Nash Bargaining over MIMO Interference Systems Zengmao Chen † , Sergiy A. Vorobyov †† , Cheng-Xiang Wang † , and John Thompson ††† † Joint Research Institute for Signal and Image Processing, Heriot-Watt University, Edinburgh, EH14 4AS, UK. †† Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, T6G 2V4, Canada. ††† Joint Research Institute for Signal and Image Processing, University of Edinburgh, Edinburgh, EH9 3JL, UK. Email: zc34@hw.ac.uk, vorobyov@ece.ualberta.ca, cheng-xiang.wang@hw.ac.uk, john.thompson@ed.ac.uk Abstract—In this paper, the source covariance matrices of multiple-input multiple-output (MIMO) interference channels (IFCs) are investigated from a game-theoretic perspective. It is proved that the requirement of sufficiently small interference-to- noise ratio (INR) is the sufficient condition for the uniqueness of the Nash bargaining (NB) solution. The structure of the source covariance matrices, which constitute the feasible set of NB solution, is analyzed by comparing them with the covariance matrices leading to the Nash equilibrium (NE). The existence of the NB solution and concavity of the rate product for MIMO IFCs are also studied. Index Terms – MIMO interference channel, Nash bargaining, Nash equilibrium, source covariance matrix, game theory. I. I NTRODUCTION With the philosophy of sharing the available spectrum with either peer networks or legacy systems, dynamic spectrum access (DSA) [1] holds the potential to significantly improve the spectrum utilization. The newly emerged cognitive radio technology [2] is one of its promising implementations, where all secondary users try to share the spectrum holes and operate in an interference channel (IFC). Besides traditional information-theoretic approaches, there has recently been a new research trend to study IFCs from a game-theoretic perspective. Various IFCs have been investigated in the literature, where the Nash equilibrium (NE) and Nash bargaining (NB) were applied to interference games stemming from these IFCs. The NB was theoretically examined for single-input single-output (SISO) and multiple-input single-output (MISO) IFCs in [3] and [4] in terms of its existence, respectively. The practical algorithms finding the NB solution for SISO and MISO IFCs were proposed in [5] and [6], respectively. The NB over IFCs was extended to the MIMO case in [7], where a practical suboptimal algorithm for finding the NB solution was designed by exploring the gradient projection method [8]. To the best of the authors’ knowledge, there have been no research attempts to theoretically study the NB in MIMO IFCs in terms of its uniqueness and existence, which motivates the research in this paper. In this paper, we fill this gap by deriving a sufficient condition for the uniqueness of the NB solution in MIMO interference systems. The feasible NB set is also analyzed by studying the structure of the source covariance matrices. Finally, the existence of the NB solution and concavity of the rate product for MIMO IFCs are investigated in terms of simulations. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Model Consider an L-link MIMO interference system with L trans- mitters and L corresponding receivers, where the transmitter and receiver for each link are equipped with N t and N r antennas, respectively. The N r × 1 complex base-band signal vector received by link i(i =1, 2,...,L) can be written as y i = √ ρ i H ii x i + L  j=1,j=i √ η ij H ij x j + n i (1) where ρ i is the normalized signal-to-noise ratio (SNR) for link i; η ij is the normalized interference-to-noise radio (INR) from transmitter j to receiver i; H ii and H ij are the N r × N t channel matrices from transmitter i and transmitter j to receiver i, respectively; x i is the N t × 1 transmitted signal vector for link i, and n i is the N r ×1 independently identically distributed (i.i.d.) additive white Gaussian noise (AWGN) vec- tor perceived by link i with zero mean and identity covariance matrix E[n i n H i ]= I, E[·] is the expectation operator, (·) H denotes the Hermitian transpose operation, and I is the identity matrix. We assume that: 1) each transmitter/receiver trans- mits/receives symbols independently; 2) the co-channel inter- ference from other links is unknown and treated as noise (i.e., no interference cancellation (IC) techiniques are employed by receivers); 3) the channel varies sufficiently slow and can be considered as time invariant during the period of each symbol transmission. The mutual information for user i can be expressed as [9] I i (Q) = log 2 det ( I + ρ i Q i H H ii R -1 -i H ii ) i =1,...,L (2) where Q i = E[x i x H i ] is the Hermitian positive semi-definite (PSD) source covariance matrix of the input signal vector for link i and R -i = I+ ∑ L j=1,j=i η ij H ij Q j H H ij is the covariance matrix of the interference-plus-noise received by receiver i. We define Q  (Q 1 ,..., Q L ) as a set of source covariance matrices. Since the transmission of each user is power limited, the following trace constraint applies to Q i tr(Q i ) ≤ p i (3) where tr(·) denotes the trace of a matrix. We also assume that each transmitter i has the full knowledge of channel, This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings 978-1-4244-3435-0/09/$25.00 ©2009 IEEE Authorized licensed use limited to: Heriot-Watt University. Downloaded on February 21,2010 at 11:56:22 EST from IEEE Xplore. Restrictions apply.