An Information-Theoretic Framework for Receiver Quantization in Communication

Abstract

We investigate information-theoretic limits and design of communication under receiver quantization. Unlike most existing studies, this work is more focused on the impact of resolution reduction from high to low. We consider a standard transceiver architecture, which includes i.i.d. complex Gaussian codebook at the transmitter, and a symmetric quantizer cascaded with a nearest neighbor decoder at the receiver. Employing the generalized mutual information (GMI), an achievable rate under general quantization rules is obtained in an analytical form, which shows that the rate loss due to quantization is $\log\left(1+γ\mathsf{SNR}\right)$, where $γ$ is determined by thresholds and levels of the quantizer. Based on this result, the performance under uniform receiver quantization is analyzed comprehensively. We show that the front-end gain control, which determines the loading factor of quantization, has an increasing impact on performance as the resolution decreases. In particular, we prove that the unique loading factor that minimizes the MSE also maximizes the GMI, and the corresponding irreducible rate loss is given by $\log\left(1+\mathsf {mmse}\cdot\mathsf{SNR}\right)$, where mmse is the minimum MSE normalized by the variance of quantizer input, and is equal to the minimum of $γ$. A geometrical interpretation for the optimal uniform quantization at the receiver is further established. Moreover, by asymptotic analysis, we characterize the impact of biased gain control, showing how small rate losses decay to zero and providing rate approximations under large bias. From asymptotic expressions of the optimal loading factor and mmse, approximations and several per-bit rules for performance are also provided. Finally we discuss more types of receiver quantization and show that the consistency between achievable rate maximization and MSE minimization does not hold in general.