Balancing Egoism and Altruism on the MIMO Interference - Eurecom

24 Pages·2009·0.21 MB·English

submittedtoIEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS 1
Balancing Egoism and Altruism on the MIMO
Interference Channel
Zuleita K. M. Ho, Student Member, IEEE, David Gesbert Senior Member, IEEE,
Abstract
This paper considers the so-called multiple-input-multiple-output interference channel (MIMO-IC)
which has relevance in applications such as multi-cell coordination in cellular networks as well as
spectrum sharing in cognitive radio networks among others. We address the design of precoding (i.e.
beamforming) vectors at each sender with the aim of striking a compromise between beamforming
gainattheintendedreceiver(Egoism)andthemitigationofinterferencecreatedtowardsotherreceivers
(Altruism).Combiningegoisticandaltruisticbeamforminghasbeenshownpreviouslytobeinstrumental
to optimizing the rates in a multiple-input-single-output interference channel MISO-IC (i.e. where
receivers have no interference canceling capability) [6], [8]. Here we explore these game-theoretic
concepts in the more general context of MIMO channels and using the framework of Bayesian games
[17],allowingustoderive(semi-)distributedprecodingtechniques.Wedrawparallelswithexistingwork
on the MIMO-IC, including rate-optimizing and interference-alignment precoding techniques, showing
how such techniques may be improved and re-interpreted through a common prism based on balancing
egoistic and altruistic beamforming. Our analysis and simulations attest the improvements in terms of
complexity or performance, especially in scenario where existing IA-based methods fail to approach
sum rate optimally.
Index Terms
multi-cell, MIMO, distributed beamforming, Pareto boundary, game theory, Bayesian equilibrium,
interference channels, distributed bargaining, egoistic, altruistic, interference alignment
I. INTRODUCTION
The mitigation of interference in multi-point to multi-point radio systems is of utmost im-
portance and has relevance in several practical contexts. Among the more popular cases, we
This paper is funded by European Project COOPCOM.
October6,2009 DRAFT
2 submittedtoIEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS
may cite the optimization of multi cell MIMO systems with full frequency reuse and cognitive
radio scenarios featuring two or more service providers sharing an identical spectrum license
over overlapping coverage areas. In all these cases, the system may be modeled as a network of
N interfering radio links where each link consists of a sender trying to communicate messages
c
to a unique receiver in spite of the interference arising from or created towards other links.
Recently, the attention of the research community was drawn to the so-called coordinated
transmission methods where interference effects are mitigated or even exploited in exchange
for an additional overhead in exchanging data symbols and channel state information (CSI)
between the transmitters. Bringing coordination to its fullest (i.e. assuming a complete sharing
of data and CSI), multiple transmit antennas can be exploited as a virtual MIMO array and
optimal forms of precoding allow the system designer to effectively exploit interference [7],
[10], [20], [23], [26]. In contrast, in a scenario where the backhaul network cannot support a
complete sharing of data symbols across all transmitters, the channel then remains a so-called
interference-channel whereby the senders can resort to a milder form of coordination that does
not require joint encoding of data packets. Coordination over the interference channel may take
place over one or several domains characterizing the transmission parameters of each sender such
as the choice of power levels [11], beamforming vectors [10], [13], [23], [24], [27], assigned
subcarriers in OFDMA [25], scheduling [1], [12] etc to cite a few.
Recently an interesting framework for beamforming-based coordination was proposed for
the MISO case by which the transmitters (e.g. the base stations) seek to strike a compromise
between selfishly serving their users while ignoring the interference effects on the one hand,
and altruistically minimizing the harm they cause to other non-intended receivers on the other
hand. An important result in this area was the characterization of all so-called Pareto optimal
beamforming solutions for the two-cell case in the form of positive linear combinations of the
purely selfish and purely altruistic beamforming solutions [8], [9], [14]. Unfortunately, how or
whether at all this analysis can be extended to the context of MIMO interference channels
(i.e. where receivers have themselves multiple antennas and interference cancelling capability)
remains an open question.
