Crystallized Rates Region of the Interference Channel via Correlated Equilibrium with Interference As Noise

Abstract

Abstract-Treating the interference as noise in the n−user interference channel, the paper describes a novel approach to the rates region, composed by the time-sharing convex hull of 2 n − 1 corner points achieved through On/Off binary power control. The resulting rates region is denoted crystallized rates region. By treating the interference as noise, the n−user rates region frontiers has been found in the literature to be the convex hull of n hyper-surfaces. The rates region bounded by these hypersurfaces is not necessarily convex, and thereby a convex hull operation is imposed through the strategy of time-sharing. This paper simplifies this rates region in the n−dimensional space by having only an On/Off binary power control. This consequently leads to 2 n − 1 corner points situated within the rates region. A time-sharing convex hull is imposed onto those corner points, forming the crystallized rates region. The paper focuses on game theoretic concepts to achieve that crystallized convex hull via correlated equilibrium. In game theory, the correlated equilibrium set is convex, and it consists of the time-sharing mixed strategies of the Nash equilibriums. In addition, the paper considers a mechanism design approach to carefully design a utility function, particularly the Vickrey-Clarke-Groves auction utility, where the solution point is situated on the correlated equilibrium set. Finally, the paper proposes a self learning algorithm, namely the regretmatching algorithm, that converges to the solution point on the correlated equilibrium set in a distributed fashion.

Crystallized rate regions in the secondary interference channels

“…Let us stress that any solution point on the crystallized rate borderline (frontier) will lie somewhere on the straight lines connecting any of the neighboring characteristic points. Similar conclusions can be drawn for the precoded MIMO systems, where (7), which defines the achievable rate in a time-sharing approach, has to be rewritten in order to include the transmit and receive beamforming set (see (8)). …”

“…tr(Q i ) = P i . Following the approach proposed in [8], [15] we state that instead of power control problem in finding the metrics Q i , the problem becomes finding the appropriate time-sharing coefficients of the (N t + 2) n − 1 corner points. For the two-user 2 × 2 TSD-MIMO case we will obtain 15 points, i.e.…”

“…The idea of the Crystallized Rate Regions has been introduced in [8] and can be understood as an approximation of the achievable rate regions by a convex time-sharing hull, where the potential curves between characteristic points are replaced with the straight lines connecting these points. Let us denote each point in the rate region as Φ(Q 1 , Q 2 ).…”

“…In [7] it has been shown that if one treats interference as noise, the n-user achievable rates region can be defined as the convex hull of n hyper-surfaces. A novel strategy to represent this rate region in the n-dimensional space, by having only binary switching power control has been proposed in [8]. Based on this approach one can illustrate the achievable rateregion in form of a crystal with the straight lines connecting any two adjacent corner points.…”

“…However, in case of MIMO systems the number of available game strategies is high and the linear programming solution becomes very complicated and costly. Thus, a distributed solution can be applied, such as the regret matching learning algorithm proposed in [8], to achieve the correlated equilibrium at lower computational cost. Moreover, recently some papers have been presented, where the so-called satisfaction equilibrium has been used instead of the Nash or Pareto equilibria.…”