Capacity-Achieving Input Distribution of the Additive Uniform Noise Channel With Peak Amplitude and Cost Constraint

TL;DR

This work addresses the capacity-achieving input distribution for an additive channel with uniform noise under a peak-amplitude constraint and a cost constraint c(x) = x^α. Using Smith's optimality conditions, the authors derive a complete, parameter-dependent characterization of the optimal input: when the cost constraint is inactive, the optimal input is discrete with a fixed set of mass-point locations determined by r = 1/(2b); when the constraint is active and α  1, the optimal input remains discrete with weights m_j ∝ e^{-λ^* c_j}; and when the constraint is active with α > 1, the optimal input has full support on [0,1]. The results reveal a phase transition from discrete to continuous capacity-achieving inputs driven by the cost function's convexity and the tightness of the cost constraint, extending classical discreteness results to a bounded-noise, peak-constrained setting. This provides fundamental insights for signaling design under amplitude and power-like constraints and suggests directions for extending to softer constraints or higher dimensions. The analysis leverages analytic constructions of mass points, uniqueness of the λ^* parameter, and linearity properties of the marginal information density to establish necessary and sufficient conditions for discreteness.

Abstract

Under which condition is quantization optimal? We address this question in the context of the additive uniform noise channel under peak amplitude and power constraints. We compute analytically the capacity-achieving input distribution as a function of the noise level, the average power constraint and the exponent of the power constraint. We found that when the cost constraint is tight and the cost function is concave, the capacity-achieving input distribution is discrete, whereas when the cost function is convex, the support of the capacity-achieving input distribution spans the entire interval.

Figures (3)

• Figure 1: The different cases discussed in Theorem \ref{['thm:main']}. In the left column $r\in\mathbb{N}$ ($r=4$) and in the right column $r\notin\mathbb{N}$ ($r=4.4$). Top: Phase diagram in the $\alpha$-$\bar{c}$-plane. Green and red background indicate $p_X^{\ast}$ with discrete support and support on the entire interval $[0,1]$, respectively. Ia,b and IIa,b: discrete $p_X^{\ast}$ with masses and positions indicated by the heights and the positions of the blue arrows, corresponding $p_N(y\mid x)$ by dashed boxes in Ia/IIa and by dotted (j odd) and dashed (j even) boxes in Ib/IIb. The black line is the resulting $p_Y^{\ast}$. IIIa and IIIb: numerical result for $p_X^{\ast}$ (blue) using the Blahut-Arimoto algorithm blahut_1972arimoto_1972 and corresponding $p_Y^{\ast}$ in black.
• Figure 2: a) The r.h.s. and the l.h.s. of (\ref{['eq:ineq_constr']}), illustrating the linear interpolation between the points of support, where (\ref{['eq:eq_constr']}) ensures equality. Other parameters: $r=2.4$ and $\bar{c}=0.54<\bar{c}^\ast$. b) $p_X^\ast\left(x\right)$ as a function of $\alpha$ obtained numerically by means of the Blahut-Arimoto algorithm blahut_1972arimoto_1972. For $\alpha\leq1$, $p_X$ is discrete and for $\alpha>1$, it has support on the entire interval $[0,1]$. Other parameters: $r=2.4$ and $\bar{c}=0.35<\bar{c}^\ast$.
• Figure 3: Capacity-achieving input distribution $p_X^{\ast}\left(x\right)$ as a function of $r$ for inactive (panel a)) and tight (panel b)) cost constraint. The diameter of the dots represents the mass. Other parameters: $\bar{c}=3$ in b) and $\alpha=0.7$ in both.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Remark 3
• Lemma 4
• Lemma 5
• Remark 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10