1

MIMO Capacity with Average Total and

Per-Antenna Power Constraints
Giorgio Taricco

Abstract
arXiv:1902.01817v1 [cs.IT] 5 Feb 2019

MIMO capacity with a joint total and per-antenna average power constraint is considered in this
work. The problem arises when, besides having a limited available power at the transmitter, also the
individual antennas cannot radiate power beyond the limits of their corresponding RF chains. Closed-
form results are illustrated in specific cases and, in particular, in the unit-rank channel matrix case. Lower-
complexity optimization problems are derived for the other cases. Numerical complexity is derived in
terms of the number of equivalent real variables for the optimization problem and a general formula is
provided depending on the channel matrix rank. Numerical results are included to validate and illustrate
the application of the proposed optimization algorithms and also to evaluate the time complexity of its
implementation.

Index Terms

MIMO, Channel capacity, Sum power constraint, Per-antenna power constraint,

I. I NTRODUCTION

The capacity of multipe-input multiple-output (MIMO) channels is commonly evaluated under

a total average power constraint [1]–[3] in order to account for the limited availability of
transmission power to be sent to the transmitting antennas. Under the assumption of known
channel matrix at both the transmitter and receiver endpoints, i.e., with perfect Channel State
Information at the Transmitter (CSIT) and at the Receiver (CSIR), the capacity is obtained by
applying the water-filling algorithm [1] to the equivalent eigen-channels describing the MIMO
channel itself. The resulting solution is very simple and elegant and its derivation is quite
straightforward after the application of a singular value decomposition (SVD) to the channel
matrix itself. The water-filling algorithm implementation is very simple and efficient and it solves
completely the optimization problem behind the evaluation of the channel capacity. However,
 2

some implementations of MIMO transmitters rely on separate RF chains feeding the different
transmit antennas so that the average transmit power from every antenna is subject to a specific
constraint [4], [5]. Also in he case of distributed MIMO systems, where transmitting antennas
are located at different places, power limitations affect different antennas separately and there
is no overall power constraint [5]. These scenarios are opposite, for example, to the case of
an implementation based on a common amplifier followed by a passive beamforming network.
To summarize, power limitation on a MIMO transmitter can be of two different types: i) total
average power (TP) constrained or ii) per-antenna average power (PAP) constrained. We can
also figure out a hybrid situation where both constraints have to be enforced (TP and PAP):
the individual antenna power amplifiers may trade-off some of their available power or may
drain their power supply from a common source [4], [9]. This motivates the study of the MIMO
channel capacity under TP and PAP constraints.
The existing literature offers many works on the MIMO channel capacity with TP and/or PAP
constraints. One of the earliest results is due to Vu, who developed an iterative algorithm for the
PAP-only MIMO case in [5], [6]. Her approach is based on the solution of the KKT equations
relevant to the PAP constrained optimization problem addressing the maximization of the MIMO
channel’s mutual information. The approach is ingenious and relies on the particular structure
of the optimizing covariance matrix, which is split into different signature components leading
to an efficient iterative algorithm. Nevertheless, the TP constraint is not considered and there
is limited discussion of complexity and convergence issues. More recently, in [10], the PAP-
only constrained MIMO capacity has also been considered and a closed-form solution for the
optimal input covariance matrix has been derived when the channel matrix and the optimal input
covariance matrix have full rank. This result was anticipated in [5, eq. (23)] in an equivalent
form without giving the explicit conditions required. Finally, Loyka [4] and [8] considered the
joint TP and PAP constrained capacity problem in the multiple-input single-output (MISO) case
and obtained a closed-form solutions for the capacity-achieving input covariance matrix in this
case.
In this work we address the evaluation of the capacity of a MIMO channel with perfect
CSIT/CSIR under a joint TP and PAP constraint. The paper extends in a nontrivial way the
results of [5] and provides an algorithmic solution for this problem, which is still open for
MIMO channels under a joint TP and PAP constraint (see [4]). The contributions of the paper
can be summarized as follows.
 3

We emphasize the different type of optimization problems arising when the channel matrix
has full rank (equal to the number of transmit antennas) or not. We refer to the former as
the full-rank case and to the latter as the singular case. In both cases we show that the
capacity-achieving covariance matrix has a specific form depending on a number of real positive
parameters not greater than the number of transmit antennas and on the Gramian of the channel
matrix. Moreover, we show that in the full-rank case the TP constraint is active at the optimum
(i.e., the constraint is met with equality) while it may be inactive in the singular case.
The solution of the KKT equations leading to the main results discussed above is different
from [5] for several reasons: i) the Lagrangian function is extended to encompass the TP
constraint; ii) we provide a robust justification of the positivity of the diagonal Lagrange
multiplier matrix D, corresponding to the set of real positive parameters determining the optimum
covariance matrix.1 iii) we simplify the derivation of the optimality conditions and obtain a
specific relationship between the matrix F to the matrix R (matrix notation is consistent with [5]).
The derivation of a specific structure of the capacity-achieving covariance matrix is a key
contribution of this work. This reduces dramatically the complexity of the optimization prob-
lem. In fact, we are considering the minimization of a strictly convex function (the negative
mutual information) over a convex set (intersection of the TP, PAP, and positive semidefiniteness
constraints on the input covariance matrix), which corresponds to a convex optimization problem.
As such, its solution is global and unique [11]. Then, any standard optimization algorithm (e.g.,
interior point) provides the capacity over the variable space corresponding to the input covariance
matrix. A general closed-form solution appears to be out of question, except in the case when
specific conditions are satisfied, as also evidences in [10] for the PAP-only case. We derive
specific conditions for the existence of a closed-form solution in the joint TP and PAP case.
Since we resort to numerical algorithms for the derivation of the channel capacity, a metric
of interest is complexity. We note that the proposed algorithm for the full-rank case reduces
dramatically complexity because i) the optimization is based on a diagonal matrix determined
by nT real parameters instead of the n2T real parameters characterizing the input covariance matrix;
and ii) positive semidefiniteness of the input covariance matrix is automatically enforced.

