite; denote determinant, trace and rank of , respectively; is

-th eigenvalue of

; unless indicated otherwise, eigenvalues are in decreasing order, ; denotes ceiling, while is the positive part of ; and denote the range and null space of while is its Moore-Penrose pseudo-inverse; is statistical expectation.

Ii Channel Model

Let us consider the standard discrete-time model of the Gaussian MIMO channel:

(1)

where and are the received and transmitted signals, noise and channel matrix, of dimensionality , , , and , respectively, where

are the number of Rx and Tx antennas. This is illustrated in Fig. 1. The noise is assumed to be complex Gaussian with zero mean and unit variance, so that the SNR equals to the signal power. A complex-valued channel model is assumed throughout the paper, with full channel state information available both at the transmitter and the receiver. Gaussian signaling is known to be optimal in this setting

[7]-[9] so that finding the channel capacity amounts to finding an optimal transmit covariance matrix , which can be expressed as the following optimization problem (P1):

(2)

where , , is the Tx covariance and is the constraint set. In the case of the total power constraint (TPC) only, it takes the form

(3)

where is the maximum total Tx power. The solution to this problem is well-known: optimal signaling is on the eigenmodes of , so that they are also the eigenmodes of optimal covariance , and the optimal power allocation is via the water-filling (WF). This solution can be compactly expressed as follows:

(4)

where is the ”water” level found from the total power constraint and denotes positive eigenmodes of Hermitian matrix :

(5)

where are -th eigenvalue and eigenvector of . Note that this definition allows in (4) to be singular, since the respective summation (as in (5)) includes only strictly-positive modes: , where are -th eigenvalue and eigenvector of , so that are excluded from the summation due to .

In a multi-user system, there is a 2nd channel from the Tx to an external (primary) user U, see Fig. 1,

(6)

where is the matrix of the Tx-U channel (see Fig. 1); all vector/matrix dimensions are equal to the respective number of antennas. There is a limit on how much interference the Tx can induce (via ) to the user U:

(7)

where is the maximum acceptable interference power and the left-hand side is the actual interference power at user U. In this setting, the constraint set becomes

(8)

where ; this is extended to multiple IPCs in Section IX. The IPC in (8) is consistent with the respective per-user constraints in the CR setting in [13]-[17], with the relay system setting in [18] as well as with the wiretap channel setting in [12]. The Gaussian signalling is still optimal in this setting and the capacity subject to the TPC and IPC can still be expressed as in (2) but the optimal covariance is not anymore, as discussed in the next section.

Fig. 1: A block diagram of Gaussian MIMO channel under interference constraint. and are the channel matrices to the Rx and an external (primary) user U respectively.

Iii Optimal Signalling Under Interference Constraint

The following Theorem gives a closed-form solution for the optimal Tx covariance matrix under the TPC and the IPC in (8) in the general case.

Theorem 1.

Consider the capacity of the Gaussian MIMO channel in (2) under the joint TPC and IPC in (8),

(9)

The optimal Tx covariance matrix to achieve the capacity can be expressed as follows:

(10)

where ; is Moore-Penrose pseudo-inverse of ; are Lagrange multipliers (dual variables) responsible for the total Tx and interference power constraints found as solutions of the following non-linear equations:

(11)

subject to , . The capacity can be expressed as follows:

(12)

where .

Proof.

See Appendix. ∎

As indicated in Section IX, this solution can be easily extended to multiple IPCs. When is full-rank, [31] and in (10) reduces to the respective solutions in [17][18].

Next, we explore some general properties of the capacity. It is well-known that, without the IPC, grows unbounded as increases, as (assuming ). This, however, is not necessarily the case under the IPC with fixed . The following proposition gives sufficient and necessary conditions when it is indeed the case.

Proposition 1.

Let be fixed. Then, the capacity grows unbounded as increases, i.e. as , if and only if .

Proof.

See Appendix. ∎

Since the condition of this Proposition is both sufficient and necessary for the unbounded growth of the capacity, it gives the exhaustive characterization of all the cases where such growth is possible. In practical terms, those cases represent the scenarios where any high spectral efficiency is achievable given enough power budget. On the other hand, if , then very high spectral efficiency cannot be achieved even with unlimited power budget, due to the dominance of the IPC. It can be seen that the condition holds if , and hence the capacity grows unbounded with under the latter condition.

In the standard Gaussian MIMO channel without the IPC, if either or/and , i.e. in a trivial way. On the other hand, in the same channel under the TPC and IPC, the capacity can be zero in non-trivial ways, as the following proposition shows. In practical terms, this characterizes the cases where interference constraints of primary users rule out any positive rate of a given user and, hence, spectrum sharing is not possible.

