Precoding Strategy Selection for Cognitive MIMO
Multiple Access Channels using Learning Automata
1
2
Wei Zhong1,2 , Youyun Xu2,1 , Meixia Tao1
Dept. of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Institute of Communication Engineering, PLA University of Science & Technology, Nanjing 210007, China
Email: {[email protected], [email protected], [email protected]}
Abstract—In this paper, we study the quantized precoding
strategy selection for multiple-input multiple-output (MIMO)
multiple access channels (MAC) in cognitive radio (CR) networks
through a game-theoretic perspective. Since the secondary users
in such system are difficult to be coordinated by a centralized
authority, they are noncooperative and attempt to maximize their
own payoffs selfishly in a distributed method. We propose a
noncooperative precoding strategy selection game and find that
it is a potential game which possesses at least one pure strategy
Nash equilibrium. A decentralized learning algorithm with a
small amount of feedback is proposed to obtain Nash equilibrium.
We prove that the proposed algorithm can converge to a pure
strategy Nash equilibrium. Simulation results are provided to
verify our analysis.
Index Terms—MIMO, cognitive radio, precoding, potential
game, learning automata.
I. I NTRODUCTION
Cognitive multiple-input multiple-output (MIMO) radio [1]
[2] is drawing more or more attention for its significant
promises in increasing spectral efficiency. On the one hand,
it possesses the merits of being cognitive in spectrum access.
On the other hand, it enjoys the dramatic increase in channel
capacity by using multiple antennas. To deliver these promises,
it is essential to coordinate the transmission schemes for each
secondary user, in particular, in the multiple-access channel.
This paper is concerned with the precoding matrix selection for
MIMO multiple-access channels (MAC) in a cognitive radio
(CR) network. In a typically MIMO MAC in CR network,
multiple secondary users, coexisting with primary users and
sharing the same spectrum, attempt to access a common
receiver under certain interference temperature constraints (to
protect primary users).
It is known that the optimal transmission strategy in traditional non-cognitive MIMO MAC requires full knowledge
of channel state information (CSI) at all links [3] and is
hence difficult to be implemented in practice. The interference
temperature constraints imposed by the cognitive environment
further complicates the problem. Thus, we are motivated
to consider a quantized precoding technique for the MIMO
MAC in a CR network where each secondary user simply
chooses a precoding strategy from a finite set (i.e. codebook).
Furthermore, given that the common receiver may have no
prior information of the secondary users, we assume that no
interference cancelation techniques are used and the multiuser
interference is treated as additive, albeit colored, noise. There-
fore, our considered MIMO MAC is noncooperative and the
codebooks for each secondary user can only be designed locally. More specifically, we consider that each secondary user
selfishly selects a precoding strategy from a predetermined,
finite, and discrete set to maximize its own payoff. In this case,
centralized algorithm is usually impractical as the codebooks
of the users are not known at the receiver. Furthermore,the
exhaustive search algorithm is too complex to be used for its
complexity increasing exponentially with the number of users.
Thus, distributed precoding strategy selection is of practical
interest.
In this paper, we formulate the secondary user behaviors,
i.e., distributed precoding strategy selection, as a noncooperative game [4]. It is shown that the proposed game falls into the
framework of potential games introduced in [5]. By analyzing
the Nash equilibrium (NE), we find that our proposed game
has at least one pure strategy NE. To obtain the NE in a selforganized method, a decentralized learning algorithm based on
the concept of learning automata is designed. This learning
algorithm is shown to be computationally simple and efficient
in [6] [7]. The learning algorithm is proposed for each user
to modify its current mixed strategy using the random payoff
of successive plays of the game. We prove that the users can
use the decentralized algorithm to efficiently learn one of the
pure strategy NEs of the proposed game.
Precoding or beamforming in CR networks has been studied
from different aspects in the literature. Zhang et al studied
the joint beamforming and power allocation for a single-input
multiple-output (SIMO) MAC in CR networks in [8]. We
extend SIMO MAC to MIMO MAC as the terminals in future
networks are likely to be equipped with multiple antennas.
