A High-Resolution Analysis of Receiver Quantization in Communication

TL;DR

This work analyzes the impact of uniform output quantization with moderate-to-high resolution on communication performance using generalized mutual information ($\mathrm{GMI}$) under an i.i.d. complex Gaussian codebook and nearest-neighbor decoding. It derives an analytical expression for the achievable rate, $I_{\mathrm{GMI}} = \log(1+\mathrm{SNR}) - \log(1+\gamma\mathrm{SNR})$, where $\gamma$ depends on the quantizer via parameters $\mathcal{A}$ and $\mathcal{B}$, and shows that the effective rate saturates at high SNR as $\overline{I}_{\mathrm{GMI}} = \log\frac{1}{\gamma}$. The paper then provides high-resolution insights: overload distortion causes an exponentially decaying rate loss in the loading factor $L$ (as $\Theta(e^{-L^2/2}/L^3)$), while granular distortion yields a quadratic decay in step size $\ell$ (as $(\ell^2/12)\mathrm{SNR}$). Importantly, the optimal loading factor scales as $L^* \sim 2\sqrt{\ln(2K)}$ with $2K$-level quantizers, and an accurate finite-resolution estimator $\hat{L}$ from a transcendental equation offers practical design guidance. These results inform robust receiver quantization design and gain-control strategies for modern communicators facing non-negligible quantization effects.

Abstract

We investigate performance limits and design of communication in the presence of uniform output quantization with moderate to high resolution. Under independent and identically distributed (i.i.d.) complex Gaussian codebook and nearest neighbor decoding rule, an achievable rate is derived in an analytical form by the generalized mutual information (GMI). The gain control before quantization is shown to be increasingly important as the resolution decreases, due to the fact that the loading factor (normalized one-sided quantization range) has increasing impact on performance. The impact of imperfect gain control in the high-resolution regime is characterized by two asymptotic results: 1) the rate loss due to overload distortion decays exponentially as the loading factor increases, and 2) the rate loss due to granular distortion decays quadratically as the step size vanishes. For a $2K$-level uniform quantizer, we prove that the optimal loading factor that maximizes the achievable rate scales like $2\sqrt{\ln (2K)}$ as the resolution increases. An asymptotically tight estimate of the optimal loading factor is further given, which is also highly accurate for finite resolutions.

Figures (5)

• Figure 1: Achievable rate (\ref{['main']}) with $b$-bits uniform output quantization and the optimal loading factor.
• Figure 2: Impact of loading factor: $b=9$ bits.$\mspace{40mu}$ Four-sigma rule of thumb is convenient.
• Figure 3: Impact of loading factor: $b=6$ bits.$\mspace{40mu}$ Four-sigma rule of thumb causes some loss.
• Figure 4: Impact of loading factor: $b=3$ bits.$\mspace{40mu}$ Four-sigma rule of thumb causes considerable loss.
• Figure 5: Numerical results and asymptotics of $L^\ast$.