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On the Capacity of Free-Space Optical Intensity Channels
Amos Lapidoth, Stefan M. Moser, Michele A. Wigger
TL;DR
This work studies the capacity of an optical communication channel with nonnegative inputs and develops upper and lower capacity bounds under average- and peak-power constraints. The bounds converge at high power, while low-power asymptotics are characterized for both constraint settings.
Problem
The paper studies the capacity of an optical communication channel whose input is nonnegative and represents optical intensity, under power constraints.
Method
The paper presents firm upper and lower bounds for cases with simultaneous average- and peak-power constraints and with only an average-power constraint.
Results
At high power with a fixed average-to-peak-power ratio, the upper and lower bounds differ by 0; at low power, the peak-constrained asymptotics are exact, while average-only asymptotics are given up to a constant factor.
Takeaways & Limitations
The bounds determine the channel capacity asymptotically at high power, while low-power behavior is characterized exactly for peak-constrained cases and within a constant factor for average-only constraints.
Abstract
from arXiv · showhide
New upper and lower bounds are presented on the capacity of the free-space optical intensity channel. This channel is characterized by inputs that are nonnegative (representing the transmitted optical intensity) and by outputs that are corrupted by additive white Gaussian noise (because in free space the disturbances arise from many independent sources). Due to battery and safety reasons the inputs are simultaneously constrained in both their average and peak power. For a fixed ratio of the average power to the peak power the difference between the upper and the lower bounds tends to zero as the average power tends to infinity, and the ratio of the upper and lower bounds tends to one as the average power tends to zero. The case where only an average-power constraint is imposed on the input is treated separately. In this case, the difference of the upper and lower bound tends to 0 as the average power tends to infinity, and their ratio tends to a constant as the power tends to zero.
I. INTRODUCTION
The paper models free-space optical intensity communication with nonnegative inputs, additive Gaussian noise, and simultaneous average- and peak-power constraints. It develops upper and lower capacity bounds using duality, entropy methods, and low-power binary inputs.
- Channel model: Nonnegative optical-intensity inputs are corrupted by additive Gaussian noise modeled from many independent disturbance sources.The receiver measures incident optical intensity through an electrical current proportional to detected intensity.
- Constraints: Battery and safety considerations motivate simultaneous average-power constraint E and maximum peak-power constraint A.The separate average-only case is also analyzed.
- Capacity bounds: New upper and lower bounds characterize the capacity C(A, E) and separately the capacity C(E) under only an average-power constraint.Capacity is formulated through mutual information maximized over nonnegative input laws satisfying the relevant constraints.
- Results: The upper–lower-bound gap never exceeds 1 nat when α exceeds 0.03 or when only average power is constrained.At high power with fixed α, the bounds coincide as their difference tends to zero.
- Constraints: The ratio α = E/A describes the relative strength of the average-power constraint, with α = 1 representing only a peak-power constraint.Small α corresponds to a dominant average-power constraint and a weak peak-power constraint.
- Methods: Upper bounds use a dual mutual-information expression with relative entropy, while firm lower bounds rely on the entropy power inequality.Low-power lower bounds additionally use binary inputs, and peak-constrained cases use asymptotic mutual-information results for weak signals.
II. RESULTS
The paper presents firm upper and lower capacity bounds across three power-constraint regimes. Their gap vanishes at high power with fixed average-to-peak ratio, while low-power asymptotics are exact with peak constraints and known up to a constant factor with average power only.
- Strong peak-power constraint: The optimal input distribution has average power equal to half the peak power, making the average-power constraint inactive in the strong-peak-power regime.This property holds irrespective of α in the stated regime.
- Results: The results cover simultaneous average- and peak-power constraints, an inactive average-power regime, and an average-power-only constraint.The cases are organized by the average-to-peak ratio and whether the average-power constraint is active.
- Asymptotics: The upper and lower bounds converge at high power with fixed α; low-power asymptotics are exact with peak constraints and determined up to a constant factor with average power only.The average-power-only bounds have a maximum gap of 0.64 nats near E/σ ≈ 1.8 dB.
- Average- and peak-power constraints: Theorem 2 supplies two capacity bounds for average-to-peak ratios such as α = 0.1 and α = 0.4.The corresponding figures report maximum gaps of 0.72 nats and 0.56 nats, respectively.
- Strong peak-power constraint: For α > 1/2, the average-power constraint is inactive, so capacity equals the capacity with only a peak-power constraint.Bounds for this regime can therefore be obtained from the α = 1 case.
- Strong peak-power constraint: The α ≥ 1/2 regime exhibits the known high-power asymptotic behavior of a Gaussian channel with only a peak-power constraint.The paper identifies this behavior as well known.
C. Bounds on Channel Capacity with an Average-Power Constraint
The section develops upper and lower capacity bounds for the optical intensity channel when only average power is constrained, and characterizes their low-power relationship.
- Theorem 6 provides a lower bound and multiple upper bounds for capacity C(E) without a peak-power constraint.The bounds depend on free parameters subject to different restrictions.
- The bounds of Theorem 6 are depicted in Figure 4 for selected parameter choices.The parameter choices are described as suboptimal but useful.
- Theorem 7 states the asymptotic behavior of capacity in the only-average-power case.
- At low SNR, the asymptotic upper-to-lower-bound ratio is 2 rather than 1, although the bounds exhibit similar behavior.
III. DERIVATION
The derivation constructs lower bounds through selected input distributions and upper bounds through output-distribution duality, with parameter restrictions tailored to each bound.
- A lower bound is obtained by selecting an input distribution Q(·) and evaluating the resulting mutual information.Tightness depends on choosing Q(·) near capacity while keeping the evaluation tractable.
- The lower-bound input distribution is chosen to maximize differential entropy under the channel constraints.
- The upper-bound derivation uses the duality approach, specifying an output distribution R(·) and evaluating relative entropy.
- The output distributions used for the upper bounds are specified through their densities.
- The derivations assign free parameters to the bounds and impose nonnegative or upper-bounded restrictions where required.For bounds (27) and (28), δ is restricted differently: nonnegative for (28) and δ ≤−σe−1 for (27).