Furthermore, the work in [8] is a global optimization problem
which requires full CSI at all transmitters and the receiver
while we focus on the competitive optimality which is a more
practical consideration in a noncooperative system. Scutari et
al studied the linear precoding strategies for noncooperative
systems through a game theoretic approach in [9] [10]. The
extension to the cognitive MIMO radio system is considered
in [2]. Therein, both theoretic analysis and the algorithm
are carefully investigated. However, unlike our work, the
receivers of the secondary users are not the same and the
precoding strategies are not chosen from a finite set. Moreover,
MIMO asynchronous IWFA in [2], which needs to perform
the waterfilling procedure using the overall interference-plus-
noise covariance matrix, and hence still has a considerable
complexity.
Notations: s†i denotes the conjugate transpose of si , tr(·)
stands for trace, |·| denotes determination, IM denotes the M ×
M identical matrix, max(·) denotes the maximum operator,
min(·) denotes the minimum operator.
II. S YSTEM M ODEL
Consider a cognitive MIMO MAC with n secondary users.
The receiver is equipped with Mr antennas, each secondary
users is equipped with Mti ≤ Mr transmit antennas for
i ∈ N , where N = {1, · · · , n}. Assume that n secondary
users attempt to access the channel simultaneously and the
channel is static. The receiver has no priori information of the
secondary users. Each secondary user interferes each other.
Let Li denote the number of transmission data substreams (i.e.
modes) of secondary user i. The main difference of the mode
Li between our work and the previous works in single user
MIMO system is that Li is allowed to be “0”. When Li = 0,
the secondary user i will be shut down, i.e. no substream will
be transmitted. In this paper, we assume that Li ∈ Li , where
Li ⊆ {0, 1 · · · , Mti } is the set of the supported modes. As
we only focus on precoding strategy selection, power control
is not considered in this paper.
A precoding strategy Fi , chosen from codebook Fi , maps
Li data substreams to Mti transmit antennas of secondary
user i. The codebook Fi = {ḞiLi }Li ∈Li is a finite set of
precoding strategies. The codebooks of the secondary user
are independent and locally designed. Note that, Ḟi0 = 0
Mt
and Ḟi i = {IMti }. When Li = 0 and Li = Mti ,
Li
Li
ḞiLi = {Ḟi1 , · · · , ḞiK Li }, where KiLi is the cardinality
i
Li
Ki , the cardinality of Fi , is equal to
of
Ḟi . Obviously,
Li
Li ∈Li Ki .
The cognitive MIMO MAC with quantized precoding is
modeled as
n
n
y=
Hi Fi xi + z + v =
(1)
H̃i xi + z + v
i=1
i=1
where y (Mr ×1) is the channel output, xi (Li ×1) is the
channel input of secondary user i, z (Mr ×1) is a vector of
additive white Gaussian noise (AWGN) with zero mean and
covariance equal to IMr , v is the interference vector from the
primary users, Hi is the Mr ×Mti channel matrix of secondary
user i, and H̃i = Hi Fi is the Mr ×Li matrix associated
with the selected precoding matrix Fi of secondary user i.
We assume that v is zero-mean circularly symmetric complex
Gaussian with arbitrary covariance matrix V. For ∀i ∈ N , the
entries of Hi are independent, identically distributed (i.i.d.)
and zero-mean unit-variance circularly symmetric complex
Gaussian.
In order to protect the communications of primary users
in the CR network, the total received power at a specified
measurement point must satisfy the following interference
temperature constraint
Co.1 : P ≤ ρth
(2)
where
P =
n
i=1
tr
ρi
Gi0 Fi F†i G†i0
Li
(3)
Gi0 is the Mg × Mti channel matrix between secondary user
i and the measurement point, ρi is the transmit signal to noise
ratio (SNR) of secondary user i when it is active, ρth is a
predefined threshold. Here, as quantized precoding is applied,
the power is assumed to be uniformly allocated among the
selected substreams for each secondary user. Assume that
{Gi0 Fi }i∈N are perfectly known by the measurement point.
For convenience, we assume that the system just has one measurement point. However, our model can be easily extended
to the scenario where multiple measurement points exist. In
that case, multiple interference temperature constraints should
be satisfied simultaneously.
Since the total number of spatial degrees of freedom
is
n
L
),
limited by min(Mr , Mtotal ) (where Mtotal =
i
i=1
increasing the number of transmit antennas at the secondary
users will not increase the total number of spatial degrees of
freedom further when Mtotal > Mr . Hence, the portion of
the data streams more than min(Mr , Mtotal ) can give just a
little contribution to the gain of the sum rate. Moreover, in
the perspectives of the link performance such as bit error rate
(BER), it is difficult to achieve the reliable communication
when Mr < Mtotal . As a result, in order to tradeoff the
complexity, reliability and performance of the system, the total
number of transmission streams should satisfy the following
constraint
(4)
Co.2 : Mtotal ≤ Mr .
