Source-linked AI summary
Capacity of the Gaussian Two-way Relay Channel to within 1/2 Bit
Wooseok Nam, Sae-Young Chung, Yong H. Lee
TL;DR
The paper studies capacity in a full-duplex Gaussian two-way relay channel without a direct source link. It proposes nested lattice coding on the uplink and structured binning on the downlink, achieving a 1/2-bit cut-set gap for all channel parameters with a vanishing gap at high uplink SNR.
Problem
The capacity region of the general two-way relay channel remains unknown, motivating capacity analysis for the Gaussian model.
Method
The paper uses nested lattice codes for the uplink and structured binning for the downlink.
Results
1/2 bit is the achievable scheme's gap to the cut-set bound for all channel parameters, and the gap vanishes as uplink SNRs increase.
Takeaways & Limitations
The achievable region asymptotically approaches the capacity region of the Gaussian two-way relay channel.
Abstract
from arXiv · showhide
In this paper, a Gaussian two-way relay channel, where two source nodes exchange messages with each other through a relay, is considered. We assume that all nodes operate in full-duplex mode and there is no direct channel between the source nodes. We propose an achievable scheme composed of nested lattice codes for the uplink and structured binning for the downlink. We show that the scheme achieves within 1/2 bit from the cut-set bound for all channel parameters and becomes asymptotically optimal as the signal to noise ratios increase.
2 Bit
The paper concerns Gaussian two-way relay channels, with network coding and lattice codes as central topics.
- Gaussian two-way relay channels are the paper's subject.
- Network coding and lattice codes are identified as key topics.
I. INTRODUCTION
The introduction frames the Gaussian two-way relay channel as a network-building block with unresolved general capacity, then presents a lattice-based scheme approaching the cut-set bound.
- The general two-way relay channel capacity region remains unknown.
- Classical AF, DF, and CF relaying strategies have been extended to the two-way relay channel, with DF network coding improving achievable rates but generally incurring multiplexing loss.
- The considered Gaussian model uses full-duplex nodes and no direct communication link between the source nodes.
- Nested lattice codes exploit computation coding on the uplink, while structured binning addresses the downlink broadcast channel with receiver side information.
- 1/2 bit is the stated gap to the cut-set bound for general channel parameters, and the gap vanishes as uplink SNRs increase.
II. SYSTEM MODEL
The system model is a full-duplex Gaussian two-way relay channel in which independent source messages are exchanged through the relay without a direct source-to-source path.
- The Gaussian two-way relay channel has full-duplex source and relay nodes and no direct path between the source nodes.
- Each source message W_i ranges over 2^(nR_i) possibilities, with n channel uses and rate R_i.
- The source messages W_1 and W_2 are independent, and each source transmits subject to power constraint P_i.
- The relay receives the uplink transmissions and transmits on the downlink using signals determined by past channel outputs.
- A rate pair is achievable when a sequence of encoding and decoding functions has error probability vanishing as n tends to infinity.
III. AN UPPER BOUND FOR THE CAPACITY REGION
The cut-set bound limits each source rate by the weaker of an uplink and downlink mutual-information term, with Gaussian independent inputs maximizing the relevant terms.
- The Gaussian model's lack of a direct source-to-source path induces the simplified bounds in (2a) and (2b).
- R_1 and R_2 are each bounded by the minimum of two mutual-information terms involving the relay uplink and opposite-node downlink.
- Independent zero-mean Gaussian inputs with variances P_1, P_2, and P_R maximize all terms under the minimizations.
IV. AN ACHIEVABLE RATE REGION FOR THE GAUSSIAN TRC
The paper combines nested lattice codes on the uplink with structured binning on the downlink to construct an achievable rate region for the Gaussian two-way relay channel. The resulting region is within 1/2 bit of the upper bound for all channel parameters and approaches capacity as uplink SNRs increase.
- Nested lattice codes are used for the uplink, while structured message binning is used for the downlink.Destination nodes use their own transmitted messages as side information when decoding.
- Theorem 1 gives an achievable rate region for the Gaussian two-way relay channel.