1
The argument used in [5] to show that the Lagrange multipliers di are real and positive is incorrect. The paper claims that the
positivity of the di ’s derives from the complementary slackness condition and the fact that the optimum solution corresponds to
active constraints (inequalities met as equalities). However, if di ≥ 0, di ((Q)ii − Pi ) = 0, and (Q)ii = Pi , we cannot be sure
that di > 0 because di = 0 is also compatible with all the equations. This property is crucial to the existence of Ď , D −1 .
 4

We also deal extensively with the singular case (when the channel matrix rank is lower than
the number of transmit antennas). It is worth mentioning that this case was not addressed in [5]
for the PAP-only case though some progress was made in an unpublished work by the same
author [7]. The singular case was addressed, only in the MISO case, in [4], [8].
The remainder of the paper is organized as follows. Section II introduces the channel model
along with the definition of TP and PAP constraints. A basic form of the general optimization
problem based on real variables is reported in (1). Section III derives the equivalent optimization
problems in the full-rank and singular cases (Theorems 1 and 2, respectively). Here, full-rank
means that the channel matrix rank is equal to the number of transmit antennas and singular
corresponds to the opposite condition. Explicit conditions for the existence of a closed-form
solution are given in Corollary 1. A general complexity measure defined in terms of the number
of equivalent real variables required by the optimization problem is provided in Corollary 2. The
rank of the capacity-achieving covariance matrix is shown to be smaller than the channel matrix
rank in Corollary 3. Finally, Theorem 3 provides an explicit solution for the joint TP and PAP
case applicable to the case of unit rank channel matrix. This theorem extends the results from
[4] applicable to the MISO case. Algorithm 1 describes how to solve the optimization problem
in this case and obtain the corresponding capacity-achieving input covariance matrix. Section
IV provides numerical results to illustrate the capacity evaluation. First, to validate the methods,
numerical results are reported to reproduce results from [5] and [4]. Next, a few MIMO scenarios
are presented to illustrate the applicability of the optimization proposed, in particular concerning
the PAP constraint increase required to attain the water-filling capacity corresponding to the
TP-only constraint. Then, complexity evaluation is reported by considering a large number of
channel matrix instances and measuring the complexity of the proposed form of the optimization
problem (6) against the basic general form 1. Concluding remarks are reported in Section V.

A. Notation

We denote matrices by uppercase boldface letters (e.g., A) and column vectors by lowercase
boldface letters (e.g., a). We denote the entry at the ith row and jth column of A as (A)ij .
We denote by ai and Ai the ith column and the ith row of the matrix A, respectively, unless
otherwise explicitly stated. We denote by 0m×n an all-zero matrix of dimensions m × n. We
denote by In the n × n identity matrix. Matrix inequalities imply that the matrices involved
are Hermitian and follow the standard definitions from [12]: A > B ⇔ (A − B) is positive
 5

definite; A ≥ B ⇔ (A − B) is positive semidefinite. The Frobenius norm of a matrix A is

denoted by kAk (similarly for a vector). Notation diag(a1 , . . . , an ) denotes a diagonal matrix
whose elements on the main diagonal are a1 , . . . , an ; diag(a) denotes a diagonal matrix whose
elements on the main diagonal are taken from the vector a; diag(A) denotes a diagonal matrix
whose elements are taken from the diagonal elements of the matrix A. U(A) and L(A) are
the upper and lower triangular matrices corresponding to A, i.e., (U(A))ij = (A)ij if i < j
and 0 otherwise and (L(A))ij = (A)ij if i > j and 0 otherwise. We denote the minimum
and maximum eigenvalues of a Hermitian matrix A as λmin (A) and λmax (A), respectively. We
denote (x)+ , max(0, x) and extend this definition to diagonal matrices by applying the scalar
definition to each diagonal entry. We also define (A)+ for Hermitian matrices A in the following
way: if A has the unitary factorization A = U DU H for some unitary matrix U and diagonal
matrix D, then (A)+ , U (D)+ U H .

II. C HANNEL M ODEL

We consider an nT ×nR MIMO channel with nT transmit and nR receive antennas, characterized
by the standard channel equation
y = Hx + z

where H is the nR ×nT channel matrix, x is the transmitted symbol vector, z is the received noise
sample vector with joint circularly symmetric complex Gaussian distribution z ∼ CN (0, InR ),
and y is the received signal sample vector.
The channel matrix is assumed to be known exactly at both the transmitter and the receiver
endpoints (i.e., we have perfect CSIT/CSIR). Every column of H contains at least one nonzero
element and hence has positive norm (otherwise the corresponding transmitted signal would be
useless).
In accordance with the previous assumptions, the mutual information corresponding to an
input covariance matrix Q is given by [1]

I(x; y) = log det(InR + HQH H )

when x ∼ CN (0, Q), and represents the maximum achievable rate over the MIMO channel
under the input covariance constraint.
Hereafter, we consider the joint TP and PAP constrained MIMO channel capacity

C= max log det(InR + HQH H ) (1)

Q:Q≥0,tr(Q)≤Ptot ,diag(Q)≤P
 6

where Ptot characterizes the TP constraint, i.e., the maximum average transmitted power from
all the antennas, which is represented by the trace of the input covariance matrix Q. Moreover,
P is a diagonal matrix containing the PAP upper bounds to the average power transmitted by
every antenna. If P = diag(P1 , . . . , PnT ), the constraints are given by (Q)ii ≤ Pi , i = 1, . . . , nT ,
individually, or by diag(Q) ≤ P in compact matrix form. We shall assume that
nT
X
Pi ≥ Ptot . (2)
i=1

because otherwise the TP constraint would be unattainable.

