Underdetermined DOA Estimation Under the Compressive Sensing Framework: A Review

Direction of arrival (DOA) estimation from the perspective of sparse signal representation has attracted tremendous attention in past years, where the underlying spatial sparsity reconstruction problem is linked to the compressive sensing (CS) framework. Although this is an area with ongoing intensive research and new methods and results are reported regularly, it is time to have a review about the basic approaches and methods for CS-based DOA estimation, in particular for the underdetermined case. We start from the basic time-domain CSbased formulation for narrowband arrays and then move to the case for recently developed methods for sparse arrays based on the co-array concept. After introducing two specifically designed structures (the two-level nested array and the co-prime array) for optimizing the virtual sensors corresponding to the difference coarray, this CS-based DOA estimation approach is extended to the wideband case by employing the group sparsity concept, where a much larger physical aperture can be achieved by allowing a larger unit inter-element spacing and therefore leading to further improved performance. Finally, a specifically designed ULA structure with associated CS-based underdetermined DOA estimation is presented to exploit the difference co-array concept in the spatio-spectral domain, leading to a significant increase in DOFs. Representative simulation results for typical narrowband and wideband scenarios are provided to demonstrate their performance

Array Signal Processing Based on Traditional and Sparse Arrays

Array signal processing is based on using an array of sensors to receive the impinging signals. The received data is either spatially filtered to focus the signals from a desired direction or it may be used for estimating a parameter of source signal like direction of arrival (DOA), polarization and source power. Spatial filtering also known as beamforming and DOA estimation are integral parts of array signal processing and this thesis is aimed at solving some key probems related to these two areas. Wideband beamforming holds numerous applications in the bandwidth hungry data traffic of present day world. Several techniques exist to design fixed wideband beamformers based on traditional arrays like uniform linear array (ULA). Among these techniques, least squares based eigenfilter method is a key technique which has been used extensively in filter and wideband beamformer design. The first contribution of this thesis comes in the form of critically analyzing the standard eigenfilter method where a serious flaw in the design formulation is highlighted which generates inconsistent design performance, and an additional constraint is added to stabilize the achieved design. Simulation results show the validity and significance of the proposed method. Traditional arrays based on ULAs have limited applications in array signal processing due to the large number of sensors required and this problem has been addressed by the application of sparse arrays. Sparse arrays have been exploited from the perspective of their difference co-array structures which provide significantly higher number of degrees of freedoms (DOFs) compared to ULAs for the same number of sensors. These DOFs (consecutive and unique lags) are utilized in the application of DOA estimation with the help of difference co-array based DOA estimators. Several types of sparse arrays include minimum redundancy array (MRA), minimum hole array (MHA), nested array, prototype coprime array, conventional coprime array, coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS) and super nested array. As a second contribution of this thesis, a new sparse array termed thinned coprime array (TCA) is proposed which holds all the properties of a conventional coprime array but with \ceil*{\frac{M}{2}} fewer sensors where $M$ is the number of sensors of a subarray in the conventional structure. TCA possesses improved level of sparsity and is robust against mutual coupling compared to other sparse arrays. In addition, TCA holds higher number of DOFs utilizable for DOA estimation using variety of methods. TCA also shows lower estimation error compared to super nested arrays and MRA with increasing array size. Although TCA holds numerous desirable features, the number of unique lags offered by TCA are close to the sparsest CADiS and nested array and significantly lower than MRA which limits the estimation error performance offered by TCA through (compressive sensing) CS-based methods. In this direction, the structure of TCA is studied to explore the possibility of an array which can provide significantly higher number of unique lags with improved sparsity for a given number of sensors. The result of this investigation is the third contribution of this thesis in the form of a new sparse array, displaced thinned coprime array with additional sensor (DiTCAAS), which is based on a displaced version of TCA. The displacement of the subarrays generates an increase in the unique lags but the minimum spacing between the sensors becomes an integer multiple of half wavelength. To avoid spatial aliasing, an additional sensor is added at half wavelength from one of the sensors of the displaced subarray. The proposed placement of the additional sensor generates significantly higher number of unique lags for DiTCAAS, even more than the DOFs provided by MRA. Due to its improved sparsity and higher number of unique lags, DiTCAAS generates the lowest estimation error and robustness against heavy mutual coupling compared to super nested arrays, MRA, TCA and sparse CADiS with CS-based DOA estimation

Direction Finding With Mutually Orthogonal Antennas

Estimating the direction-of-arrival of incident electromagnetic plane waves (a.k.a. direction finding or DF) has typically been accomplished in the past using arrays of spatially separated antennas. The spatial separation produces a delay in each antenna\u27s measured voltage due to the finite propagation time as the wave strikes each antenna in succession. In this thesis, we approach the problem differently by using three antennas that have been oriented in orthogonal directions but are co-located at the origin of a coordinate system. Being co-located, this mutually orthogonal arrangement of antennas cannot detect the propagation phase delay and must rely solely on the polarization properties of the incident waves. Using the vector effective height concept, three algorithms are formulated. The first algorithm estimates the direction-of-arrival by computing a vector that is perpendicular to the locus of the instantaneous electric field vector. The second and third algorithms are based on the well-known maximum likelihood and MUSIC algorithms. Simulation results show that each algorithm can estimate the direction-of-arrival with a root-mean-squared error within 1° or less when the incident wave is circularly polarized, the antennas are small compared to wavelength, and the signal-to-noise ratio is above 20dB