Coordination on the MIMO interference channel has emerged as a very popular topic recently,
with several important contributions shedding light on rate-scaling optimal precoding strategies
based on so-called interference alignment, subspace optimization, alternated maximum SINR
DRAFT October6,2009
3
optimization, [4], [15], [19] and rate-maximizing precoding strategies [21], [27]. In this paper,
our contributions are as follows:
We re-visit the problem of precoding on the MIMO-IC through the prism of game-theoretic
•
egoistic and altruistic beamforming methods. For doing so, we derive analytically, for two
levels of receiver-to-transmitter feedback, the equilibria of so-called egoistic and altruistic
Bayesian games [5] which are a class of games where players (transmitters) do not have
access to complete channel state information, which is the situation in distributed precoding.
The first level corresponds to no feedback while in the second level, the users are allowed
•
to feedback the coefficients of the receive beamforming coefficients. For each case, we
suggest a precoding technique based on balancing the egoistic and the altruistic behavior
at each transmitter with the aim of approaching the Pareto boundary of the rate region or
maximizing the sum rate.
In the case of no feedback, the precoding scheme is shown to achieve the Pareto Boundary
•
(see later) in MISO-IC.
Inthecaseofreceivebeamformerfeedback,theprecodingschemeprovidesagame-theoretic
•
interpretation of previous work aimed at maximizing the sum-rate over the MIMO-IC, such
as [21].
– Theprecodingtechniqueaimsatsumratemaximization,basedonbalancingtheegoistic
and the altruistic behavior at each transmitter, where the balancing weights are derived
fromstatisticalparameters,hencerequiringlessfeedbackthansumrateoptimalmethods
[21].
– We show that our algorithm exhibits the same optimal rate scaling (when SNR grows)
as shown by recent interesting iterative interference alignment, alternated subspace
optimization and iterative maximum SINR precoding [4], [15], [19].
– At finite SNR, we show improvements in terms of sum rate, especially in the case
of asymmetric networks where interference-alignment methods are unable to properly
weigh the contributions on the different interfering links to maximize the sum rate. This
situation is particularly relevant in practical contexts where, for complexity limitation
reasons, only a subset of cells (links) is coordinated across, while other uncoordinated
links constribute to additional unequal amounts of unstructural interference.
October6,2009 DRAFT
4 submittedtoIEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS
A. Notations
The lower case bold face letter represents a vector whereas the upper case bold face letter
represents a matrix. (.)H represents the complex conjugate transpose. I is the identity matrix.
V(max)(A) (resp. V(min)(A)) is the eigenvector corresponding to the largest (resp. smallest)
eigenvalue of A. is the expectation operator over the statistics of the random variable B.
B
E
S B define a set of elements in S excluding the elements in B.
\
II. SYSTEM MODEL
We study a wireless network of N cells, where a subset of N N transmitters will form a
c
≤
coordination cluster (i.e. will be coordinated across) and are especially considered, while other
transmitters will contribute to uncoordinated interference. The transmitters could be the base
stations (BS) in the cellular downlink. Each transmitter is equipped with N antennas and the
t
receivers (e.g. mobile stations) with N antennas. In each cell of the N cells, an orthogonal mul-
r c
tiple access scheme is assumed, e.g. each transmitter (Tx) communicates with a unique receiver
(Rx) at a time. Transmitters are not allowed or able to exchange user message information,
giving rise to an interference channel over which we seek some form of beamforming-based
coordination. The channel from Tx i to Rx j H Nr Nt is given by:
ji ×
∈ C
H = √α H¯ , i,j = 1,...,N (1)
ji ji ji c
EachelementinchannelmatrixH¯ isanindependentidenticallydistributedcomplexGaussian
ji
random variable with zero mean and unit variance and α denotes the slow-varying shadowing
ji
and pathloss attenuation.
