Achieving the capacity of the N-relay Gaussian diamond network within logn bits

Abstract

We consider the N -relay Gaussian diamond network where a source node communicates to a destination node via N parallel relays through a cascade of a Gaussian broadcast (BC) and a multiple access (MAC) channel. Introduced in 2000 by Schein and Gallager, the capacity of this relay network is unknown in general. The best currently available capacity approximation, independent of the coefficients and the SNR's of the constituent channels, is within an additive gap of 1.3N bits, which follows from the recent capacity approximations for general Gaussian relay networks with arbitrary topology.In this paper, we approximate the capacity of this network within 2 log N bits. We show that two strategies can be used to achieve the information-theoretic cutset upper bound on the capacity of the network up to an additive gap of O(log N ) bits, independent of the channel configurations and the SNR's. The first of these strategies is simple partial decode-and-forward. Here, the source node uses a superposition codebook to broadcast independent messages to the relays at appropriately chosen rates; each relay decodes its intended message and then forwards it to the destination over the MAC channel. A similar performance can be also achieved with compress-and-forward type strategies (such as quantize-map-and-forward and noisy network coding) that provide the 1.3N -bit approximation for general Gaussian networks, but only if the relays quantize their observed signals at a resolution inversely proportional to the number of relay nodes N . This suggest that the rule-of-thumb to quantize the received signals at the noise level in the current literature can be highly suboptimal.I. INTRODUCTION Consider a Gaussian relay network where a source node communicates to a destination with the help of intermediate relay nodes. Characterizing the capacity of this network is a long-standing open problem in network information theory. The seminal work of Cover and El-Gamal [2] has established several basic achievability schemes for the single relay channel, such as decode-and-forward and compress-andforward. Recently, significant progress has been made by generalizing the compress-and-forward strategy to achieve the capacity of any Gaussian relay network within an additive gap that depends on the network only through the total number of relay nodes N (or the total number of transmit and receive antennas when nodes are equipped with multiple antennas) [3], [4], [5], [6], [7]. The fact that the gap to capacity is independent of the channel gains, the SNR's and the exact topology of the network suggests that compressand-forward can be universally good for relaying across different channel configurations, SNR regimes and topologies. However, the dependence of the gap to N limits the applicability of these results to small networks with few relays. The best currently available capacity approximation in [4] is within 1.3N bits (per second per Hz) of the information-theoretic cutset upper bound on the capacity of the network. For typical spectral...

Gaussian half-duplex relay networks: Improved gap and a connection with the assignment problem

“…Our derived gap is valid for a general fully connected HD relay network. In the FD literature, it is well known that the gap of 1.26K bits for a general network [6] can be reduced to 2 log(K − 1) bits for a diamond network [7]. In a diamond network the network topology is restricted compared to a general network, i.e., the source can not communicate directly with the destination and the relays can not communicate among themselves.…”

Partial decode-Forward relaying for the Gaussian two-hop relay network

“…Moreover, the computation of (19) or (20) involves computing Gaussian capacity functions for L(N + 1) cuts, which is a significant savings from the directed computation of the cutset bound with all L2 N possible cuts as in (2). We summarize this result as follows.…”

“…Following a similar (and in some sense dual) development for noisy network coding in [8], we setẐ j ∼ N(0, N ) in (22). Then, the first term of (21) becomes …”

“…1, these coding schemes achieve the cutset bound within O(N ) bits for any channel parameters (recall that N is the number of relays). Recently, Chern andÖzgür [8] provided a more refined analysis on the performance of NNC and showed that it achieves within (1/2) log 2N (N + 1) bits from the cutset bound regardless of the number of destinations. In the same paper [8], Chern andÖzgür extended the partial decodeforward (PDF) scheme for the relay channel by Cover and El Gamal [4] to the Gaussian diamond network (L = 1).…”

Efficient subnetwork selection in relay networks

“…We resort instead to approximate schemes, such as Quantize-Map-and-Forward (QMF) [2] and Noisy Network Coding (NNC) [7], that are proved to achieve all rates within a constant additive gap (independent of the channels in the network) from the cutset upper bound on capacity for arbitrary single-source singledestination wireless relay networks. We are encouraged in this choice by the fact that initial concerns about the additive gap affecting QMF/NNC performance at moderate SNRs have also been addressed in [11], [4], with appropriate choices of quantizer distortions demonstrating excellent moderate SNR performance. We thus focus our attention on QMF/NNC-based relay networks and hence, use the achievable rate expressions for these schemes [7], or indeed their close siblings-the cutset upper bound expressions on network capacity (which differ from the QMF/NNC rates by a mere constant), as the network objective functions for our problem.…”