[Paper Review] Bounds on the Achievable Rate of Noisy feedback Gaussian Channels under Linear Feedback Coding Scheme

Chong Li, Nicola Elia|arXiv (Cornell University)|Aug 13, 2011

Wireless Communication Security Techniques12 references4 citations

TL;DR

This paper derives computable upper and lower bounds on the maximum achievable rate for additive Gaussian noise channels with noisy feedback under a linear feedback coding scheme (Cover-Pombra scheme). By formulating the bounds as convex optimization problems, the authors show that feedback enhances capacity even when noisy, but performance degrades significantly with increasing feedback noise—especially in low-noise regimes where their upper bound outperforms prior work.

ABSTRACT

In this paper, we investigate the additive Gaussian noise channel with noisy feedback. We consider the setup of linear coding of the feedback information and Gaussian signaling of the message (i.e. Cover-Pombra Scheme). Then, we derive the upper and lower bounds on the largest achievable rate for this setup. We show that these two bounds can be obtained by solving two convex optimization problems. Finally, we present some simulations and discussion.

Motivation & Objective

• To address the lack of computable bounds on the capacity of Gaussian channels with noisy feedback, particularly under linear feedback coding.
• To quantify how feedback noise impacts the maximum achievable rate in such systems, especially in contrast to ideal (noiseless) feedback.
• To provide informative, computable upper and lower bounds on the largest achievable rate under the linear feedback coding scheme.
• To analyze the sensitivity of achievable rate to feedback noise and channel correlation, offering insight into system design trade-offs.

Proposed method

• Derives the n-block achievable rate using directed information and mutual information under the Cover-Pombra linear feedback coding scheme.
• Models the system with forward and feedback Gaussian noise, assuming independent message, forward noise, and feedback noise processes.
• Transforms the upper bound derivation into a convex optimization problem by introducing auxiliary variables and leveraging matrix inequalities.
• Derives a lower bound via a similar convex optimization framework, using a different auxiliary matrix to characterize achievable rate under linear feedback.
• Uses the Schur complement lemma to express the constraints in convex form, enabling efficient numerical computation.
• Validates bounds via simulations on a first-order moving average (1st-MV) forward channel with varying feedback noise power and correlation parameters.

Experimental results

Research questions

• RQ1 How does feedback noise affect the maximum achievable rate in Gaussian channels when feedback is linearly encoded?
• RQ2 Can tight, computable upper and lower bounds on the achievable rate be derived for linear feedback coding under noisy feedback?
• RQ3 How does the correlation structure of the forward channel noise influence the system's sensitivity to feedback noise?
• RQ4 How do the proposed bounds compare to existing bounds in the literature, especially in low-feedback-noise regimes?
• RQ5 Under what conditions does noisy feedback still provide a rate enhancement over non-feedback systems?

Key findings

• The proposed upper bound outperforms the bound in [12], especially in low feedback noise regions, and converges to the ideal feedback capacity as feedback noise vanishes.
• The lower bound demonstrates that even noisy feedback can enhance the achievable rate, with the bound becoming tighter when feedback noise is small.
• The achievable rate decreases sharply with increasing feedback noise power σ, and for σ ≥ 0.8, feedback provides negligible rate gain over non-feedback systems.
• Channels with higher forward noise correlation (larger α in 1st-MV model) show reduced sensitivity to feedback noise, indicating that correlated noise mitigates feedback corruption.
• The bounds are derived as solutions to convex optimization problems, enabling efficient numerical computation and practical evaluation.
• The lower bound becomes invalid (negative or below non-feedback capacity) when feedback noise is large, indicating the bound is only meaningful in low-noise feedback regimes.