# Performance Analysis for Channel Estimation with 1-bit ADC and Unknown Quantization Threshold

## TL;DR

This paper investigates the limits of channel estimation accuracy using 1-bit ADCs with unknown thresholds, analyzing the impact of hardware imperfections and proposing optimal algorithms for low SNR regimes.

## Contribution

It provides a comprehensive analysis of channel estimation with unknown 1-bit quantization thresholds, including error expressions and conditions for minimal performance loss.

## Key findings

- Establishes conditions for negligible loss due to unknown thresholds in low SNR.
- Derives analytic error bounds for nonlinear channel estimation algorithms.
- Validates theoretical results with experimental performance comparisons.

## Abstract

In this work, the problem of signal parameter estimation from measurements acquired by a low-complexity analog-to-digital converter (ADC) with $1$-bit output resolution and an unknown quantization threshold is considered. Single-comparator ADCs are energy-efficient and can be operated at ultra-high sampling rates. For analysis of such systems, a fixed and known quantization threshold is usually assumed. In the symmetric case, i.e., zero hard-limiting offset, it is known that in the low signal-to-noise ratio (SNR) regime the signal processing performance degrades moderately by ${2}/{\pi}$ ($-1.96$ dB) when comparing to an ideal $\infty$-bit converter. Due to hardware imperfections, low-complexity $1$-bit ADCs will in practice exhibit an unknown threshold different from zero. Therefore, we study the accuracy which can be obtained with receive data processed by a hard-limiter with unknown quantization level by using asymptotically optimal channel estimation algorithms. To characterize the estimation performance of these nonlinear algorithms, we employ analytic error expressions for different setups while modeling the offset as a nuisance parameter. In the low SNR regime, we establish the necessary condition for a vanishing loss due to missing offset knowledge at the receiver. As an application, we consider the estimation of single-input single-output wireless channels with inter-symbol interference and validate our analysis by comparing the analytic and experimental performance of the studied estimation algorithms. Finally, we comment on the extension to multiple-input multiple-output channel models.