# Signal Amplitude Estimation and Detection from Unlabeled Binary Quantized Samples

## TL;DR

This paper develops methods for estimating signal amplitude and detecting signals from unlabeled binary quantized samples, addressing unknown ordering and providing algorithms with theoretical guarantees.

## Contribution

It introduces polynomial-time ML estimators for permutation and amplitude, analyzes model identifiability, and derives probability bounds for permutation recovery.

## Key findings

- ML estimators effectively recover signal amplitude and permutation.
- Explicit probability expressions relate sample size and quantizer count.
- Numerical results confirm theoretical accuracy and robustness.

## Abstract

Signal amplitude estimation and detection from unlabeled quantized binary samples are studied, assuming that the order of the time indexes is completely unknown. First, maximum likelihood (ML) estimators are utilized to estimate both the permutation matrix and unknown signal amplitude under arbitrary, but known signal shape and quantizer thresholds. Sufficient conditions are provided under which an ML estimator can be found in polynomial time and an alternating maximization algorithm is proposed to solve the general problem via good initial estimates. In addition, the statistical identifiability of the model is studied. Furthermore, the generalized likelihood ratio test (GLRT) detector is adopted to detect the presence of signal. In addition, an accurate approximation to the probability of successful permutation matrix recovery is derived, and explicit expressions are provided to reveal the relationship between the number of signal samples and the number of quantizers. Finally, numerical simulations are performed to verify the theoretical results.