The transmit beamforming vector of Tx i is w Nt 1 and the receive beamforming vector
i ×
∈ C
of Rx i is v Nr 1. As in several important contributions dealing with coordination on the
i ×
∈ C
interferencechannel[1],[3],[4],[9],[13],[22],[28],weassumelinearprecoding(beamforming).
With the noise variance σ2 and individual transmit power P for the N transmitters, the received
i c
signal-to-noise ratio of Rx i is
vHH w 2P
γ = | i ii i| . (2)
i Nc vHH w 2P +σ2
j=i| i ij j| i
6
P
DRAFT October6,2009
5
Tx 1
Tx 2
Tx 3
Tx 4
N = 7;Nc = 4
Fig. 1. This figure illustrates a system of N = 7 cells where Nc = 4 form a coordination cluster. Empty squares represent
transmitters whereas filled squares represent receivers. The noise power (which includes out of cluster interference) undergone
in each cell varies from link to link.
A. Receiver design
The receivers are assumed to employ maximum SINR (Max-SINR) beamforming throughout
the paper so as to also maximize their rates [18]. The receive beamformer is classically given
by:
C 1H w
v = Ri− ii i (3)
i C 1H w
Ri− ii i
| |
where C is the covariance matrix of received interference and noise at Rx i.
Ri
C = H w wHHHP +σ2I (4)
Ri ij j j ij i
j=i
X6
Importantly, the noise will in practice capture thermal noise effects but also any interference
originating from the rest of the network, i.e. coming from transmitters located beyond the
coordination cluster. Thus, depending on path loss and shadowing effects, the σ2 may be
{ i}
quite different from each other [16]. Fig. 1 illustrates a system of N = 7 cells where N = 4
c
form a coordination cluster.
October6,2009 DRAFT
6 submittedtoIEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS
Tx1 Tx2 Tx3 Tx4 Tx5 TxNc
H11 H12 H13 H14 H15
M
1
B
Nc
H
H1Nc H2NcH3NcH4NcH5Nc NcNc
Rx1 Rx2 Rx3 Rx4 Rx5 RxNc
Fig.2. Limitedchannelknowledgemodelforanexampleoftransmitter,hereTxN ,indicatedbydottedlines,andanexample
c
of receiver, here Rx 1, indicated by solid lines.
B. Limited Channel knowledge
To allow for overhead reduction and a better scalability of multi cell coordination techniques
when the number of links N is large, we seek solutions which can operate based on limited,
c
preferably local, CSI. Although there may exist various ranges and definitions of local CSI, we
assume the devices (Tx and Rx alike) are able to gain knowledge of those channel coefficients
directly connected to them, as illustrated in Fig. 2.
The set of CSI locally available (resp. not available) at Tx i denoted by B (resp. B ) is
i ⊥i
defined as:
B = H ; B = H B (5)
i { ji}j=1,...,Nc ⊥i { kl}k,l=1...Nc \ i
Similarly, define the set of channels known (resp. unknown) at Rx i denoted by M (resp. M )
i ⊥i
as:
M = H ; M = H M (6)
i { ij}j=1,...,Nc ⊥i { kl}k,l=1...Nc \ i
Additional receiver feedback: Because the local CSI above is insufficient to exploit all the
degrees of freedom of the MIMO interference channel [4], some additional limited feedback will
be considered where indicated, in the form of binary feedback or feedback of the beamforming
vectors v used at the receiver. In the case of reciprocal channels, the feedback requirement
i
DRAFT October6,2009
7
can be replaced by a channel estimation step based on uplink pilot sequences. Additionally, it
will be classically assumed that the receivers are able to estimate the covariance matrix of their
interference signal, based on, say, transmit pilot sequences.