- The achievable rate region is within 1/2 bit of the upper bound for any transmit powers and noise variances.
- As the uplink SNRs increase, the gap vanishes and the achievable region asymptotically approaches the Gaussian TRC capacity region.
A. Lattice scheme for the uplink
The uplink scheme uses nested lattice codes built from a lattice chain, enabling the relay to decode a function of both source messages rather than reconstructing them separately. Lattice-goodness properties and Euclidean decoding provide reliable recovery under the stated rate conditions.
- Lattice construction: Nested lattice coding is adopted for the uplink, with full construction details omitted because of page limitations.The paper refers readers to prior comprehensive treatments for lattice codes.
- Lattice construction: Two nested lattice codes are designed, one for each source node, assuming P1 ≥ P2 without loss of generality.The code construction relies on lattice chains and Voronoi regions.
- Lattice construction: The lattice-chain theorem provides Λ1^n ⊆ Λ2^n ⊆ ΛC^n with Λ1^n and Λ2^n simultaneously Rogers-good and Poltyrev-good, and ΛC^n Poltyrev-good.
- Encoding: Each source maps its message one-to-one to a codeword Wi in Ci and transmits a dithered modulo-lattice signal Xi.The dithers make Xi uniformly distributed over the relevant Voronoi region and independent of Wi, satisfying the power constraint asymptotically.
- Relay decoding: The relay decodes a lattice combination T of the source codewords instead of decoding W1 and W2 separately.This computation-coding step avoids requiring full message recovery at the relay and is implemented through Euclidean lattice decoding.
- Reliability: The decoding error probability vanishes as n →∞ when the coding rate is below the relevant threshold R1*.The bound uses the Poltyrev exponent, which is positive in the stated operating regime.
B. Downlink phase
The downlink uses structured binning and receiver side information so each destination can recover the other source’s message through relay-message decoding.
- B. Downlink phase: Node 1 decodes a relay message from a jointly typical codeword in its bin-specific codebook.Its codebook contains relay codewords indexed by message combinations involving W1 and possible W2 values.
- B. Downlink phase: Given the correct relay message, each destination estimates the other source’s message using its own transmitted message as side information.Node 1 uses W1 and its relay-message estimate to recover node 2’s message.
- B. Downlink phase: Node 2 performs the analogous relay-message decoding using a codebook indexed by W2 and possible W1 values.The construction mirrors node 1’s decoding operation with the source roles reversed.
- B. Downlink phase: The downlink broadcast does not reduce either destination’s point-to-point capacity because message binning exploits each node’s knowledge of its own transmitted message.The message pair is binned to the relay message through the scheme’s lattice-induced relation.
C. Achievable rate region
The achievable-region analysis relates overall message errors to uplink and downlink decoding errors, then shows these terms vanish under the stated rate conditions.
- C. Achievable rate region: The overall error probability is bounded by errors in relay decoding and the two destinations’ relay-message decoding.The message estimates are exact when both destination estimates and the relay estimate are correct.
- C. Achievable rate region: The second and third error terms vanish as n →∞ when conditions (12) and (15) hold.Together with the first-term condition, these bounds yield the achievable rate region.
- C. Achievable rate region: The first error term vanishes as n →∞ when Ri < R∗i, for i ∈{1, 2}.This condition is supplied by Theorem 3 in the rate-region derivation.
V. CONCLUSION
The paper combines nested lattice coding on the uplink with structured binning on the downlink for the Gaussian TRC. Its achievable region is within 1/2 bit of the cut-set bound across channel parameters, with the gap vanishing at high SNR.
- V. CONCLUSION: The proposed Gaussian TRC scheme uses nested lattice codes for the uplink and structured binning for the downlink.These components form the paper’s achievable scheme.
- V. CONCLUSION: 1/2 bit is the gap between the resulting achievable rate region and the cut-set bound for all channel parameters.The conclusion states this bound uniformly over the channel parameters considered.
- V. CONCLUSION: The gap eventually vanishes in the high SNR regime, while the exact Gaussian TRC capacity region remains open.Thus the achievable region approaches the cut-set bound asymptotically without resolving exact capacity.