## I Introduction

Aggressive frequency re-use and hybrid (non-orthogonal) access schemes envisioned as key technologies in 5G systems [1] can potentially generate significant amount of inter-user interference and hence should be designed and managed carefully. In this respect, multi-antenna (MIMO) systems have significant potential due to their significant signal processing capabilities, including interference cancellation and precoding, which can also be done in an adaptive and distributed manner [2][3]

. 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)

[2]-[5]. 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 [6][7].Much less is known under the added interference power constraint (IPC), which limits the power of interference induced by a secondary transmitter to a primary receiver in a spectrum-sharing system. A game-theoretic approach to this problem was proposed in [8], where a fixed-point equation was formulated from which the optimal transmitt covariance matrix can in principle be determined. Unfortunately, no closed-form solution is known for this equation and the considered settings require the channel to the primary receiver to be full-rank hence excluding the important 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 and, consequently, no interplay between the TPC and the IPC can be studied.

Cognitive radio MIMO systems under interference constraints have been also studied in [9][10][11], where a number of numerical optimization algorithms were developed but no closed-form solutions are known to the underlying optimization problems. Optimal signaling for the Gaussian MIMO channel under the TPC and the IPC has been also studied in [12]-[14] using the dual problem approach, and was later extended to multi-user settings in [15]. 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. This limits insights significantly.

In this paper, we study the spectrum-sharing potential of Gaussian MIMO channels and concentrate on analysis rather than numerical algorithms. This provides deeper understanding of the problem and a number of insights unavailable from numerical algorithms alone. Specifically, we obtain novel closed-form solutions for an optimal transmit covariance matrix for the Gaussian MIMO channel under the TPC and multiple IPCs. All constraints are included explicitly and hence anyone is allowed to be inactive. This allows one 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 (including massive MIMO settings). In some cases, our KKT-based approach leads to closed-form solutions for the optimal dual variables as well, including full-rank and rank-1 (beamforming) solutions and the conditions for their optimality. 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.

Under the added IPC(s), the unitary-invariance of the feasible set is lost and hence many known solutions and standard ”tricks” (e.g. Hadamard inequality) of the analysis under the TPC alone cannot be used. This has profound impact on optimal signaling strategies as well as on analytical techniques to solve the underlying optimization problem. In particular, unlike the standard water-filling solution, (i) signaling on the channel eigenmodes is not optimal anymore (unless all IPCs are inactive or if their channel eigenmodes are the same as those of the main MIMO channel); (ii) the rank of an optimal covariance matrix can exceed that of the channel; (iii) an optimal covariance matrix is not necessarily unique; (iv) the channel capacity can be zero for a non-zero Tx power and channel; (v) the channel capacity may stay bounded under unbounded growth of the Tx power. All these phenomena have major impact on the spectrum-sharing capabilities of MIMO channels. We demonstrate that capacity scaling with Tx power under multiple IPCs can be understood in terms of a natural linear-algebraic structure of MIMO channels of different users.

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. 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

[2]-[5] 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:

where is the ”water” level found from the total power constraint , are -th eigenvalue and eigenvector of , so that are excluded from the summation due to ; denotes positive eigenmodes of Hermitian matrix : , where are -th eigenvalue and eigenvector of .

In a spectrum-sharing multi-user system, there is a limit on how much interference the Tx can induce (via ) to primary user , see Fig. 1,

(4) |

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

(5) |

where and represent channel to user and respective interference constraint power, , is the number of users, see Fig. 1.

The Gaussian signalling is still optimal in this setting and the capacity subject to the TPC and IPCs can still be expressed as in (2) but the optimal covariance is not anymore. In particular, the unitary-invariance of the feasible set under the TPC alone is lost due to the presence of the IPCs in so that well-known results and ”tricks” (based on unitary invariance of the feasible set) cannot be used anymore. Since the ”shape” of the feasible set affects significantly optimal , this results in a number of new properties of optimal signaling and of the capacity, as we show below.

One may also consider the total (rather than individual) interference power constraint so that

In this case, all the results of this paper will apply with , , and .

## Iii Optimal Signalling Under the TPC and IPCs

To characterize fully the capacity, a closed-form solution for the optimal signaling problem (P1) in (2) under the joint constraints in (5) is given below in the general case, i.e. are allowed to be singular and any of the constraints are allowed to be inactive. This extends the known results in [12]-[14] to the general case.

###### Theorem 1.

Consider the capacity of the Gaussian MIMO channel in (2) under the joint TPC and IPC in (5). The optimal Tx covariance matrix to achieve the capacity can be expressed as follows:

(6) |

where ; is the Moore-Penrose pseudo-inverse of ; are Lagrange multipliers (dual variables) responsible for the TPC and IPCs, found from

(7) |

subject to , . The respective capacity is

(8) |

where .

###### Proof.

See Appendix. ∎

Based on (6), one observes that plays a role of a ”whitening” filter, which disappears when all IPCs are inactive. When is full-rank, i.e. , then is unique, which is not necessarily the case in general - a remarkable difference to the TPC-only case, where is always unique.

A number of known special cases follow from (6): If and is full-rank, then (see e.g. [17]) and in (6) reduces to the respective solutions in [12]-[14]. If all IPCs are inactive, then , and , as it should be. Below, we will give explicit conditions when this is the case.

### Iii-a General properties

Next, we explore some general properties of the capacity related to its unbounded growth with and its being strictly positive. It turns out that those properties induce a natural linear-algebraic structure for the set of channels of all users.

