Capacity of Uniform Noise Channels Under Average Input Power Constraints

Technical Summary: Capacity of Uniform Noise Channels Under Average Input Power Constraints

Problem Statement
This paper addresses a long-standing open problem in information theory regarding the channel capacity of an additive noise channel $Y = X + Z$ where the noise $Z$ follows a uniform distribution on a centered interval $[-v, v)$, subject to an average input power constraint $E[X^2] \le u$. While Shannon's foundational work established the capacity for Gaussian noise, and subsequent research solved the case for uniform noise under amplitude constraints ($|X| \le u$), the capacity under average power constraints for uniform noise remained unresolved. Existing bounds for general additive channels are not sharp in this setting, and standard analytical techniques relying on noise regularity (e.g., tail positivity, analyticity) fail because the uniform distribution lacks these properties.

Methodology
The core methodological innovation is the identification and exploitation of a specific periodization identity for the output density of a uniform noise channel. The authors observe that for any input $X$ independent of $Z \sim \text{Unif}([-v, v))$, the periodization of the output density $f_Y$ with respect to the lattice $\Lambda_v = 2v\mathbb{Z}$ is constant. Specifically:
$$\sum_{k \in \mathbb{Z}} f_Y(s + 2kv) = \frac{1}{2v}, \quad \forall s \in [-v, v).$$
This implies that the output modulo $2v$ is uniformly distributed, regardless of the input distribution.

This observation transforms the mutual information maximization problem (which is equivalent to maximizing the output differential entropy $h(Y)$ subject to a second-moment constraint) into a constrained optimization problem with a linear equality constraint on the periodized density. The authors utilize Fourier analytic techniques, specifically Poisson summation, to analyze the function $\Theta_{\lambda, v}(s)$, defined as a sum of Gaussian kernels over the lattice. They define a function $Q_v(\lambda)$ representing the second moment of a specific distribution derived from $\Theta_{\lambda, v}$ and prove its strict monotonicity to establish a unique mapping between the power constraint and a Lagrange multiplier $\lambda$.

Key Contributions and Results
The paper provides an explicit, closed-form characterization of the channel capacity and the unique capacity-achieving distributions:

1. Capacity Expression: The capacity $C(u, v)$ is given by:
   $$C(u, v) = \lambda(u + v^2/3) + \frac{1}{2v} \int_{-v}^{v} \log(\Theta_{\lambda, v}(s)) \, ds$$
   where $\lambda = \lambda(u, v)$ is the unique positive solution to $Q_v(\lambda) = u + v^2/3$.

2. Output Distribution: The unique capacity-achieving output density $g_{\lambda, v}(y)$ is absolutely continuous and takes the form:
   $$g_{\lambda, v}(y) = \frac{\exp(-\lambda y^2)}{2v \Theta_{\lambda, v}([y]_v)}$$
   where $[y]_v$ denotes the residue of $y$ modulo $2v$ in $[-v, v)$.

3. Input Distribution: The paper derives the unique capacity-achieving input density $f_X$, which is also absolutely continuous (contrasting with the discrete distributions often found under amplitude constraints). The density is expressed as a series involving the parameters of the output distribution.

4. Regularity: The authors prove that both the optimal input and output distributions are absolutely continuous with respect to the Lebesgue measure, resolving questions about the structural nature of the optimal input for this specific noise type.

Significance
The paper claims to settle the specific question of determining the capacity and optimal distributions for uniform noise under average power constraints, a problem explicitly labeled as open in prior literature. The significance lies in:

• Exact Characterization: Providing precise analytical expressions for the capacity and distributions, rather than relying on loose bounds.
• Structural Insight: Demonstrating that the optimal input is absolutely continuous, distinguishing it from the discrete solutions found in amplitude-constrained scenarios.
• Methodological Advancement: Introducing a periodization identity that allows the application of Fourier analytic techniques to a noise distribution (uniform) that typically resists such analysis due to its lack of smoothness and tail properties.
• Contextual Relevance: While the resulting densities appear visually Gaussian-like (consistent with known bounds stating Gaussian inputs are within 0.254 bits of capacity for log-concave noise), the paper emphasizes the importance of identifying the precise non-Gaussian densities that achieve capacity.

The authors note that these specific distributions appear unique to this problem and do not naturally arise in other areas of probability theory. The work is presented as a step toward a broader quantitative understanding of additive channels, though the authors acknowledge that a general theory for all symmetric log-concave noise distributions remains an open question.