In this paper, we assume that the system can support at
least one secondary user to transmit under Co.1 and Co.2
simultaneously.
III. N ONCOOPERATIVE Q UANTIZED P RECODING
S TRATEGY S ELECTION G AME
As the MIMO MAC is noncooperative, the coordination
mechanism is unavailable. Then each secondary user’s transmission is a source of interference for the others. When
a secondary user selfishly chooses a precoding strategy to
increase its own payoff, it may increase the interference of
some other users. Thereby, the strategies chosen by different
secondary users depend on each other. Such problem can be
modeled as a noncooperative game as below.
Let G E =[N , ζ, {ui }i∈N ] denote the noncooperative quantized precoding strategy selection game for the cognitive
MIMO MAC, where the player set N ={1, . . . , n} is the index
set for the rational secondary users currently in the system, the
pure strategy set is the set of all possible precoding strategies
ζ = F1 × · · · × Fn (× denotes the Cartesian product here),
and the payoff function of secondary user i is defined as:
ui = Ri + β P Θ(ρth − P ) + β M Θ(Mr − Mtotal ) (5)
†
where Ri = 12 log2 Lρii H̃i H̃i + Ni − 12 log2 |Ni | is the mutual
information of secondary user i when the inference of the other
n ρ
†
secondary users are regarded as noise, Ni = j=i Ljj H̃j H̃j +
IMr + V, function Θ(s) is defined as Θ(s) = s for s < 0 and
Θ(s) = 0 otherwise, β P and β M are non-negative scalars,
the second term in (6) is an indication of the interference
temperature constraint, the third term in (6) is an indication
of the maximum transmission number constraint.
Then each secondary user maximizes its individual payoff
depending on the precoding strategies of all the other terminals
in the system. This can be expressed as:
GE :
max ui (Fi , F−i ) ,
Fi ∈Fi
f or all i ∈ N
(6)
where F−i is the precoding strategies of all the secondary users
excluding the i-th secondary user.
Defining a function RP as:
RP = R + β P Θ(ρth − P ) + β M Θ(Mr − Mtotal )
where
n
ρ
1
†
i
H̃i H̃i + IMr + V
R = log2 2
L
i
i=1
ρi
1
†
= log2 H̃i H̃i + Ni .
2
Li
(7)
(8)
In this paper, we assume that β P is chosen to be sufficiently
large to ensure that ui < 0, ∀i ∈ N and RP < c when
B > ρth . In the special case where there is no interference temperature constraint (e.g. secondary users share the
spectrum in unlicensed band or the primary users stop to
transmit signals), β P = 0. Furthermore, β M is assumed to
be sufficiently large to guarantee that ui < 0, ∀i ∈ N and
RP < c when Mtotal > Mr . If this constraint Co.2 is not
considered, we can set β M = 0.
Theorem 1: G E is a potential game with the potential
function RP .
Proof: Assume that Fi ∈ Fi is an arbitrary strategy of
secondary user i with mode Li , Fi ∈ Fi is an alternate strategy
of secondary user i with mode Li , the strategies of the other
secondary users stay unchanged. Let
P = tr
ρi
†
Gi0 Fi F†
i Gi0
Li
+
tr
j∈N ,j=i
and
Mtotal
= Li +
ρj
† †
Gj0 Fj Fj Gj0
Lj
(9)
Lj .
(10)
j∈N ,j=i
Then, we can have
ui (Fi , F−i ) − ui Fi , F−i
ρi
1
ρi †
1
†
= log2 H̃i H̃i + Ni − log2 H̃i H̃i + Ni 2
Li
2
Li
th
th
P
P
+ β Θ ρ − P − β Θ ρ − P
+ β M Θ(Mr − Mtotal ) − β M Θ(Mr − Mtotal
)
= RP (Fi , F−i ) − RP Fi , F−i .
(11)
Thus according to definition of potential game in [5], G E is
a potential game and RP is the potential function of G E .
We propose another game G C =[N , ζ, {uC
i }i∈N ], where the
payoff function of each secondary user i is the same and
C
is a game
defined as uC
i = RP for all i ∈ N . Then G
with common payoff and the following theorem is provided.
Theorem 2: The NEs of G C coincide with the the NEs of
G E and G E which possesses at least one pure strategy NE.
