[Paper Review] Capacity of The Discrete-Time Non-Coherent Memoryless Rayleigh Fading Channels at Low SNR

Zouheir Rezki, David Haccoun+1 more|ArXiv.org|Jan 3, 2008

Wireless Communication Security Techniques14 references3 citations

TL;DR

This paper derives a closed-form expression for the capacity-achieving input distribution and exact capacity of a discrete-time non-coherent memoryless Rayleigh fading channel at low SNR. It resolves the non-coherent capacity penalty and provides tight analytical bounds on the optimal input's non-zero mass point, offering deeper insight into energy-efficient communication in fading environments without CSI at either end.

ABSTRACT

The capacity of a discrete-time memoryless channel, in which successive symbols fade independently, and where the channel state information (CSI) is neither available at the transmitter nor at the receiver, is considered at low SNR. We derive a closed form expression of the optimal capacity-achieving input distribution at low signal-to-noise ratio (SNR) and give the exact capacity of a non-coherent channel at low SNR. The derived relations allow to better understanding the capacity of non-coherent channels at low SNR and bring an analytical answer to the peculiar behavior of the optimal input distribution observed in a previous work by Abou Faycal, Trott and Shamai. Then, we compute the non-coherence penalty and give a more precise characterization of the sub-linear term in SNR. Finally, in order to better understand how the optimal input varies with SNR, upper and lower bounds on the capacity-achieving input are given.

Motivation & Objective

• To analytically characterize the capacity-achieving input distribution for non-coherent Rayleigh fading channels at low SNR.
• To resolve the observed peculiar behavior of the optimal input distribution reported by Abou Faycal et al. in prior work.
• To derive exact expressions for the non-coherent capacity and the non-coherence penalty at low SNR.
• To provide tight upper and lower bounds on the location of the non-zero mass point in the optimal input distribution.
• To offer a precise characterization of the sub-linear term in SNR that governs capacity scaling at low SNR.

Proposed method

• Derives a closed-form expression for channel mutual information at low SNR, valid as a lower bound for all SNR values.
• Uses the Kuhn-Tucker conditions to characterize the optimal input distribution, which is discrete with a finite number of mass points including one at zero.
• Applies the Lambert W function to solve transcendental equations arising from the optimality conditions, particularly involving the non-zero mass point location.
• Establishes a fundamental relation between the SNR and the optimal input distribution parameters, enabling exact capacity computation at low SNR.
• Develops iterative lower and upper bounds on the non-zero mass point via self-mapping techniques on the Lambert W function.
• Derives analytical bounds on the capacity by bounding the input distribution's support, leading to tighter capacity estimates.

Experimental results

Research questions

• RQ1 What is the exact form of the capacity-achieving input distribution for non-coherent Rayleigh fading channels at low SNR?
• RQ2 How does the optimal input distribution vary with SNR, particularly the location of its non-zero mass points?
• RQ3 What is the precise value of the non-coherence penalty at low SNR, and how does it affect capacity scaling?
• RQ4 Can tight analytical bounds be derived for the optimal input distribution's non-zero mass point location?
• RQ5 How does the sub-linear term in SNR influence the capacity behavior in the low-SNR regime?

Key findings

• The paper derives a closed-form expression for the channel mutual information at low SNR, which serves as a lower bound on mutual information for all SNR values.
• An exact expression for the non-coherent capacity is derived at low SNR, resolving prior ambiguities in capacity scaling.
• The optimal input distribution is shown to be discrete with a finite number of mass points, one at zero, and the non-zero points are analytically characterized via the Lambert W function.
• Tight upper and lower bounds are established on the location of the non-zero mass point in the optimal input, improving upon prior numerical estimates.
• The non-coherence penalty is precisely quantified, and the sub-linear term in SNR is characterized analytically, confirming its dominance at low SNR.
• The derived bounds on the input distribution enable tighter estimates of non-coherent capacity, enhancing understanding of energy efficiency in fading channels.