On Direction of Arrival Estimation with 1-bit Quantizer (Radar SIgnal Processing)

Ilia Yoffe 1,2

ilia@alephzero.ai

Nir Regev 1,2

nir@alephzero.ai

Dov Wulich 1

1 EE Dept., Ben-Gurion University of the Negev, Beer-Sheva, Israel

2 AlephZero Consulting, LLC, LA, US

Abstract—Designing and implementing analog front-end circuits is a complex problem and thus, is the cornerstone of any radar system design. We propose removing the gain control block, as well as reducing the complexity by introducing a 1-bit analog to digital converter (ADC) at the receiving path. Nevertheless, this nonlinear quantization operation distorts the signal in a way that does not preserves its Gaussianity, rendering the common Maximum Likelihood (ML) based Direction of Arrival (DOA) estimation methods non-optimal. We derive the ML optimal DOA estimator for the 1-bit ADC and propose sub-optimal, yet, effective estimator to reduce the complexity of the ML estimator. We benchmark the performance of the proposed estimators derived in this paper against the derived Cramér–Rao lower bound and investigate the case of a known and unknown transmitted signals. We show that the proposed algorithms attain the bound under various conditions as well as outperform a naïve ML approach for the 1-bit ADC problem.

I. Introduction

The implementation complexity of analog front-end circuits is a major hurdle in the design of sensors array. In order to reduce this complexity, it is proposed to apply 1-bit quantizer at the receiving path of each antenna, thus, enabling the removal of the gain control block as well as simplification of quantizer’s structure, as it can be implemented by a single comparator. This simplification enables simple and inexpensive implementation of receiving array with a very large number of antennas.

Nevertheless, a serious drawback of a 1-bit Analog to Digital Converter (ADC) is that it performs a highly nonlinear and noninvertible operation on the original analog receive signal. As a result of this operation, the common assumption of the Gaussianity of the received signal is violated. In fact, the received signal is turned into a binary correlated sequence. The distribution of this sequence is known as multivariate orthant probability. Unfortunately, closed-form expressions are available only for maximal sequence length of 4 [1].

Several studies related to 1-bit quantization are found in the literature. The issue of frequency estimation with a 1-bit quantizer is studied in [2]. This problem has a similar formulation as Direction of Arrival (DOA) estimation. The presented analysis assumes a deterministic signal model with unknown parameters.

DOA with 1-bit quantizer with random signals is studied in [3]. Due to the complexity of the problem the full analytical analysis is only presented for 2 sensors. Moreover, several sub-optimal methods for DOA estimation were proposed for an M-sensors Uniform Linear Array (ULA). These methods are based on the reconstruction of the covariance matrix of the original signal before the clipping.

In [4] the DOA estimation problem with a 1-bit quantizer of random signals is considered. Here, a nonlinear transformation is applied on the received signal after quantization. This transformation is re-generating an approximately Gaussian distributed signal, which is then used in a Maximum Likelihood (ML) framework for estimation and performance evaluation.

In [5] the authors reported the results of 1-bit DOA estimation for sparse nested and coprime arrays while in [6] a generalized sparse Bayesian learning algorithm is used iteratively as a two-step estimator, namely, a sparse Bayesian learner followed by a naïve Minimum Mean Squared Error (MMSE) estimator of a linear problem. The same problem is studied in [7] for the Sparse MIMO array setting and is employing a support vector machine (SVM) approach.

In this work, we derive DOA estimation methods for ULA with 1-bit quantizer for deterministic known and unknown signal. We do not assume sparsity. In Section II we assume a single source and deterministic known signal. We derive the ML DOA estimator and propose a sub-optimal low complexity estimator based on 1 bit FFT. The performance of the estimators is benchmarked against the numerically calculated Cramér–Rao Bound (CRB) for an ideal quantizer case (ML and CRB). In Section III we assume a single source and deterministic unknown signal. We derive an Expectation Maximization (EM) based method for DOA estimation. Moreover, we study the performance degradation of the existing methods, as naïve ML and MUSIC, in the case of 1-bit quantization.

Full paper: https://www.alephzero.ai/papers

About the Authors:

Nir Regev is a co-founder of AlephZero Consulting. He is a senior research scientist, with over 21 years of experience in developing algorithms. Loves engineering problems that are unsolvable. Specializing in computer vision, radar signal processing, multi-target tracking and deep learning, classification, radar micro-Doppler, deep learning based target classification, optimization and statistical signal processing. 

Ilia Yoffe is a principal research scientist in AlephZero Consulting. He is an Algorithms scientist with extensive industry experience. Holds a Ph.D. in electrical engineering. Published more than 15 papers in top journals and conferences. Specializing in digital communication (physical layer), spatial signal processing and radar signal processing. Experienced with machine learning, deep learning, computer vision. Experienced with communications standards including 802.11 (Wi-Fi and Wigig) and 5G, as well as reverse engineering communications protocol based on air-recordings on downlink/uplink.