Proof: The first statement of this theorem is easily proved
according to Lemma 2.1 in [5]. Next we shall focus on the
proof of the second statement. Suppose that {Foi }i∈N is a
maximizer of RP . ∀i ∈ N , Fi ∈ Fi , Fi = Foi is an alternate
strategy of secondary user i, we can get
o o o C
(12)
uC
i Fi , F−i < ui Fi , F−i .
That is, if an arbitrary secondary user changes its own strategy
unilaterally, it can never get a payoff which is larger than the
payoff obtained by {Foi }i∈N . Thus, according to the definition
of NE in [4], the precoding strategy tuple {Foi }i∈N is a pure
strategy NE of game G C . Then {Foi }i∈N is also a pure strategy
NE of game G E , according to Lemma 2.1 in [5]. Hence the
theorem is proved.
Theorem 3: The precoding strategy tuple that does not
satisfy Co.1 or Co.2 cannot be the pure strategy NE of G E with
the payoff function defined in (6), if for an arbitrary strategy
Fi = 0, Fi ∈ Fi , ∀i ∈ N ,
⎞
⎛⎧
⎫
⎬
⎨
R (F , F )
⎟
⎜
i i −i β P > max ⎝
⎠
⎩ tr ρi G F F† G† ⎭
i0
i
i i0
Li
i∈N ,Fi ∈Fi ,Fi =0
and
β M > max
Ri (Fi , F−i )
Li
(13)
.
(14)
i∈N ,Li ∈Li ,Li =0
Proof: Assume that there exists an NE that {Fi }i∈N
does
Co.1 or Co.2. Then, {Fi }i∈N = {0}i∈N and
not satisfy
ui Fi , F−i < ui (0, · · · , 0) = 0, ∀i ∈ N .
For an arbitrary strategy Fi = 0, then, based on (14) and
(15), we can have
†
(15)
G
Ri Fi , F−i − β P tr ρ̄i Gi0 Fi F†
i
i0 < 0
and
Ri Fi , F−i − β M Li < 0.
(16)
Thus, ui Fi , F−i < ui 0, F−i . This contradicts the
assumption that {Fi }i∈N is a NE of G E by definition. Hence,
{Fi }i∈N can never be a NE of G E .
Based on Theorem 3, it is easy to know that the precoding
strategy tuple which does not satisfy both Co.1 and Co.2
cannot be the pure strategy NE of G E if (15) and (16) are
satisfied.
IV. L EARNING AUTOMATA AND D ECENTRALIZED
L EARNING A LGORITHM
In this section, a decentralized stochastic learning algorithm
which employs a team of learning automata to evolve to the
NE of G E with only a small amount of information exchange
[6] is designed to obtain the NE.
To characterize the learning algorithm, we extend the game
G E to a mixed strategy form. Let pi = [pi1 , · · · , piKi ] be the
mixed strategy of the secondary user i, where pik denotes the
probability with which ith secondary user chooses the kth pure
strategy. The expected payoff g i for secondary user i is given
by [6] [7]
g i (p1 , · · · , pn ) = E[uC
i | jth user employs strategy
pj , 1 ≤ j ≤ n] =
{Ḟik }i∈N
uC
i
n
psk
(17)
s=1
If the mixed strategy precoding game is played successively,
it could be modeled as a stochastic game of learning automata.
Each secondary user (i.e. player) is represented by a learning
automaton and the actions of the automaton are the pure
strategies of the secondary user. The mixed strategy pi (t) is the
action probability distribution of the ith automaton at instant
t. pik (t) denotes the probability with which ith automaton
chooses the kth pure strategy at instant t. The normalized
payoff to the ith secondary user will be the reaction to the ith
uC (t)−A
automaton which is denoted as ri (t). We let ri (t) = iB−A
C
be the normalized uC
i (t), where B = maxt (ui (t)) and
C
A = mint (ui (t)). The value of ri (t) lies in the interval [0, 1).
Then, at each iteration, r1 (t) = · · · = rn (t) = r(t). When one
of the automata chooses an action independently according to
its current action probabilities, a play of the game comes into
being. The game is played repeatedly to learn the NE. Let
pi (0) denote the initial mixed strategy of secondary user i.
The learning algorithm used by each of the secondary user is
given as below.
