[Paper Review] Asymmetric Compute-and-Forward with CSIT

Jingge Zhu, Michael Gastpar|arXiv (Cornell University)|Jan 14, 2014

Cooperative Communication and Network Coding5 references21 citations

TL;DR

This paper proposes an asymmetric compute-and-forward scheme that leverages Channel State Information at Transmitters (CSIT) to achieve larger computation rate regions by allowing different users to transmit at varying rates using tailored lattice codes. By decoupling the shaping lattice from the decoding lattice and introducing power scaling factors βk, the scheme enables significant rate region gains—arbitrarily better than conventional compute-and-forward in certain MIMO settings—especially when channel conditions are asymmetric.

ABSTRACT

We present a modified compute-and-forward scheme which utilizes Channel State Information at the Transmitters (CSIT) in a natural way. The modified scheme allows different users to have different coding rates, and use CSIT to achieve larger rate region. This idea is applicable to all systems which use the compute-and-forward technique and can be arbitrarily better than the regular scheme in some settings.

Motivation & Objective

• To address the limitation of conventional compute-and-forward, which uses identical lattice codes for all users and does not exploit CSIT.
• To enable asymmetric message rates across users in interference networks by allowing different users to use different lattice codes.
• To extend the computation rate concept to a rate tuple, allowing flexible control over individual user rates.
• To demonstrate that CSIT can be used effectively in compute-and-forward to achieve larger achievable rate regions than the standard scheme.
• To show that the proposed scheme can outperform regular compute-and-forward arbitrarily in certain MIMO settings, especially with strong channel asymmetries.

Proposed method

• Introduces a modified compute-and-forward scheme where each transmitter uses a shaping lattice Λks with second moment βk²P, distinct from the decoding lattice Λm(k) used at relays.
• Uses nested lattice chains where Λk^s ⊆ Λc (coarsest lattice), ensuring simultaneous goodness for shaping and decoding.
• Employs power scaling factors βk to control the effective rate of each user independently, enabling asymmetric rates.
• Decodes linear combinations ∑k amk tk at relays using lattice decoding over Λm, with the decoding success dependent on the effective signal-to-noise ratio.
• Derives a new computation rate tuple bound using the modified lattice structure, where the rate for user k depends on βk and the channel gains.
• Applies the scheme to MIMO integer-forcing receivers, showing that the modified rate expression scales as log P even when the original scheme is bounded by O(1).

Experimental results

Research questions

• RQ1 Can CSIT be used to improve the achievable rate region in compute-and-forward beyond the symmetric-rate assumption of the original scheme?
• RQ2 How can different users be assigned different transmission rates in a compute-and-forward framework while maintaining reliable decoding?
• RQ3 What is the maximum achievable rate tuple when transmitters have access to CSIT and can adapt their lattice codes accordingly?
• RQ4 In what settings does the asymmetric scheme outperform the original compute-and-forward scheme by an arbitrarily large margin?
• RQ5 Can the modified scheme achieve sum-rate scaling of log P in MIMO channels where the original integer-forcing scheme is bounded by O(1)?

Key findings

• The proposed asymmetric compute-and-forward scheme achieves a strictly larger computation rate region than the conventional scheme when CSIT is available, especially in asymmetric interference networks.
• For a two-way relay channel with unequal power constraints (P₁=1, P₂=20), the modified scheme achieves a larger rate region than the regular scheme, with the relay decoding different linear functions yielding improved performance.
• In a 2×2 MIMO channel with H = [1,1; 0,ε], the modified scheme achieves a rate scaling as log P for any ε > 0, while the original integer-forcing scheme is bounded by O(1) when ε ~ 1/√P.
• The modified scheme's rate expression in MIMO systems scales as log P, demonstrating it can be arbitrarily better than the original scheme in high-SNR, low-rank scenarios.
• By using simultaneous Diophantine approximation, the scheme can find βk and integer coefficients amk such that the effective noise is minimized, enabling high-rate decoding even in ill-conditioned channels.
• The rate region improvement is demonstrated numerically in Figure 3 and Figure 4, showing that the black convex hull (modified scheme) strictly dominates the red curve (regular scheme) in achievable rates.