A New Restriction on Low-Redundancy Restricted Array and Its Good Solutions

In array signal processing, a fundamental problem is to design a sensor array with low-redundancy and reduced mutual coupling, which are the main features to improve the performance of direction-of-arrival (DOA) estimation. For a $N$-sensor array with aperture $L$, it is called low-redundancy (LR) if the ratio $R=N(N-1)/(2L)$ is approaching the Leech's bound $1.217\leq R_{opt}\leq 1.674$ for $N\rightarrow\infty$; and the mutual coupling is often reduced by decreasing the numbers of sensor pairs with the first three smallest inter-spacings, denoted as $\omega(a)$ with $a\in\{1,2,3\}$. Many works have been done to construct large LRAs, whose spacing structures all coincide with a common pattern ${\mathbb D}=\{a_1,a_2,\ldots,a_{s_1},c^\ell,b_1,b_2,\ldots,b_{s_2}\}$ with the restriction $s_1+s_2=c-1$. Here $a_i,b_j,c$ denote the spacing between adjacent sensors, and $c$ is the largest one. The objective of this paper is to find some new arrays with lower redundancy ratio or lower mutual coupling compared with known arrays. In order to do this, we give a new restriction for ${\mathbb D}$ to be $s_1+s_2=c$ , and obtain 2 classes of $(4r+3)$-type arrays, 2 classes of $(4r+1)$-type arrays, and 1 class of $(4r)$-type arrays for any $N\geq18$. Here the $(4r+i)$-Type means that $c\equiv i\pmod4$. Notably, compared with known arrays with the same type, one of our new $(4r+1)$-type array and the new $(4r)$-type array all achieves the lowest mutual coupling, and their uDOFs are at most 4 less for any $N\geq18$; compared with SNA and MISC arrays, the new $(4r)$-type array has a significant reduction in both redundancy ratio and mutual coupling. We should emphasize that the new $(4r)$-type array in this paper is the first class of arrays achieving $R<1.5$ and $\omega(1)=1$ for any $N\geq18$

Multi-source parameter estimation and tracking using antenna arrays

This thesis is concerned with multi-source parameter estimation and tracking using antenna arrays in wireless communications. Various multi-source parameter estimation and tracking algorithms are presented and evaluated. Firstly, a novel multiple-input multiple-output (MIMO) communication system is proposed for multi-parameter channel estimation. A manifold extender is presented for increasing the degrees of freedom (DoF). The proposed approach utilises the extended manifold vectors together with superresolution subspace type algorithms, to achieve the estimation of delay, direction of departure (DOD) and direction of arrival (DOA) of all the paths of the desired user in the presence of multiple access interference (MAI). Secondly, the MIMO system is extended to a virtual-spatiotemporal system by incorporating the temporal domain of the system towards the objective of further increasing the degrees of freedom. In this system, a multi-parameter es- timation of delay, Doppler frequency, DOD and DOA of the desired user, and a beamformer that suppresses the MAI are presented, by utilising the proposed virtual-spatiotemporal manifold extender and the superresolution subspace type algorithms. Finally, for multi-source tracking, two tracking approaches are proposed based on an arrayed Extended Kalman Filter (arrayed-EKF) and an arrayed Unscented Kalman Filter (arrayed-UKF) using two type of antenna arrays: rigid array and flexible array. If the array is rigid, the proposed approaches employ a spatiotemporal state-space model and a manifold extender to track the source parameters, while if it is flexible the array locations are also tracked simultaneously. Throughout the thesis, computer simulation studies are presented to investigate and evaluate the performance of all the proposed algorithms.Open Acces

Cooperative Position and Orientation Estimation with Multi-Mode Antennas

Robotic multi-agent systems are envisioned for planetary exploration and terrestrial applications. Autonomous operation of robots requires estimations of their positions and orientations, which are obtained from the direction-of-arrival (DoA) and the time-of-arrival (ToA) of radio signals exchanged among the agents. In this thesis, we estimate the signal DoA and ToA using a multi-mode antenna (MMA). An MMA is a single antenna element, where multiple orthogonal current modes are excited by different antenna ports. We provide a first study on the use of MMAs for cooperative position and orientation estimation, specifically exploring their DoA estimation capabilities. Assuming the agents of a cooperative network are equipped with MMAs, lower bounds on the achievable position and orientation accuracy are derived. We realize a gap between the theoretical lower bounds and real-world performance of a cooperative radio localization system, which is caused by imperfect antenna and transceiver calibration. Consequentially, we theoretically analyze in-situ antenna calibration, introduce an algorithm for the calibration of arbitrary multiport antennas and show its effectiveness by simulation. To also improve calibration during operation, we propose cooperative simultaneous localization and calibration (SLAC). We show that cooperative SLAC is able to estimate antenna responses and ranging biases of the agents together with their positions and orientations, leading to considerably better position and orientation accuracy. Finally, we validate the results from theory and simulation by experiments with robotic rovers equipped with software-defined radios (SDRs). In conclusion, we show that DoA estimation with an MMA is feasible, and accuracy can be improved by in-situ calibration and SLAC