Algorithm 1(LR−I Algorithm at the transmitter):
1 Set the initial probability vector pi (0).
2 At every time instant t, each secondary user i chooses an
action Fti according to his action probability vector pi (t).
Then the secondary user informs the receiver its selected
mode (i.e. Li ).
3 The measurement point evaluates whether the interference temperature constraint is satisfied and then informs
the receiver. The receiver calculates Θ(Mr − Mtotal ) and
r(t). Then it broadcasts r(t) to all secondary users. Each
secondary user obtains the same reaction r(t) based on
the set of all actions.
4 Each secondary user updates his action probability according to the rule
Ḟik = Fti
pik (t + 1) = pik (t) − bri (t)pik (t),
pik (t + 1) = pik (t) + bri (t)(1 − pik (t)), Ḟik = Fti .
(18)
i = 1, · · · , n, k = 1, · · · , Ki .
where 0 < b < 1 is a parameter (the step size).
5 If ∀i ∈ N , there exists a component of pi (t) which
is larger than a value approaching one, say 0.99, stop.
Otherwise, go to step 2.
As we can see that the learning algorithm determines the
mixed strategies for the secondary users by the history of
play. The advantages of Algorithm 1 lie in: 1) Decentralized
and self-organized. 2) Limited feedback. The secondary user
needs to know only the action chosen by himself and his
random payoff at that instant. None of the secondary users
has any information regarding the other secondary users. The
feedback information is only a real number of normalized
common payoff and is not relative to the number of the
secondary users and antennas; 3) Computationally simple
and efficient. Each secondary user just need to compute the
probabilities of different actions at each instant.
Theorem 4: Algorithm 1 converges to one of the pure
strategy NEs of G E with sufficiently small value for b.
Proof: As G C is a game with common payoff which
possesses at least one pure strategy NE, then the Algorithm
1 can converge to one of the pure strategy NEs of G C with
sufficiently small value for b according to the Theorem 4.1
and Remark 4.1 in [6]. From Theorem 2, we know that the
set of NEs of G E coincides with the set of NEs of G C . Thus
when Algorithm 1 converges to a pure strategy NE of G C , it
also finds a pure strategy NE of G E . Hence the theorem is
proved.
V. S IMULATION R ESULTS
In this section, we give numerical results via computer
simulations. We consider a CR network with interference
temperature constraint parameter β P = 102 , maximum transmit substreams number constraint parameter β M = 102 ,
n = 4, N = {1, 2, 3, 4}, Mr = 4, Mt1 = Mt2 = 4,
Mt3 = Mt4 = 2, L1 = L2 = {0, 1, 4}, L3 = L4 = {0, 1, 2},
K1 = K2 = K3 = K4 = 6, Mg = 2, ρth = 10dB, b = 0.1,
V = 0 and SNR=12dB. Since the value of A and B can not
be known in advance, thus we dynamically update the value
of A and B. That is, let A = 0 and B = 10, at each iteration
C
C
t, if A > uC
i (t) or B < ui (t), then we let A = ui (t) or
C
B = ui (t).
Moveover, without loss of generality, we assume that
ρi = ρj , ∀i, j ∈ N and the codebook of each secondary
user is randomly generated using random vector quantization
Li
(RVQ) technique proposed in [11]. That is, Ḟil ∈ CMti ×Li
(l = 1, · · · , KiLi ) is a Mti × Li random unitary matrix, i.e.
Li † Li
Ḟil Ḟil = ILi . Then the precoding matrix of each secondary
user is selected from a random codebook containing i.i.d.
entries. However, the model in our paper is not limited to the
RVQ scheme. We note that using different codebooks does not
affect our theoretical results in this paper.
Fig.1 shows the evolution of the probability of different
precoding strategies of secondary user i = 1 when b = 0.1
for a specific channel realization. It is shown in Fig.1 that
secondary user i = 1 converges to a pure strategy F1 = Ḟ14
VI. C ONCLUSION
1
0.9
p11
p12
p13
p14
p15
p16
0.8
The probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
300
350
400
Iterations
Fig. 1. The evolution of the mixed strategy of a choice secondary user (a
special channel realization).
ACKNOWLEDGMENT
This work was supported in part by the National High-Tech
R&D Program of China (863 Program) (No. 2009AA01Z249),
by the Major State Basic Research Development Program
of China (973 Program) (No. 2009CB3020402), and by
the National Science Key Special Project of China (No.
2008ZX03003-004 and 2008BAH30B09).
