Partial decode-and-forward rates for the Gaussian MIMO relay channel: Inner approximation of non-convex rate constraints

Abstract

In this paper, we consider achievable partial decodeand-forward (PDF) rates for the Gaussian multiple-input multipleoutput (MIMO) relay channel. The PDF scheme generalizes the decode-and-forward (DF) scheme as it allows to optimize the amount of information that is transmitted in cooperation with the relay. For the Gaussian channel case, the optimal channel inputs for the PDF scheme are unknown, and even if Gaussian channel inputs are used, the resulting optimization problem (OP) is non-convex. Thus, suboptimal approaches are necessary in order to efficiently evaluate PDF rates. In this paper, we apply the so-called Inner Approximation Algorithm (IAA). By successively refining an approximation of the original OP, we are able to evaluate PDF rates by solving a sequence of convex OPs. Our simulation results show that these suboptimal PDF rates are able to outperform the rates achieved by point-to-point (P2P) transmission and the DF scheme. For scenarios where the source is equipped with more antennas than the relay, the suboptimal PDF rates even approach the cut-set bound (CSB).

Maximizing the Partial Decode-and-Forward Rate in the Gaussian MIMO Relay Channel

“…For the numerical simulations, we adopt the line network example from [23], where the relay lies on a line between source and destination. The The first important observation of the simulations we performed is that Algorithm 2 always terminated with ∆ (n) ≤ ǫ CP for ǫ CP = 10 −3 .…”

“…First, we can evaluate the performance of previously proposed suboptimal algorithms, and second, we can study the gap between the PDF rate and the CSB. As a suboptimal method, we have chosen the inner approximation approach (IAA) from [23], which was reported to outperform other suboptimal heuristics (see the simulations in [23]).…”

“…In case of DF, these covariance matrices can be obtained by solving a convex program [12], [16], but the additional interference term makes the rate maximization problem for PDF nonconvex. Therefore, previous approaches to maximize the PDF rate based on successive convex approximation [23] or zero-forcing [22] provide only suboptimal solutions.…”

The Optimal Input Distribution for Partial Decode-and-Forward in the MIMO Relay Channel

“…In fact, the PDF rate maximization problem (53) was already derived in [25], [26], but when those papers were written, it was not yet clear whether Gaussian channel inputs maximize the achievable PDF rate. Now, on the other hand, we can conclude from Theorem 2 that (53) yields the maximum achievable PDF rate for the Gaussian MIMO relay channel.…”

“…Another approach uses the so-called inner approximation algorithm (IAA) [27], which solves a sequence of approximating convex optimization problems instead of the original nonconvex one [26]. For Rayleigh fading channels, both of these suboptimal PDF schemes can achieve considerable gains compared to DF and approach the CSB if the source is equipped with more antennas than the relay [25], [26]. Furthermore, it has been shown in [13] that the ZF scheme yields the maximum achievable PDF rate R PDF if…”