Proposition 2.

Consider the Gaussian MIMO channel under the TPC and IPC and let . Its capacity is zero if and only if and .

Proof.

To prove the ”if” part, observe that implies that (since has positive eigenvalues, ) so that and hence so that for any feasible . Hence, .

To prove the ”only if” part, assume first that and set , where . Note that is feasible: and . Furthermore,

(13)

and hence is necessary for . To show that is necessary as well, assume that , which implies that . Now set , for which , so it is feasible and

(14)

so that is necessary for . ∎

Note that the condition is equivalent to zero-forcing transmission, i.e. the capacity is zero only if the ZF transmission is required; otherwise, . The condition cannot be satisfied if and hence under the latter condition, which is also sufficient for unbounded growth of the capacity with . This is summarized below.

Corollary 1.

If , then

1. and .

2. as

Thus, the condition represents favorable scenarios where spectrum sharing is possible for any and arbitrary large capacity can be attained given enough power budget.

It should be pointed out that (11) in Theorem 1 allows anyone of the dual variables to be inactive (i.e. or , but not simultaneously), unlike the standard WF solution, where the TPC is always active. While it is not feasible to find dual variables in a closed form in general (since (11) is a system of coupled non-linear equations), they can be found in such form in some special cases, as the next sections show. Section VIII will develop an iterative bisection algorithm (IBA) to find the optimal dual variables in the general case with any desired accuracy and prove its convergence.

Iv Full-rank solutions

While Theorem 1 establishes a closed-form solution for optimal covariance in the general case, it is expressed via dual variables for which no closed-form solution is known in general so they have to be found numerically using (11). This limits insights significantly. In this section, we explore the cases when the optimal covariance is of full rank and obtain respective closed-form solutions. First, we consider a transmit power-limited regime, where the IPC is redundant and hence the TPC is active.

Proposition 3.

Let and be bounded as follows:

(15)

then , i.e. the IPC is redundant111i.e. can be ommited without affecting the capacity, which corresponds to ., is of full-rank and is given by:

(16)

where . The capacity can be expressed as

(17)
Proof.

It follows from (10) that, if is full rank, then so is and

(18)

(since it is of full rank) and hence

(19)

The full-rank condition is equivalent to

(20)

where we used the standard tools of matrix analysis [31][32]. When the IPC is redundant, from which (16) and

(21)

follow. It remains to establish (15). To this end, implies (active TPC) and (from (IV)) together with (21) implies 1st inequality in (15). 2nd inequality in (15) ensures that the IPC is redundant: and . Indeed, the capacity under the joint constraints (TPC+IPC) cannot exceed that under the TPC alone, for which the optimal covariance is as in (16). However, under 2nd inequality in (15), satisfies and hence is feasible under the joint constraints as well so that (i) both capacities are equal and (ii) in (16) is also optimal under the joint constraints, which corresponds to (so that the IPC can be removed without affecting the capacity). (17) is obtained by using (16) in . ∎

Next, we consider an interference-limited regime, where the TPC is redundant and hence the IPC is active.

Proposition 4.

Let and be bounded as follows:

(22)

then , i.e. the TPC is redundant, is of full-rank and is given by:

(23)

where . The capacity can be expressed as

(24)
Proof.

When the TPC is redundant, and (23) with

(25)

follow from (10) in the same way as for Proposition 3. 1st condition in (22) follows from , which, using (IV), is equivalent to

(26)

2nd condition in (22) ensures that the TPC is redundant since the Tx power is sufficiently large: and . ∎

It is clear from (17) and (24) that the latter represents an interference-limited scenario while the former - a Tx power-limited one.

It follows from the proof of Proposition 3 that a general full-rank solution is given by

(27)

and the respective capacity is , which holds if . If neither (15) nor (22) hold, then both power constraints are active: ; closed-form solutions for in this case are not known and they have to be found numerically from (11). Note that in (27) generalizes the respective WF full-rank solution (no IPC) to the interference-constrained environment, where takes the role of .

While the optimal covariance in (16) is the same as the standard WF over the eigenmodes of (subject to the power constraint only), its range of validity is different: while the standard WF solution is of full rank under only the lower bound in (15) (this can be obtained by setting so that only the lower bound remains), i.e. the WF optimal covariance is of full rank for all sufficiently high power/SNR, the interference constraint also imposes the upper bound in (15) and hence its full-rank solution will not hold if is too high, a remarkable difference to the standard WF.