III. BAYESIAN GAMES ON INTERFERENCE CHANNEL
Bayesian games are a class of games in which players must optimize their strategy based on
incomplete state information [5]. Below we provide a few useful definition for this framework
in the context of the MIMO interference channel.
A Bayesian game is defined as the following,
G =< ,Ω,< ,u ,B > > (7)
N Ai i ⊥i i∈N
where
1) is the set of players in the game, here refers to the set of transmitters 1,...,N .
c
N { }
2) Ω is the set of all possible global channel states CNr Nt Nc.
×
3) i is the action set of player i, here refers to all c(cid:8)hoices o(cid:9)f beamforming vectors wi such
A
that the power constraint is fulfilled w 2 1.
i
| | ≤
4) u : Ω R is the utility function of player i. In the next section we define the
i i
× A →
egoistic and altruistic utilities.
5) B is the missing channel state information at player i.
⊥i
Definition 1: A strategy of player i, s : B is a deterministic choice of action given
i i i
→ A
information B of player i.
i
Definition 2: A strategy profile s = (s ,s ) achieves the Bayesian Equilibrium if s is the
∗ ∗i ∗i ∗i
−
best response of player i given strategy tuple s for all other players and is characterized by
∗i
−
i s = argmax u (s ,s ,B ,B ) (8)
∀ ∗i EB⊥i i i ∗−i i ⊥i
Note that, intuitively, the player’s strategy is opti(cid:8)mized by averaging(cid:9)over the distribution of all
missing state information.
In the following sections, we derive the equilibria for egoistic and altruistic bayesian games
respectively. These equilibria contribute extreme strategies which do not perform optimally in
terms of the overall network performance, yet can be exploited as components of a more general
beamforming-based coordination technique.
October6,2009 DRAFT
8 submittedtoIEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS
A. Egoistic Bayesian Game
In the egoistic game, we assume that each Tx seeks to maximize the received SINR cor-
responding to its user based on the incomplete information defined in (5). Denote the set of
transmit beamforming vectors of users j,j = i, by w . The egoistic utility function for link i
i
6 −
is defined as its received SINR ui(wi,w−i,Bi,B⊥i ) = PNj6=ci|v|viHiHHHiiijwwi|j2|2PP+σi2. And thus the Egoistic
Bayesian equilibrium is a strategy profile s = (w ,...,w ) such that
∗ 1 Nc
vHH w 2P
i ,w = arg max | i ii i|
∀ ∈ N i |wi|≤1EB⊥i Nj=ci|viHHijwj|2P +σi2
6
Assuming Max-SINR receivers at all links, we oPbtain the following result:
Theorem 1: The best-response strategy of Tx i in the egoistic Bayesian game is
wEgo = V(max)(HHH ) (9)
i ii ii
and the corresponding Max-SINR receiver i, according to (3) is v = CRi−1HiiwiEgo
i |CRi−1HiiwiEgo|
Proof:
Lemma 1 (The Expected Inverse Covariance): The expected value of the inverse interference
and noise covariance matrix of Rx i is given by
C 1 = cI (10)
EB⊥i R−i
where c is some positive real number.
Proof: See section IX-A in Appendix.
The SINR at the Max-SINR receiver can be rewritten to wHHHC 1H w . By Lemma 1, the
i ii R−i ii i
transmit beamforming solution yielding the egoistic equilibrium is therefore given by:
maxwHHH (cI)H w = cmaxwHHHH w (11)
i ii ii i i ii ii i
wi wi
Thus, the optimal beamformer is given by V(max)(HHH ) (maximum right singular vector of
ii ii
direct channel matrix).
Note that in the special MISO case (i.e. receivers have only 1 antenna), the egoistic solution
HH
(9) becomes the maximum ratio combining solution: w = ii which coincides with the non-
i Hii
| |
Bayesian game equilibrium shown in [8]. This approach can be shown to be sum-rate optimal in
low SNR regime for the MISO [2] and also the MIMO case, since in this regime the interference
is negligible.