The function log det(InR + HQH H ) is strictly concave over the convex set of positive
semidefinite matrices Q ≥ 0 [1]. Since also the TP and PAP constraints lead to convex sets, the
optimization problem is convex and can be solved by solving the KKT equations. Moreover, since
the objective function is strictly concave and the optimization problem is convex, its solution is
global and unique [11].

Remark 1 Optimization problem (1) can be solved directly by using one of the several optimiza-
tion algorithms proposed in the literature for nonlinear convex optimization (e.g., interior-point,
sequential quadratic programming, active-set optimization algorithms) [14]. Since the matrix Q
is Hermitian and may have complex elements, one may consider the following equivalent real
optimization problem:

min − log det(InR + HQH H ) (3-a)

X

s.t. Q = diag(X) + U(X) + U(X)T

+ j [L(X) − L(X)T ] ≥ 0 (3-b)

diag(Q) ≤ P = diag(P1 , . . . , PnT ) (3-c)

tr(Q) ≤ Ptot (3-d)

A closed-form general solution is out of question in the general MIMO case, whereas it is
feasible in the MISO case [4] and in a special full-rank case [5], [10]. Nevertheless, the KKT
equations can be used to decrease the complexity of the optimization problem itself by turning
it into an equivalent one with a smaller number of unknowns.
 7

III. S OLUTION OF THE O PTIMIZATION P ROBLEM

The optimization problem corresponding to the derivation of the MIMO channel capacity
under a joint TP and PAP constraint can be written in the following standard form [11]:

min − log det(InR + HQH H ) (4-a)

Q

s.t. Q≥0 (4-b)

diag(Q) ≤ P = diag(P1 , . . . , PnT ) (4-c)

tr(Q) ≤ Ptot (4-d)

This optimization problem is convex because of the strict convexity of the objective function
− log det(InR +HQH H ) [1] and of the convexity of the domain (intersection of convex domains).
The solution of the optimization problem (4) depends on the rank of the channel matrix, which
can be classified in two different cases.
1) The full-rank case, when the channel matrix rank is equal to the number of transmit
antennas.
2) The singular case, when the channel matrix rank is smaller than the number of transmit
antennas.
The latter case occurs whenever, though not exclusively, nR < nT , as in the MISO case. Both
cases will be addressed in the following sections.

A. Full-Rank Case (rank(H) = nT )

The solution of the optimization problem (4) depends on the rank of the channel matrix H.
In the case of full rank of H it can be characterized by the following theorem.

Theorem 1 (rank(H) = nT ) The capacity-achieving input covariance matrix for a MIMO chan-
nel with a joint TP and PAP constraint is given by

Qopt = K −1 (K Ďopt K − InT )+ K −1 , (5)

 8

where K , (H H H)1/2 is the full-rank matrix square root of the Gramian of the channel matrix
H and Ďopt is derived by the solution of the optimization problem

min − log det[InT + (K ĎK − InT )+ ] (6-a)

Ď

s.t. Ď > 0 (6-b)

tr[K −1 (K ĎK − InT )+ K −1 ] = Ptot (6-c)

diag[K −1 (K ĎK − InT )+ K −1 ] ≤ P (6-d)

Proof: See Appendix A

This optimization problem can be solved by a standard convex optimization algorithm (e.g.,
interior point). In order to initialize the matrix Ď we notice that, from the proof of Theorem 1
in Appendix A, since ΛF − InT ≤ (ΛF − InT )+ , and hence, from (24-d), we get

Ď − K −2 ≤ Q ⇒ tr(Ď) ≤ Ptot + tr(K −2 ).

Thus, we can initialize Ď by

Ď0 , (Ptot + α tr(K −2 )InT

for some α < 1. The choice of α affects the convergence properties of the iterative algorithm.
As noted in [5], and subsequently in [10], for the PAP-only case, a special case occurs when
it is possible to solve in closed form the optimization problem of Theorem 1. With the joint TP
and PAP constraints we have the the following result.

Corollary 1 (rank(H) = nT ) The capacity-achieving input covariance matrix for a MIMO chan-
nel with joint TP and PAP constraints is
Ptot + tr(K −2 )
Qopt = InT − K −2 , (7)
nT
where K , (H H H)1/2 , if the following conditions hold:

Ptot ≥ nT λmax (K −2 ) − tr(K −2 ) (8-a)

n o
−2
Ptot ≤ nT min (K )ii + Pi − tr(K −2 ) (8-b)
1≤i≤nT

The capacity is
Ptot + tr(K −2 )
 
C = nT log2 + log2 (K 2 ). (9)
nT
Proof: Assume that Ď > K −2 and maximize det(Ď) by the scaled identity matrix in the
proof of Theorem 1.
 9

B. Singular Case (rank(H) < nT )

Here, we consider the case of a channel matrix with rank strictly lower than the number
of columns, i.e., the number of transmit antennas nT . The following theorem characterizes the
capacity-achieving covariance matrix in the singular case.

Theorem 2 (rank(H) < nT ) The capacity-achieving input covariance matrix for a MIMO chan-
nel with joint TP and PAP constraints and channel matrix with rank ν < nT , the number of
transmit antennas, is obtained by solving the following optimization problem.

min − log det[InR + HQH H ] (10-a)

Ď,Q

s.t. Ď > 0, Q ≥ 0 (10-b)

diag(Q) ≤ P (10-c)

tr(Q) ≤ Ptot (10-d)

VHH QVH = Λ−1 H H −1

H UH (H ĎH − InR )+ UH ΛH (10-e)

where we used the reduced-size SVD of the channel matrix H = UH ΛH VHH , where ΛH is a ν ×ν
positive definite diagonal matrix and UH , VH are nT × ν and nR × ν partial unitary matrices,
such that UHH UH = VHH VH = Iν .