Next, we explore the case where is of rank 1. This models the case when a primary user has a single-antenna receiver or when its channel is a keyhole channel, see e.g. [33][34].

Proposition 5.

Let be of full rank and be of rank-1, so that , where and are its active eigenvalue and eigenvector. If

(28)
(29)

then the IPC is redundant, the optimal covariance is of full rank and is given by the standard WF solution,

(30)

where .

If

(31)
(32)

then the IPC and TPC are active, the optimal covariance is of full rank and is given by

(33)

where , and are found from (11):

(34)
Proof.

See Appendix. ∎

Note that the 1st two terms in (33) represent the standard WF solution while the last term is a correction due to the IPC, which is reminiscent of a partial null forming in an adaptive antenna array, see e.g. [36].

V Inactive Constrains

It is straightforward to see both constraints cannot be inactive at the same time (since the capacity is a strictly increasing function of the Tx power without the IPC). In this section, we explore the scenarios when one of the two constraints is redundant222”inactive” implies ”redundant” but the converse is not true: for example, inactive TPC means and this implies (from complementary slackness) so that it is also redundant (can be omitted without affecting the capacity), but does not imply since is also possible in some cases..

Note that, unlike the standard WF where the TPC is always active, it can be inactive under the IPC, which is ultimately due to the interplay of interference and power constraints. The following proposition explores this in some details.

Proposition 6.

The TPC is redundant only if and is active otherwise. In particular, it is active (for any and ) if , e.g. if is full-rank and is rank-deficient.

Proof.

Use (87) and note that this condition is necessary for the TPC to be redundant (since the KKT conditions are necessary for optimality and is also necessary for the TPC to be redundant). Now, if , then

(35)

where is the dimensionality of , and hence is impossible so that the TPC is active. ∎

If is full-rank, a more specific result can be established.

Proposition 7.

If , then the TPC is redundant if and only if , where the optimal covariance is as follows

(36)

and is

(37)

where and is the rank of the optimal covariance matrix, which can be found as the largest solution of the following inequality:

(38)

In particular, this holds if

(39)
Proof.

Let be as in (36). Then, , since is the capacity under the IPC only (this follows since (36) is a special case of (10) with ). On the other hand, if , then is feasible under the joint (IPC+TPC) constraints and hence , which proves the equality and hence is optimal. (37) follows from :

(40)

(38) ensures, from (36), that the rank of is . Further note that (39) implies :

(41)

since , but the converse is not true. ∎

Note that Proposition 7 gives a closed-form solution (including dual variables) for an optimal signaling strategy in the interference-limited regime (when the TPC is redundant and hence the IPC is active). The following proposition gives a sufficient and necessary condition for the IPC to be redundant (and hence the TPC is automatically active).

Proposition 8.

The IPC is redundant and hence the standard WF solution is optimal, , if and only if

(42)

in which case the TPC is active: . In particular, this holds if

(43)
Proof.

Note that, under the joint (IPC+TPC) constraint, (since is attained by relaxing the IPC and retaining the TPC only, which cannot decrease the optimum). On the other hand, under the stated conditions, is feasible under the joint constraints and hence the upper bound is achieved, and is optimal. It is straightforward to see that (43) implies (42), since (note that (43) is sufficient but not necessary). ∎

While Proposition 8 gives sufficient and necessary conditions for the IPC to be redundant, they depend on and . The next proposition gives a sufficient condition which is independent of and .

Proposition 9.

Let . Then, the IPC is redundant for any and and the standard WF solution is optimal: .

Proof.

As in Proposition 8, . It is straightforward to verify that any active eigenvector of is orthogonal to and hence, due to , , i.e. the IPC is redundant and is feasible (for any and ) and hence , from which the desired result follows. ∎

It follows from this Proposition that ZF transmission is optimal in this case, for any and , so that there is no loss of optimality in using this popular signaling technique.

Vi Rank-1 Solutions

In this section, we explore the case when is rank-one. As we show below, beamforming is optimal in this case. A practical appeal of this is due to its low-complexity implementation. Furthermore, rank-one is also motivated by single-antenna mobile units while the base station is equipped with multiple antennas, or when the MIMO propagation channel is of degenerate nature resulting in a keyhole effect, see e.g. [33][34].

We begin with the following result which bounds the rank of optimal covariance in any case.

Proposition 10.

If the TPC is active or/and is full-rank, then the rank of the optimal covariance of the problem (P1) in (2) under the constraints in (8) is bounded as follows:

(44)

If the TPC is redundant and is rank-deficient, then there exists an optimal covariance of (P1) under the constraints in (8) that also satisfies this inequality333Under these conditions, an optimal covariance has an unusual property of being not necessarily unique, see an example in Section X, so that there may exist extra solutions that do not salsify this inequality, but all of them deliver the same capacity..

