A Game-Theoretic Approach to Power Allocation in Frequency-Selective Gaussian Interference Channels

This paper defines a rate maximization power allocation game in a frequency selective Gaussian interference channel, af ter assuming a suboptimal but pragmatic multi-user coding scheme. We show that the Nash equilibrium always exists in this game. We consider a distributed power allocation scheme [l, Section IV.], and provide a sufficient condition for the convergence of the distributed algorithm. The condition for the convergence is also a sufficient condition under which the Nash equilibrium is unique. I. PROBLEM FORMULATION Current communication channels are both frequency selective and interference-limited, with a Gaussian thermal noise at the receiver end. Coordinating all the users is costly and physically impossible in a fast-fading channel. Hence, a frequency selective Gaussian interference channel well describes a multiuser channel [I, 21. When Discrete Multi-Tone (DMT) is assumed, the channel can be modeled as a set of N parallel independent frequencyflat channels. A Gaussian interference channel per frequency tone is modeled as a linear channel between the M transmitters and M receivers with a thermal noise. For the mth user at the nth tone, the received signal to interference plus noise ratio (SINR) is SINR(m,n) = C,,, H ( m , A , n ) P ( ~ , n ) + W ( m , n ) ’ P ( m , n ) where P(m,n) is the transmit s i g h power, W(m,n) is the thermal noise variance for user m at the nth frequency tone. Here, H(m, fi, n) is the interference channel path gain power from user m to user m at the nth frequency tone. The interference channel path gain and the noise variance have been scaled to make all the main path gain power equal to 1. The sum of transmitted signal powers across the frequency tones is constrained by E,”=, P(m,n) 5 P(m) where P(m) 2 0 is a given power constraint for user m. The optimal coding scheme is not known for this channel. Hence we assume the following suboptimal but practical coding scheme; at each frequency tone, each user regards the sum of interference from all the other users and a thermal noise as a Gaussian noise [l, 21. Then, each user’s channel is equivalent to a single-user additive Gaussian noise. Here, the coding/decoding scheme to maximize the achievable rate is a Gaussian random codebook. From this coding/decodmg scheme, the maximum achievable rate R(m,n) for user m at nth tone is R(m, n) = 3 log, [1+ SINR(m, n)] . The power budget for each user needs to be allocated efficiently so that all the users perform well. The following ‘This work was supported by Intel, STmicro, SAMSUNG, and the Post-doctoral Fellowship Program of Korea Science and Engineering Foundation (KOSEF). noncooperative game is defined in order to maximize the rate of each user while minimizing the level of coordination among them. Definition 1 (Noncooperative Rate Maximization Game) Given channel, noise, and power constraints for all the users, our game is composed of the following strategy. Each player maximizes its sum rate over frequency, given the total power constraint. In particular, user m ’s strategy is,