Proof: See Appendix B

Remark 2 Notice that, in the singular case, the TP constraint is not necessarily met with equality
because the relative argument used in the case of nonsingular K (deriving from the inequality
diag(Q) ≤ P ) fails to hold in this case where the PAP constraints implies that diag(V+ Q̃+ V+H ) ≤
P . This has a major impact on the capacity, which can be substantially lower in this case because
meeting the PAP constraints prevents to exploit the total available power according to the TP
constraint.

Corollary 2 (Complexity) The complexity of the equivalent optimization problem derived in

Ths. 1 and 2, in terms of cardinality of the variable space, is given by

Nvar = 2(nT − ν)ν + nT . (11)

Proof: See Appendix B.

 10

Remark 3 It is interesting to notice that the worst-case complexity of the optimization algorithm
corresponds to the case of rank ν = nT /2 or the closest integer. Compared to the fixed complexity
of the basic form of the optimization problem (3), the complexity of the optimization algorithm
in the full-rank case is nT times lower. When ν = nT /2, the complexity is still reduced by a
factor 2 and when ν = 1, the reduction is by a factor nT /3, approximately. However, in this last
case, the use of Algorithm 1 from Appendix C dramatically reduces complexity and provides
an exact result without the need of numerical algorithms like interior-point.

Corollary 3 (Rank) The capacity-achieving input covariance matrix Q for a MIMO channel
with joint TP and PAP constraints and channel matrix with rank ν < nT satisfies the following
inequalities:
1 ≤ rank(Q) ≤ ν. (12)

Proof: See Appendix B.

C. Unit-Rank Singular Case

In this section we consider a MIMO channel with channel matrix having unit rank, i.e., ν = 1.
The MISO channel is a special case where ν = nR = 1. More generally, a unit-rank MIMO
channel has a channel matrix which can be expressed as

H = kHkuv H , (13)

where u, v are unit-norm column vectors.

The special MISO case has been dealt with in the literature (see, in particular, [4], [8]) where
closed-form expressions have been proposed. Our approach extends these results and our findings
are summarized in the following theorem.

Theorem 3 Given a MIMO channel with unit-rank channel matrix having SVD H = kHkuv H ,
where kuk = kvk = 1, the channel capacity with joint TP and PAP constraints is given by
 nT
X 
2 2
C = log 1 + kHk |vi qi | (14)
i=1

where vi , qi are the ith elements of the vectors v, q, respectively, for i = 1, . . . , nT , and the
vector q is obtained by setting
p
qi , min(α|vi |2 , Pi ) e− j ∠vi , i = 1, . . . , nT ,
 11

where α is obtained by solving the equation

nT
X
min(α|vi |2 , Pi ) = Ptot . (15)
i=1

The unit-rank capacity-achieving covariance matrix is

Qopt = qq H (16)

and the TP constraint is always attained with equality.

Proof: See Appendix C. The appendix reports an algorithm for the solution of (15), Algo-
rithm 1.
Theorem 3 extends the results of [4, Th.1,2] and [8] from the MISO case to the unit-rank
MIMO case. It obviously applies to the MISO case, as well. Algorithm 1 from Appendix C
is more easily implementable than the derivations in [4], [8]. It is also closely related to the
algorithm used to solve the water-filling equation.

IV. N UMERICAL R ESULTS

In this section we report some numerical results obtained by applying the proposed optimiza-
tion algorithm implemented in Matlab.

A. Validation

Fig. 1 validates the results of the proposed algorithm against those reported in [5]. The
diagrams show the capacity of a 2 × 2 MIMO channel with the channel matrix defined in [5, eq.
(26)] in the following cases: PAP: PAP constraint with P1 from the abscissa and P2 = 1 − P1 ;
TP: TP constraint with Ptot = 1 (water-filling solution); MC: Q = diag(P1 , 1 − P1 ). It can be
seen that the curves coincide exactly with those reported in [5].
Figs. 2–3 validate the results from Algorithm 1 against those of [4, Figs.1-2]. The former
reports the capacity versus the TP constraint Ptot and the latter reports the nT = 4 normalized
amplitudes [(Q)ii /P ∗ ]1/2 for i = 1, . . . , nT of the capacity-achieving covariance matrix after
defining P ∗ , min(Ptot , tr(P )). It can be seen that the curves coincide exactly with those
reported in [4].
 12

1.6
PAP
TP
1.5 MC

1.4

1.3
Capacity (bit/s/Hz)

1.2

1.1

0.9

0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
P1

Fig. 1. Capacity of a 2 × 2 MIMO channel with channel matrix defined in [5, eq. (26)] in the following cases: PAP: PAP
constraint with P1 from the abscissa and P2 = 1 − P1 ; TP: TP constraint with Ptot = 1; MC: Q = diag(P1 , 1 − P1 ).