DRAFT October6,2009
9
B. Altruistic Bayesian Game
The utility of the altruistic game is defined here so as to minimize the sum of interference
powers caused to other receivers.
u (w ,w ,B ,B ) = vHH w 2 (12)
i i −i i ⊥i − | j ji i|
j=i
X6
Thus, by applying the Bayesian game framework with Max-SINR receivers, we have:
Definition 3: The best response strategy of Tx i which minimizes the sum of expected inter-
ference power from Tx i to Rx j,j = i is:
6
Nc wHHHC 1H w 2
wAlt = arg min | j jj R−j ji i| (13)
i |wi|≤1EB⊥i j=i |CR−j1Hjjwj|2
X6
and the corresponding Max-SINR receiver i, according to (3) is v = CR−i1HiiwiAlt .
i |CR−i1HiiwiAlt|
1) Special case: 2-link scenario: In the 2 links scenario, the best response of Tx i w(Alt) can
i
be found in a simplified form as follows.
Lemma 2 (Altruistic Solution in 2-link scenario): The altruistic best response strategy of Tx
i in (13) for 2 links (links i and j) is given by
1+σ 2x
wAlt = arg min j− . (14)
i |wi|≤1 2+σj−2x
where x = wHHHH w .
i ji ji i
Proof: see section IX-B in Appendix.
Equation (14) is a concave function and is minimized when x is minimized. Thus, in the
2-link case, the altruistic equilibrium is given by the least-dominant right singular vector of the
interference channel matrix
wAlt = V(min)(HHH ) j = i. (15)
i ji ji 6
In the MISO case, we obtain the so-called zero-forcing beamformer (i.e. orthogonal to the
interference vector). Thus this also generalizes the solution found previously for the MISO case
in [9]. In the MISO case, the altruistic solution tends to be optimal when interference is dominant
i.e. when SNR .
→ ∞
October6,2009 DRAFT
10 submittedtoIEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS
2) Altruistic solution for more than two links: In the N > 2 case, the Max-SINR receiver
c
structure renders the expression of the interference covariance more complicated. The computa-
tion of the exact altruistic equilibrium involves the expectation of an inverse of a Wishart matrix,
which is still an open problem. Instead, we propose a heuristic approach in which the altruistic
beamforming solution is obtained by extending the 2-link case (15):
Nc
wAlt = V(min) HHH (16)
i ji ji
!
j=i
X6
This altruistic solution, while not formally established to be a game equilibrium, allows us to
derive more general precoding techniques.
IV. DISTRIBUTED BEAMFORMING WITH BINARY USER FEEDBACK (DBA-BF)
In the following section, we investigate the design of a scheme which linearly combines the
two extreme solutions. This approach finds an optimality-based justification in the MISO case
[8], [9].
The beamforming vector is initialized, say to wEgo, and is updated at each iteration t by
i
w (t) = w (t 1)+α wAlt (17)
i i − i i
w (t)
w (t) = i (18)
i
w (t)
i
| |
where α is a small constant stepsize. This scheme finds optimality in the N = 1 case as shown
i r
below.
Lemma 3: In 2-link MISO scanerio (i.e. receivers have only 1 antenna), DBA-BF reaches the
Pareto boundary.
Proof: See appendix IX-C
At each iteration t, each Rx i computes its rate r (t) = log (1+w (t)HHHC 1(t)H w (t)).
i 2 i ii R−i ii i
Rx i reports to its transmitter a single bit to inform the base about its satisfaction: increment
of data rates (bit 1) or decrement of data rates (bit 0). In practice, the transmitters cannot
know whether the Pareto boundary is reached or not. A reasonable and fairness-based stopping
condition is that each transmitter would stop cooperating and freeze its beamformer when it
encounters a decrement of transmission rate. User i, 1 i N , would stop cooperating if
c
≤ ≤
r (t) > r (t+1). (19)
i i
DRAFT October6,2009