Bayesian In-Situ Calibration of Multiport Antennas for DoA Estimation: Theory and Measurements

The direction-of-arrival (DoA) of radio waves is used for many applications, e.g. the localization of autonomous robots and smart vehicles. Estimating the DoA is possible with a multiport antenna, e.g. an antenna array or a multi-mode antenna (MMA). In practice, DoA estimation performance decisively depends on accurate knowledge of the antenna response, which makes antenna calibration vital. As the antenna surroundings influence its response, it is necessary to measure the entire device with installed antenna to obtain the installed antenna response. Antenna calibration is often done in a dedicated measurement chamber, which can be inconvenient and costly, especially for larger devices. Thus, auto- and in-situ calibration methods aim at making antenna calibration in a measurement chamber redundant. However, existing auto- and in-situ calibration methods are restricted to certain antenna types and certain calibrations. In this paper, we propose a Bayesian in-situ calibration algorithm based on a maximum a posteriori (MAP) estimator, which is suitable for arbitrary multiport antennas. The algorithm uses received signals from a transmitter, noisy external DoA observations, takes multipath propagation into account and does not require synchronization. Furthermore, we take an estimation theoretic perspective and provide an in-depth theoretical discussion of in-situ antenna calibration in unknown propagation conditions based on Bayesian information and the Bayesian Cramér-Rao bound (BCRB). Extensive simulations show that the proposed algorithm operates close to the BCRB and the achieved DoA estimation performance asymptotically approaches the case of a perfectly known antenna response. Finally, we provide an experimental validation, where we calibrate the antenna on a robotic rover and evaluate the DoA estimation performance using measurement data. With the proposed in-situ antenna calibration algorithm, DoA estimation performance is considerably improved compared to using an antenna response obtained by simulation or in a measurement chamber

A review of closed-form Cramér-Rao Bounds for DOA estimation in the presence of Gaussian noise under a unified framework

The Cramér-Rao Bound (CRB) for direction of arrival (DOA) estimation has been extensively studied over the past four decades, with a plethora of CRB expressions reported for various parametric models. In the literature, there are different methods to derive a closed-form CRB expression, but many derivations tend to involve intricate matrix manipulations which appear difficult to understand. Starting from the Slepian-Bangs formula and following the simplest derivation approach, this paper reviews a number of closed-form Gaussian CRB expressions for the DOA parameter under a unified framework, based on which all the specific CRB presentations can be derived concisely. The results cover three scenarios: narrowband complex circular signals, narrowband complex noncircular signals, and wideband signals. Three signal models are considered: the deterministic model, the stochastic Gaussian model, and the stochastic Gaussian model with the a priori knowledge that the sources are spatially uncorrelated. Moreover, three Gaussian noise models distinguished by the structure of the noise covariance matrix are concerned: spatially uncorrelated noise with unknown either identical or distinct variances at different sensors, and arbitrary unknown noise. In each scenario, a unified framework for the DOA-related block of the deterministic/stochastic CRB is developed, which encompasses one class of closed-form deterministic CRB expressions and two classes of stochastic ones under the three noise models. Comparisons among different CRBs across classes and scenarios are presented, yielding a series of equalities and inequalities which reflect the benchmark for the estimation efficiency under various situations. Furthermore, validity of all CRB expressions are examined, with some specific results for linear arrays provided, leading to several upper bounds on the number of resolvable Gaussian sources in the underdetermined case

Twenty-five years of sensor array and multichannel signal processing: a review of progress to date and potential research directions

In this article, a general introduction to the area of sensor array and multichannel signal processing is provided, including associated activities of the IEEE Signal Processing Society (SPS) Sensor Array and Multichannel (SAM) Technical Committee (TC). The main technological advances in five SAM subareas made in the past 25 years are then presented in detail, including beamforming, direction-of-arrival (DOA) estimation, sensor location optimization, target/source localization based on sensor arrays, and multiple-input multiple-output (MIMO) arrays. Six recent developments are also provided at the end to indicate possible promising directions for future SAM research, which are graph signal processing (GSP) for sensor networks; tensor-based array signal processing, quaternion-valued array signal processing, 1-bit and noncoherent sensor array signal processing, machine learning and artificial intelligence (AI) for sensor arrays; and array signal processing for next-generation communication systems