B. Applications

Next, we illustrate the application of the algorithms proposed in Theorem 1 and 2 by consid-
ering the 3 × 4 and 3 × 2 channel matrices obtained in Matlab by the following commands:
• rng(1);H1=randn(4,3)+1i*randn(4,3);
• rng(1);H2=randn(2,3)+1i*randn(2,3);
The matrices obtained are given in eqs. (17) and (18). H1 corresponds to an instance of full-rank
MIMO since rank(H1 ) = nT = 3 and H1 corresponds to an instance of singular MIMO since
rank(H1 ) = 2 < nT = 3. In both cases we assume P = diag(0.1, 0.1, 1) and plot the capacity
with respect to Ptot . The results obtained are illustrated in Fig. 4. We can see that the rank of
the capacity-achieving covariance matrices increases with the TP constraint up to the maximum
predicted by Corollary 3. Also, the capacity with the joint constraint saturates when the TP
 13

4
C (bit/s/Hz)

1
PAP+TP
TP
PAP
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
PT

Fig. 2. Capacity of a 4 × 1 MISO channel with matrix from [4, Fig. 1].

constraint exceeds the sum of the PAP constraints, which is equal to 1.2 in this example.
Another illustration of the joint TP and PAP constrained capacity calculation is given in Fig.
5. The figure considers a 3 × 3 and a 4 × 4 MIMO channel with channel matrices given in (19)
and (20), respectively. The TP constraint is Ptot = 3 and 4, respectively, and the PAP constraint
is in the abscissa, constant for every antenna. It can be seen that equal power allocation achieves
a lower capacity than variable power allocation based on water-filling and the TP constraint,
as expected. Increasing the PAP constraint increases the capacity up to the water-filling limit
imposed by the TP constraint, as illustrated by the curves of Fig. 5. Similarly, Figs. 6 and 7
illustrate the same scenario with TP constraint reduced by a factor 10 and 100, respectively. The
figures show that, as the TP constraint decreases, the PAP constraint must increase more and
more in order to achieve the maximum capacity corresponding to a TP-only constrained system.
This is a consequence of the fact that the water-filling power distribution increases its dynamic
 14

0.9

0.8

0.7

0.6
(Qii/P * )1/2

0.5

0.4

0.3

0.2
i=1
0.1 i=2
i=3
i=4
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
PT

Fig. 3. Amplitude distribution from the capacity-achieving covariance matrix normalized with respect to P ∗ , min(Ptot , tr(P ))
with channel matrix as in Fig. 2.

 
−0.6490 − 1.5094 j −0.8456 − 1.9654 j −0.1969 − 0.2752 j
 
 1.1812 + 0.8759 j −0.5727 − 1.2701 j 0.5864 + 0.6037 j 
H3×4 = (17)
 

−0.7585 − 0.2428 j −0.5587 + 1.1752 j −0.8519 + 1.7813 j 
 
−1.1096 + 0.1668 j 0.1784 + 2.0292 j 0.8003 + 1.7737 j
 
−0.6490 − 0.5587 j −0.7585 − 0.1969 j −0.8456 − 0.8519 j
H3×2 =  (18)
1.1812 + 0.1784 j −1.1096 + 0.5864 j −0.5727 + 0.8003 j
 15

4.5

3.5
C (bit/s/Hz) / rank

2.5

1.5

1
H1:cap
0.5 H2:cap
H1:rank
H2:rank
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
PT

Fig. 4. Capacity of the 3 × 4 and 2 × 4 MIMO channels corresponding to the channel matrices reported in (17) and (18) with
fixed PAP constraints given by P = diag(0.1, 0.1, 1) and variable TP constraint in abscissa. The curves labeled by H1:cap
and H2:cap report the capacity corresponding to the channel matrices H1 and H2 , respectively. The piecewise horizontal lines
labeled by H1:rank and H2:rank report the rank of the capacity-achieving covariance matrices corresponding to the channel
matrices reported in (17) and (18), respectively.

range as the TP constraint decreases.

C. Complexity

The time complexity of numerical optimization is illustrated in Fig. 8. The points report the
average CPU time to calculate the capacity of an n × n MIMO channel with iid Rayleigh
distributed channel gains. The PAP constraint is set to

P = diag(1, 0.1, . . . , 0.1)

| {z }
n−1
and the TP constraint is set to Ptot = 1. We can see that the time complexity growth linearly with
nT and the proposed optimization algorithm based on (6), which is a consequence of the fact
 16

 
 0.1038 − 0.0877 j −0.4125 − 1.6836 j 1.9318 + 0.2237 j 
H3×3 = −0.0410 − 0.0299 j −0.5255 − 0.5724 j −0.7740 − 1.4445 j  (19)
 
 
0.0074 + 0.1378 j −0.6510 + 0.4731 j 1.4142 + 0.8371 j
 
0.3576 + 0.1914 j 0.0614 + 0.1692 j −0.7338 + 1.0030 j −0.0943 − 0.3671 j
 
 0.7547 + 0.1277 j 0.5032 − 0.2920 j −0.4087 − 1.9283 j 0.3410 − 0.0309 j 
H4×4 =
 

−0.4020 + 1.1647 j 0.1404 + 0.3537 j 1.6532 + 3.0492 j −0.4247 − 0.0599 j 
 
0.7130 − 2.0329 j 0.4724 + 0.5394 j −0.2910 + 0.7012 j −0.7934 + 0.2927 j
(20)

9.5

8.5

8
C (bit/s/Hz) / rank

7.5

6.5

6
3x3 MIMO
4x4 MIMO
5.5
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
P

Fig. 5. Capacity of the 3 × 3 and 4 × 4 MIMO channels, described by the channel matrices in (19) and (20), respectively,
versus a constant PAP constraint P in abscissa and a TP constraint Ptot = 3 and 4, respectively.
 17

3.5
C (bit/s/Hz) / rank

2.5

3x3 MIMO
4x4 MIMO
1.5
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
P

Fig. 6. Same as Fig. 5, but with TP constraint Ptot = 0.3 and 0.4, respectively.

that the variable space is nT -dimensional. The basic optimization algorithm based on problem
(1) is far more expensive in terms of cpu-time complexity. The simulation results show a more
than cubic growth with nT , more than the increased dimension of the variable space, which is
n2T in this case. This excessive growth can be partly explained by noting that the basic algorithm
requires a positive definiteness test in the constraint equations, whose complexity is cubic in the
matrix size. Needless to say, the analytic complexity evaluation of optimization algorithms is
a hard topic and we limit ourselves to make a comparison based on simulation results. In the
unit-rank case, Algorithm 1 attains the minimum time complexity, several orders of magnitude
lower than the complexity of solving the optimization problems (1) and (10).