It is well-known that, without the IPC, grows unbounded as increases, as (assuming ). This, however, is not necessarily the case under the IPCs with all fixed . The following proposition gives sufficient and necessary conditions when it is indeed the case.

###### Proposition 1.

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

(9) |

or, equivalently,

(10) |

###### Proof.

See Appendix. ∎

Let us make the following observations:

Since the above conditions are 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.

The unbounded growth of the capacity with depends only on and , all other details being irrelevant.

It can be seen that the condition holds if , and hence the capacity grows unbounded with under the latter condition.

On the other hand, if , then very high spectral efficiency cannot be achieved even with unlimited power budget, due to the dominance of the IPCs. In particular, if or, equivalently, , then (9) is impossible and the capacity stays bounded, even for infinite - the whole signaling space is dominated by IPCs in this case.

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. To this end, let , i.e. a set of all primary users requiring no interference, .

###### Proposition 2.

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

(11) |

###### Proof.

See Appendix. ∎

Note that the condition is equivalent to zero-forcing transmission with respect to user , i.e. the capacity is zero only if ZF transmission is required for at least one user; otherwise, . The condition in (11) 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 propagation scenarios where spectrum sharing is possible for any and arbitrary large capacity can be attained given enough Tx power budget.

Unlike the standard WF where the TPC is always active, it can be inactive under the IPCs, which is ultimately due to the interplay of interference and power constraints. The following proposition explores this in some details. To this end, we call a constraint ”redundant” if it can be omitted without affecting the capacity^{1}^{1}1”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..

###### Proposition 3.

The TPC is redundant only if

(12) |

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

###### Proof.

See Appendix. ∎

## 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 (7). 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. To this end, we set , , , . First, 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:

(13) | ||||

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

(14) |

where . The capacity can be expressed as

(15) |

###### Proof.

See Appendix. ∎

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. [19][20].

###### Proposition 5.

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

(16) |

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

(17) |

where .

If

(18) | |||

(19) |

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

(20) |

where , and are

(21) |

###### Proof.

See Appendix. ∎

## V 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. [19][20].

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

###### Proposition 6.

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 (5) is bounded as follows:

(22) |

If the TPC is redundant and is rank-deficient, then there exists an optimal covariance of (P1) under the constraints in (5) that also satisfies this inequality.

###### Proof.

See Appendix. ∎

###### 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

(23) |

where is Moore-Penrose pseudo-inverse of ; if is full-rank [17].

###### Proposition 7.

Let be rank-1.

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

(24) |

The capacity is

(25) |

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 (25) with .

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

(26) |

where , and is found from the IPC: . The capacity is as in (25) with

(27) |

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 7 quantifies power loss due to enforcing the IPC; means no power loss.

## Vi Appendix

### Vi-a Proof of Theorem 1

Since the problem is convex and Slater’s condition holds, the KKT conditions are both sufficient and necessary for optimality [16]. They take the following form:

(28) | |||

(29) | |||

(30) | |||

(31) |

where is Lagrange multiplier responsible for the positive semi-definite constraint . We consider first the case of full-rank (i.e. either or/and ), so that . Let us introduce new variables: , , . It follows that and (28) can be transformed to

(32) |

for which the solution is

(33) |

(this can be established in the same way as for the standard WF). Transforming back to the original variables results in (6). (7) are complementary slackness conditions in (VI-A); (8) follows, after some manipulations, by using of (6) in .

The case of singular is more involved. It implies so that . It follows from the KKT condition in (28) that, for the redundant TPC (),

(34) |

where . Let , i.e. , then

(35) |

so that and , since and . Thus, and , i.e.

(36) |

and this condition is also necessary for the TPC to be redundant. Further notice that

(37) |

where is the set of users with active IPCs. Let . Using (34), (36) and introducing new variables

(38) |

where

is a unitary matrix of eigenvectors of

, one obtains(39) |

where is a diagonal matrix of strictly positive eigenvalues of , so that (34) can be transformed to

(40) |

Using

(41) |

where and adopting (33), (6), one obtains

(42) |

Since only affects the capacity, one can set, without loss of optimality, , , , and transform (42) to

(43) |

and hence, as desired,

(44) |

### Vi-B Proof of Proposition 1

To prove the ”if” part, observe that implies . Now set , for which , so it is feasible for any . Furthermore,

(45) |

as , since .

Next, we will need the following technical result, which will also establish the last claim.

###### Lemma 1.

The following holds:

(46) |

###### Proof.

To prove the ”only if” part, let and assume that . This implies that (since is the complement of for Hermitian ). Let

(47) |

where is a semi-unitary matrix of active eigenvectors of and diagonal matrix collects its strictly-positive eigenvalues. Notice that, from the IPC,

(48) |

where is the smallest positive eigenvalue of , so that

(49) |

for any . On the other hand, implies and hence

(50) |

so that

(51) |

is bounded for any , as required.

### Vi-C Proof of Proposition 2

To prove the ”if” part, observe that implies that (since ) so that for any and hence . Under the above condition, this implies and hence so that for any feasible . Hence, .

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

(52) |

and hence for some is necessary for . To show that (11) is necessary as well, assume that it does not hold, which implies that . Now set , where if ; otherwise, , where . Notice that, for this , , so it is feasible and

(53) |

so that (11) is necessary for .

### Vi-D Proof of Proposition 3

Use (36) and (37), and note that these conditions are 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

(54) |

where is the dimensionality of , and hence (12) is impossible so that the TPC is active.

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