Proof.

See Appendix. ∎

The following are immediate consequences of Proposition 10.

Corollary 2.

If is of full-rank or/and if the TPC is active, then the optimal covariance is of full-rank only if is of full-rank (i.e. rank-deficient ensures that is also rank-deficient).

Corollary 3.

If , then , i.e. beamforming is optimal.

Note that this rank (beamforming) property mimics the respective property for the standard WF. However, while signalling on the (only) active eigenvector of is optimal under the standard WF (no IPC), it is not so when the IPC is active, as the following result shows. To this end, let , i.e. it is rank-1 with be the (only) active eigenvalue and eigenvector; be the ”interference-to-signal” ratio, and

(45)

where is Moore-Penrose pseudo-inverse of ; if is full-rank [31].

Proposition 11.

Let be rank-1.

1. If , then the TPC is redundant and the optimal covariance can be expressed as follows

(46)

The capacity is

(47)

where .

2. If , then the IPC is redundant and the standard WF solution applies: . This condition is also necessary for the optimality of under the TPC and IPC when is rank-1. The capacity is as in (47) with .

3. If , then both constraints are active. The optimal covariance is

(48)

where , and is found from the IPC: . The capacity is as in (47) with

(49)

with equality if and only if is an eigenvector of .

Proof.

See Appendix. ∎

Note that the optimal signalling in case 1 is along the direction of and not that of (unless is also an eigenvector of ), as would be the case for the standard WF with redundant IPC. In fact, plays a role of a ”whitening” filter here. Similar observation applies to case 3, with replaced by . in Proposition 11 quantifies power loss due to enforcing the IPC; means no power loss.

Vii Identical Eigenvectors

In this section, we consider a scenario where and have the same eigenvectors. This may be the case when the scattering environment around the base station (the Tx) is the same as seen from the Rx and primary user U. In this case, the general solution in Theorem 1 significantly simplifies to the following:

(50)

where unitary matrix

collects eigenvectors of and diagonal matrix collects the eigenvalues of :

(51)

where are the eigenvalues of . Dual variables are determined from the following:

(52)

subject to , . The capacity can be expressed as in (12) with .

Note that, in this case, signaling on the eigenmodes of the main channel is optimal, but power allocation is not given by the standard WF, unless the IPC is redundant (). It follows from (51) that (power allocated to -th eigenmode is zero) if (redundant TPC) and (-th eigenmode of 2nd channel is inactive), in addition to the standard WF property that if under the active TPC ().

In the context of massive MIMO systems under favorable propagation, see e.g. [37], and become diagonal matrices (and thus have the same eigenvectors), and so is , i.e. independent signaling is optimal, and the solution in (51) gives the optimal power allocation in such setting. This significantly simplifies its implementation since numerical complexity of generic convex solvers can be prohibitively large for massive MIMO settings.

Viii Iterative Bisection Algorithm

While Theorem 1 gives a closed-form solution for an optimal covariance up to dual variables, no closed-form solution is known for (11) in the general case; the sections above provided complete closed-form solutions in some special cases. In this section, we develop an iterative numerical algorithm to solve (11) in the general case in an efficient way and prove its convergence.

First, we consider the standard bisection algorithm [30]. Let be a function with the following property: for any and for any , where is a solution of . Then, the following bisection algorithm (BA) can be used to solve , where are the upper and lower bounds to : , and is any desired accuracy. In fact, it is straightforward to show that this algorithm will converge in a finite number of steps such that

(53)

where denotes ceiling, so that the convergence is exponentially fast and hence the algorithm is very efficient [30].

repeat
     1. Set .
     2. If , set . Otherwise, set . Terminate if .
until .
Algorithm 1 Bisection algorithm (BA)

An alternative stopping criteria for this algorithm is and the two criteria are equivalent when is continuous.

The BA can be used to solve for , in (11) in an iterative way, as we show below. To this end, we need to establish lower and upper bounds to the solutions required by the BA.

Proposition 12.

Let be solutions of (11), i.e. the optimal dual variables. They can be bounded as follows:

(54)
(55)

where is the rank of and is the number of Tx antennas.

Proof.

From the KKT conditions in (79),

(56)

Let be a matrix with positive eigenvalues, . Since

(57)

it follows that

(58)

Now use to obtain

(59)

where and , so that 2nd inequality in (54) follows from (56). Let