V. C ONCLUSIONS

In this work we addressed the derivation of the capacity of a MIMO channel under joint
total and per-antenna average power constraints. We have considered separately the full-rank
 18

0.9

0.8

0.7
C (bit/s/Hz) / rank

0.6

0.5

0.4

0.3

3x3 MIMO
4x4 MIMO
0.2
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
P

Fig. 7. Same as Fig. 5, but with TP constraint Ptot = 0.03 and 0.04, respectively.

and singular cases, which consist, respectively, of having the channel matrix rank equal to or
lower than the number of transmit antennas.
Closed-form results have been provided in a special full-rank case (Corollary 1) and for the
general unit-rank MIMO case (Theorem 3). The unit-rank case is supported by an iterative
algorithm similar to the water-filling algorithm leading to the unit-rank capacity achieving
covariance matrix (Algorithm 1).
When the channel rank is greater than one, numerical optimization seems to be the only way
to find the channel capacity. Then, we focused on the complexity of numerical optimization,
identified by the number of equivaent real variables in the optimization problem. First, we noticed
that the general case can be always solved by an optimization problem depending on n2T real
variables, i.e., problem (1). Then, we derived equivalent optimization problems with a smaller
number of equivalent real parameters for the full-rank case (Theorem 1) and for the singular
 19

1
Time complexity
10
BASIC
PROPOSED

10 0
Average cpu time (s)

10 -1

10 -2
2 3 4 5 6 7 8 9 10
n R =n T

Fig. 8. Average cpu time versus number of antennas based on the optimization problems (1) (BASIC) and (6) (PROPOSED).
Random iid Rayleigh n × n MIMO channel.

case (Theorem 2). The total number of equivalent real parameters is given by Corollary 2. The
range of possible values of the capacity-achieving input covariance matrix rank is derived in
Corollary 3.
Finally, numerical results are reported i) to validate the optimization algorithms against existing
literature results; ii) to illustrate the application of the proposed methods to practical MIMO
scenarios where the per-antenna constraint is of major interest; and iii) to evaluate the time
complexity of the proposed algorithms.
 20

A PPENDIX A
P ROOF OF T HEOREM 1

Proof: The optimization problem leading to the capacity considered in Theorem 1 leads to
the Lagrangian function given by

L(Q) = − log det(InR + HQH H ) + λ[tr(Q) − Ptot ]

+ tr(Λ(Q − P )) − tr(M Q).

The KKT equations can be written as follows:

H H (InR + HQH H )−1 H = λInT + Λ − M (21-a)

λ≥0 (21-b)

Diagonal Λ ≥ 0 (21-c)

Hermitian M , Q ≥ 0 (21-d)

MQ = 0 (21-e)

Λ(diag(Q) − P )) = 0 (21-f)

λ[tr(Q) − Ptot ] = 0 (21-g)

diag(Q) ≤ P (21-h)

tr(Q) ≤ Ptot (21-i)

The diagonal matrix λInT + Λ is positive definite since

(λInT + Λ)ii = λ + λi

= hHi (InR + HQH H )−1 hi + (M )ii

≥ hHi (InR + HQH H )−1 hi

> 0.

The inequalities depend on the fact that M ≥ 0 (hence (M )ii ≥ 0) and that InR + HQH H > 0
(since Q ≥ 0). The last inequality holds unless hi = 0, which would imply no signal transmission
from the i-th antenna. Next, we define the diagonal matrix

D , λInT + Λ > 0. (22)

 21

From the KKT equations (21) we get the following equations:

H H (InR + HQH H )−1 HQ = DQ (23-a)

Diagonal D > 0 (23-b)

diag(Q) ≤ P (23-c)

tr(Q) ≤ Ptot (23-d)

Now, the constraint (23-d) must be met with equality. Assume, on the contrary, that the optimum
covariance matrix Qopt satisfies the inequality tr(Qopt ) < Ptot . Then, we consider a covariance
matrix Q(α) defined by
 α/2  α/2
Pi Pj
(Q(α))ij , (Q)ij .
(Qopt )ii (Qopt )jj
It is plain to see that Q(α) ≥ 0. Moreover, Q(0) ≡ Qopt and Q(1) > Qopt since at least for
one index i with 1 ≤ i ≤ nT we have (Qopt )ii < Pi (otherwise, if (Qopt )ii = Pi for every
i = 1, . . . , nT , it would be tr(Qopt ) = ni=1 (Q)ii = ni=1
PT PT
Pi ≥ Popt from (2)). Next, we define
nT  α
X Pi
τ (α) , tr[Q(α)] = (Qopt )ii .
i=1
(Qopt ) ii

We can see that τ (α) is a continuous monotonically increasing real function of α for α ≥ 0.
Moreover, τ (0) = tr(Qopt ) < Ptot by assumption, and
nT
X
τ (1) = Pi > Ptot
i=1

by inequality (2). Hence, there exists α̂ ∈ (0, 1) such that τ (α̂) = Ptot and Q(α̂) > Qopt
satisfying the total power constraint with equality and attaining a greater mutual information
log det(InR + HQ(α̂)H H ) > log det(InR + HQopt H H ), contrary to the assumed optimality of
Qopt .
As a consequence, the optimization problem to be solved to find the capacity is given by:

H H (InR + HQH H )−1 HQ = DQ (24-a)

Diagonal D > 0 (24-b)

diag(Q) ≤ P (24-c)

tr(Q) = Ptot (24-d)

where (Q)ii < Pi for at least one index i ∈ {1, . . . , nT }.

