Optimizing the Lower Bound on the Mismatch Capacity via a Double Maximization Approach

Core Concepts

A novel dual form of the Lower Bound on the Mismatch Capacity (LM rate) is proposed, which transforms the non-convex max-min optimization problem into an equivalent double maximization formulation. This enables the development of an efficient Alternating Double Maximization (ADM) algorithm to optimize the channel input probability distribution and compute the optimized LM rate.

Abstract

The paper presents a novel approach for efficiently computing the optimized Lower Bound on the Mismatch Capacity (CLM), which is the maximum of the LM rate over all feasible channel input probability distributions.
The key challenges in computing CLM are the non-convex max-min structure of the optimization problem and the coupling between the channel input and output probability distributions. To address these, the authors propose a new dual form of the LM rate, which transforms the problem into an equivalent double maximization formulation.
This new formulation enables the development of an Alternating Double Maximization (ADM) algorithm, where each step involves a closed-form iteration that can be efficiently implemented using standard optimization procedures. The ADM algorithm is proven to converge to a local optimum.
Numerical experiments on Gaussian channels with IQ imbalance show that the proposed approach leads to noticeable rate gains compared to the LM rate without input optimization, especially in the high SNR regime and for higher-order modulation schemes.

Stats

The average power constraint is Γ = 1.
The SNR values range from -10 dB to 10 dB.

Quotes

"The computation of CLM is challenging. With a prescribed channel input probability distribution, the computation of the GMI and the LM rate can be deduced into convex optimization problems which can be readily solved by solvers such as the CVX. The computation of CLM, instead, is a max-min optimization problem, which turns out to be generally non-convex."
"To address the difficulty due to the max-min problem formulation, we propose a new dual form of the LM rate. This transforms the max-min optimization problem into a double maximization problem, which enables us to develop an alternating maximization algorithm with guarantee of local convergence."

Key Insights Distilled From

A Double Maximization Approach for Optimizing the LM Rate of Mismatched Decoding

by Lingyi Chen,... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00545.pdf

Deeper Inquiries

How can the proposed ADM algorithm be extended to handle continuous channel input and output alphabets, rather than the discrete case considered in this work

To extend the proposed Alternating Double Maximization (ADM) algorithm to handle continuous channel input and output alphabets, we need to adapt the optimization process to account for the continuous nature of the variables. In the discrete case, we optimized over probability distributions, but in the continuous case, we would optimize over probability density functions.

Continuous Probability Distributions: Instead of discrete probabilities for each symbol, we would work with continuous probability density functions for the input and output alphabets. This would involve formulating the optimization problem with integrals instead of sums.

Integration in Optimization: The optimization steps involving updating the input distribution and the variables ϕ, eψ, and ζ would now involve integrals over the continuous alphabets. The derivatives and updates would be calculated using calculus techniques for continuous functions.

Numerical Integration: To solve the optimization problem numerically, we would need to use numerical integration methods to approximate the integrals. Techniques like Monte Carlo integration or numerical quadrature methods could be employed.

Convergence and Complexity: Handling continuous variables may introduce additional complexity in the optimization process. Ensuring convergence and efficiency in the algorithm would require careful consideration of the numerical methods used for integration and optimization.

By adapting the optimization steps to work with continuous probability density functions and employing numerical integration techniques, the ADM algorithm can be extended to handle continuous channel input and output alphabets effectively.

What are the potential applications of the optimized LM rate (CLM) in practical communication systems beyond the Gaussian channel with IQ imbalance considered here

The optimized LM rate (CLM) obtained through the proposed approach has various potential applications in practical communication systems beyond the specific Gaussian channel with IQ imbalance considered in this study. Some of the key applications include:

Wireless Communication Systems: In wireless communication, where channel knowledge is imperfect or hardware constraints exist, CLM can provide insights into the achievable information rates under mismatched decoding scenarios. This can lead to improved system performance in real-world wireless networks.

Fading Channels: CLM can be valuable in scenarios with fading channels, where the channel conditions vary over time. By optimizing the LM rate, communication systems can adapt to changing channel conditions and enhance reliability and efficiency.

Non-Ideal Transceiver Hardware: Practical transceiver hardware may introduce imperfections that lead to mismatched decoding. By optimizing the LM rate, systems can mitigate the impact of hardware imperfections and improve overall performance.

Security and Robustness: CLM optimization can also be applied in secure communication systems to enhance robustness against eavesdropping and interference. By maximizing the lower bound on the mismatch capacity, systems can achieve higher levels of security.

Internet of Things (IoT): In IoT networks with constrained resources and varying channel conditions, optimizing the LM rate can enable efficient and reliable communication between IoT devices, leading to improved network performance.

Overall, the optimized LM rate has broad applications in diverse communication systems, offering performance enhancements, robustness, and adaptability in challenging operational environments.

Can the insights from the dual form and the ADM algorithm be leveraged to develop efficient optimization methods for other information-theoretic quantities in mismatched decoding scenarios

The insights gained from the dual form and the Alternating Double Maximization (ADM) algorithm can indeed be leveraged to develop efficient optimization methods for other information-theoretic quantities in mismatched decoding scenarios. Here are some ways these insights can be applied:

Generalized Mismatched Decoding Metrics: The dual form approach can be extended to handle different mismatched decoding metrics beyond the specific decoding metric considered in this study. By formulating appropriate dual forms, optimization algorithms can be developed for various decoding scenarios.

Capacity Bounds: The ADM algorithm's alternating maximization framework can be adapted to compute other capacity bounds in mismatched decoding, such as the generalized mutual information (GMI) or other lower bounds. This can lead to tighter capacity bounds and improved system performance.

Channel Coding: The optimization techniques used in the ADM algorithm can be applied to optimize channel coding schemes under mismatched decoding conditions. By maximizing information rates or capacity bounds, more efficient coding strategies can be designed.

Multi-User Systems: The dual form insights can be valuable in multi-user communication systems where decoding metrics may vary across users. By extending the ADM algorithm to handle multiple users, optimized solutions for information rates in multi-user scenarios can be obtained.

By leveraging the principles and methodologies from the dual form and the ADM algorithm, efficient optimization methods can be developed for a wide range of information-theoretic quantities and scenarios in mismatched decoding.