0.25
Average mutual information (bps/Hz)
In this paper, we study the quantized precoding strategy
selection for the MIMO MAC in CR network through a gametheoretic perspective. We find that the proposed precoding
strategy selection game is a potential game which possesses
at least one pure strategy NE. And we provide the sufficient
condition for the feasibility (i.e. satisfying the Co.1 and Co.2)
of the pure strategy NE. A decentralized learning algorithm
is proposed to evolve to a pure strategy NE of the proposed
game based on the concept of learning automata. It is proved
that this learning algorithm can converge to a pure strategy
NE. Simulation results show that the proposed algorithm can
substantially improve the average mutual information of each
secondary user under the interference temperature constraint
and the maximum transmission substream number constraint
compared to the random selection scheme.
Random selection
Algorithm 1
0.2
0.15
R EFERENCES
0.1
0.05
0
1
2
3
4
The index of the secondary user
Fig. 2. The average valid mutual information of different secondary users
when SNR=10dB (1000 channel realizations) under Co.1 and Co.2.
by using Algorithm 1 after about 256 iterations (p14 = 1). We
can see in Fig.1 that the Algorithm 1 has good convergence.
Fig.2 plots the average valid mutual information of secondary users under Co.1 and Co.2 by simulating a total of
103 independent trials. In Fig.2, “Random selection” scheme
means that each secondary user i randomly selects a precoding
strategy to transmit. The valid mutual information means that
the achievable mutual information when the corresponding
precoding strategy tuple satisfies the Co.1 and Co.2. If the
corresponding precoding strategy tuple does not satisfy the
constraints, the system is infeasible, then the mutual information is equal to zero. It is shown that Algorithm 1 significantly
outperforms the random selection scheme. Furthermore, we
find that the mutual information of the secondary users obtained by random selection scheme are nearly equal to zero
in Fig.2 which means that the secondary users seldom satisfy
Co.1 or Co.2 by using random selection scheme.
[1] S. Haykin, “Cognitive Radio: Brain-Empowered Wireless Communications,” IEEE J. Select. Areas Commun., vol.23, No.2, pp. 201-220, Feb.
2005.
[2] G. Scutari, D. P. Palomar, and S. Barbarossa, “Coginitive MIMO Radio,”
IEEE Signal Processing Magazine, pp. 46-592, Nov. 2008.
[3] D.J. Love, R.W. Heath Jr, and et al, “ An Overview of Limited
Feedback in Wireless Communication Systems,” IEEE J. Select. Areas
Commun., vol.26, No.8, pp. 1341-1365, Oct. 2008.
[4] R. Myerson, Game Theory: Analysis of Conflict. Cambridge and London:
Harvard University Press, 1991.
[5] D. Monderer and L. S. Shapley, “Potential Games”, Games and Economics Behaviour, vol.14, pp. 124-143, 1996.
[6] P.S Sastry, V.V. Phansalkar, and M.A.L. Thathachar, “Decentralized earning of Nash Equilibria in Multi-Person Stochastic Games with Incomplete
Information,” IEEE Trans Systems, Man, and Cybernetics, vol.24, pp.
769-777, May 1994.
[7] Y. Xing, C.N. Mathur et al, “Dynamic Spectrum Access with Qos
and Interference Temperature Constraints.” IEEE Trans. Mobile Comput., vol.6, No.4, pp.423-432, April 2007.
[8] L. Zhang, Y.C. Liang, and Y. Xin, “ Joint Beamforming and Power
Allocation for Multiple Access Channels in Cognitive Radio Networks,”
IEEE J. Select. Areas Commun., vol.26, No.1, pp. 38-51, Jan. 2008.
[9] G. Scutari, D. P. Palomar, and S. Barbarossa, “Optimal linear precoding
strategies for non-cooperative systems based on game theory-Part I: Nash
equilibria,” IEEE Trans. Signal Process., vol.56, No.3, pp. 1230-1249,
Mar. 2008.
[10] ——, “Optimal linear precoding strategies for non-cooperative systems
based on game theory-Part II: Algorithms,” IEEE Trans. Signal Process.,
vol.56, No.3, pp. 1250-1267, Mar. 2008.
[11] W. Santipach and M. L. Honig, “Capacity of a Multiple-Antenna
Fading Channel with a Quantized Precoding Matrix,” IEEE Trans. Inform.
Thoery, vol.55, No.3, pp. 1218-1234, Mar. 2009.