 22

Now we find an equivalent expression of (24-a):

(a)
(24-a) ⇒HD −1 H H (InR + HQH H )−1 HQH H = HQH H (25-a)
(b)
⇒HD −1 H H HQH H = HQH H (InR + HQH H ) (25-b)
(c)
⇒F R = R + R2 (25-c)

where we (a) left-multiplied by HD −1 and right-multiplied by H H (24-a); (b) right-multiplied

both sides of (25-a) by (InR + HQH H ) > 0; (c) left-multiplied by (H H H)−1/2 H H and right-
multiplied by H(H H H)−1/2 (25-b), and finally applied the definitions:

K , (H H H)1/2

Ď , D −1

F , K ĎK

R , KQK

The matrices K, F , R are Hermitian, K, F are positive definite and Q ≥ 0. Since

F R = R + R2 = (F R)H = RF ,

we have from [12, Th.1.3.12] that F , R are simultaneously unitarily diagonalizable, i.e.,

F = K ĎK = U ΛF U H

R = KQK = U ΛR U H

for some unitary matrix U and diagonal matrices ΛF , ΛR . Thus, we can rewrite (25-c) as

ΛF ΛR = ΛR + Λ2R . (26)

Given ΛF we can see that ΛR = (ΛF − InT )+ . In fact, since ΛF and ΛR are diagonal matrices,
we can consider any given diagonal position where the scalar equation is

λF λR = λR + λ2R = (λR + 1)λR .

Since Q ≥ 0 and Ď > 0, λR ≥ 0 and λF ≥ 0. If λF > 1, the equation has two possible
solutions: λR = 0 or λR = λF − 1. Since our goal is maximizing the mutual information, we
choose the latter. Otherwise, if λF ≤ 1, λR = 0 is the only possible solution. Therefore, our
chosen solution is λR = (λF − 1)+ . The extension to the matrix solution is direct.
 23

Since K is nonsingular, we have

Qopt = K −1 U (ΛF − InT )+ U H K −1

= K −1 (F − InT )+ K −1

= K −1 (K ĎK − InT )+ K −1 , (27)

which proves eq. (5) of the Theorem and leads to the optimization problem (6).

A PPENDIX B
P ROOF OF T HEOREM 2

Proof: A preliminary part of this proof is equivalent to that of Theorem 1 but the sequel is
different because in this case the rank of H is smaller than nT and hence H H H is singular.
As in the proof of Theorem 6, we obtain the equivalent optimization problem (24). Then, we
have the following equivalent expression of (24-a):

F R = R + R2 (28)

with a different definition of the matrices F , R:

Ď , D −1 , F , H ĎH H , R , HQH H .

Again, the matrices F , R, Q are Hermitian positive semidefinite. From [12, Th.1.3.12], F and
R are simultaneously unitarily diagonalizable as

F = H ĎH H = U ΛF U H

R = HQH H = U ΛR U H

for some unitary matrix U and diagonal matrices ΛF , ΛR . Thus, we can rewrite (28) as

ΛF ΛR = ΛR + Λ2R . (29)

From the same arguments used after (26), we can see that ΛR = (ΛF − Iν )+ . Then, from the
reduced-size SVD H = UH ΛH VHH , we obtain, for a given matrix Ď,

Λ−1 H H −1 H
H UH (H ĎH − InR )+ UH ΛH = VH QVH . (30)

The proof of the Theorem follows.

 24

As far as concerns the rank of the capacity-achieving covariance matrix Q, we recall two of
the KKT equations (21):

H H (InR + HQH H )−1 H + M = D (31-a)

MQ = 0 (31-b)

From (31-a), since D > 0 (see Proof of Theorem 1), so that rank(D) = nT , and rank(H H (InR +
HQH H )−1 H) = ν, we can apply inequality [12, 0.4.5.1] and obtain

rank(M ) ≥ rank(D) − rank(H H (InR + HQH H )−1 H)

= nT − ν. (32)

Moreover, from (31-b) the matrices M , Q commute and hence [12, Th.1.3.12] we can write
them as
M = U ΛM U H , Q = U Λ Q U H .

Thus, we have ΛM ΛQ = 0 and then inequality (32) implies that

rank(Q) = rank(ΛQ ) ≤ ν.

Obviously, rank(Q) ≥ 1 otherwise Q would be the all-zero matrix. This proves (12) of Corollary
3.
Additionally, we note that the covariance matrix Q is not determined only by Ď, which is
specified by ν real parameters (the diagonal entries). Since the capacity-achieving Q has rank
not greater than ν it can be specified by (2nT − ν)ν real parameters. Conditions (30) correspond
to ν 2 independent real equations so that the number of real variables in this optimization problem
is given by nT + (2nT − ν)ν − ν 2 = 2(nT − ν)ν + nT , which proves (11) in Corollary 2.

A PPENDIX C
P ROOF OF T HEOREM 3

Proof: We can set ν = 1, UH = u, ΛH = kHk, VH = v, with kuk = kvk = 1, so that

H = kHkuv H .

Then, from Corollary 3, the capacity-achieving covariance matrix has unit rank and we set

Q = qq H .
 25

Then, we notice that

log det(InR + HQH H ) = log det(InR + kHk2 uv H QvuH )

= log(1 + kHk2 |v H q|2 ).

Since nT
X
H
v q= |vi qi |e j δi ,
i=1

where vi , (v)i , qi , (q)i , δi , ∠qi − ∠vi , we can see that

nT X
X nT
H 2
|v q| = |vi qi ||vj qj | cos(δi − δj )
i=1 j=1
nT X
X nT nT
X 2
≤ |vi qi ||vj qj | = |vi qi | .
i=1 j=1 i=1

The maximum is attained when δi is any constant so that we can set δi = 0. This implies that
the maximum value of |v H q| is attained when ∠qi = ∠vi .
Once we found the phases of the qi we need the absolute values wi , |qi |, which can be
found by considering the optimization problem
nT
X
min − |vi |wi (33-a)
wi ,i=1,...,nT
i=1

s.t. − wi ≤ 0, i = 1, . . . , nT (33-b)
p
wi − Pi ≤ 0, i = 1, . . . , nT (33-c)
nT
X
wi2 − Ptot ≤ 0 (33-d)
i=1

The corresponding Lagrangian function is

nT
X nT
X nT
X p
L(w) = − |vi |wi − λi w i + µi (wi − Pi )
i=1 i=1 i=1
nT
X 
+λ wi2 − Ptot
i=1
 26

The corresponding KKT equations are

−|vi | − λi + µi + 2λwi = 0 i = 1, . . . , nT (34-a)

p
0 ≤ wi ≤ Pi i = 1, . . . , nT (34-b)
p
λi wi = µi (wi − Pi ) = 0 i = 1, . . . , nT (34-c)
nT
X
wi2 ≤ Ptot (34-d)
i=1
nT
X 
λ wi2 − Ptot =0 (34-e)
i=1

λ ≥ 0, λi , µi ≥ 0 i = 1, . . . , nT (34-f)

From (34-a) we get

|vi | + λi − µi
wi =
2λ
if λ > 0. The case λ = 0 implies that the solution of (34-a) is undefined so that it must
be discarded. As a consequence, from (34-e), the TP constraint (34-d) must be attained with
equality. Next, we can see from (34-a) that wi = 0 is not possible since |vi | > 0, λi ≥ 0, µi = 0.
√
If wi < Pi , then
|vi |
wi = .
2λ
√
If wi = Pi , then
|vi | − µi
wi = .
2λ
These conditions can be summarized by setting
 
|vi | p
wi = min , Pi .
2λ
Defining the unknown α , 1/(2λ)2 , we have wi2 = min(αi |vi |2 , Pi ) and get equation (15). The
lhs of this equation is a monotonically nondecreasing function of α, which is equal to 0 when
α = 0 and is equal to ni=1
PT
Pi when α → ∞. Thus, we have a single solution for α, which can
be found by Algorithm 1, which is based on rewriting (15) in the equivalent form
XnT
min(α, ρπ(i) )|vπ(i) |2 = Ptot
i=1
2
where ρi , Pi /|vi | and π is the permutation of the set {1, . . . , nT } such that ρπ(i) ≤ ρπ(i+1)
for i = 1, . . . , nT − 1. The solution is based on successive trials based on the assumption that
α ≤ ρπ(i) for i = 1, . . . , nT .
 27

Algorithm 1 Algorithm for the solution of eq. (15)

1: procedure C ALCULATE -α(vi , Pi , Ptot , i = 1, . . . , nT )

2: Define ρi , Pi /|vi |2 .
3: Sort ρi increasingly and let π be the permutation such that ρπ(i) ≤ ρπ(i+1) , i =
1, . . . , nT − 1.
4: for i = 1, . . . , nT do
5:
Ptot − i−1
P
Pπ(k)
α = PnT k=1 2
k=i |vπ(k) |

6: if α ≤ ρπ(i) then
7: return α
8: end if
9: end for
10: end procedure

R EFERENCES

[1] T.M. Cover and J.A. Thomas, Elements of Information Theory. New York: Wiley, 2006.
[2] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge Univ. Press, 2005.
[3] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: Information-theoretic and communications aspects,” IEEE
Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
[4] S. Loyka, “The capacity of Gaussian MIMO channels under total and per-antenna power constraints,” IEEE Trans. on
Commun., vol. 65, no. 3, pp. 1035–1043, March 2017.
[5] M. Vu, “MIMO capacity with per-antenna power constraint,” in Proc. IEEE GLOBECOM 2011, Houston, TX, USA, Dec.
2011.
[6] “System, method and apparatus for multi-input multi-output communications over per-transmitter power-constrained
channels.” U.S. Patent No. 9,503,170.22, issued Nov. 2016.
[7] M. Vu, “The capacity of MIMO channels with per-antenna power constraint, Arxiv preprint arXiv:1106.5039, 2011.
[8] P.L. Cao, T.J. Oechtering, R.F. Schaefer, and M. Skoglund, “Optimal transmit strategy for MISO channels with joint sum
and per-antenna power constraints,” IEEE Trans. Signal Process., vol. 64, no. 16, pp. 4296–4306, Aug. 2016.
[9] M. Khoshnevisan and J.N. Laneman, “Power allocation in multi-antenna wireless systems subject to simultaneous power
constraints,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3855–3864, Dec. 2012.
[10] D. Tuninetti, “On the capacity of the AWGN MIMO channel under per-antenna power constraints,” in Proc. IEEE Int.
Conf. Commun. (ICC), Sydney, NSW, Australia, Jun. 2014.
[11] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
[12] R. Horn and C. Johnson, Matrix Analysis (2nd ed.). New York: Cambridge University Press, 2013.
[13] G. Taricco, “On the beamforming capacity of MISO channels,” IEEE Wireless Commun. Letters, vol. 1, no. 2, pp. 141–144,
April 2012.
 28

[14] J.-F. Bonnans, J.C. Gilbert, C. Lemarechal, C.A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects
(2nd ed.). Springer, 2006.