Cramer-Rao Research Articles

Impact of Antenna Pattern on TOA Based 3D UAV Localization Using a Terrestrial Sensor Network

In this article, we explore the fundamental limits of the 3-dimensional (3D) localization of unmanned aerial vehicles (UAVs) in conjunction with the effects of 3D antenna radiation patterns. Although localization of UAVs has been studied to some extent in the literature, effects of antenna characteristics on 3D localization remains mostly unexplored. To study such effects, we consider a scenario where a fixed number of radio-frequency (RF) sensors equipped with single or multiple dipole antennas are placed at some known locations on the ground, and they derive the time-difference-of-arrival (TDOA) measurements from the time-of-arrival (TOA) data collected for the UAV that is also equipped with a dipole antenna. We then use these measurements to estimate the 3D location of the UAV, and to derive the Cramer-Rao lower bounds (CRLBs) on the localization error for various orientations of the dipole antennas at the transmitter and the receiver. Namely, we consider vertical-vertical (VV), horizontal-horizontal (HH), and vertical-horizontal (VH) radiation patterns in a purely line-of-sight (LoS) environment and a mixed LoS/Non-line-of-sight (NLoS) environment. We show that the localization accuracy changes in a non-monotonic pattern with respect to the UAV altitude and identify the respective critical altitudes for each of the VV, VH and HH orientations. Subsequently, we propose a multi-antenna signal acquisition technique that mitigates the accuracy degradation due to the antenna pattern mismatches, and we derive the localization CRLB for the multi-antenna scenario. Our numerical results characterize achievable localization accuracy for various antenna configurations, UAV heights, and propagation conditions for representative UAV scenarios.

Two-Stage Channel Estimation for Hybrid RIS Assisted MIMO Systems

Reconfigurable intelligent surfaces (RISs) have been proposed as a key enabler to improve the coverage of the signals and mitigate the frequent blockages in millimeter wave (mmWave) multiple-input multiple-output (MIMO) communications. However, the channel state information (CSI) acquisition is one of the major challenges for the practical deployment of the RIS. The passive RIS without any baseband processing capabilities brings difficulty on the channel estimation (CE), since the individual channels or the cascaded one can be estimated only at base station (BS) via uplink training or mobile station (MS) via downlink training. In order to facilitate the CSI acquisition, we focus on the hybrid RIS architecture, where a small number of elements are active and able to receive and process the pilot signals at the RIS. The CE is performed in two stages by following the atomic norm minimization to recover the channel parameters, i.e., angles of departure (AoDs), angles of arrival (AoAs), and propagation path gains. Simulation results show that the proposed scheme can outperform the passive RIS CE under the same training overhead. Furthermore, we also study the theoretical performance limits in terms of mean square error (MSE) via Cramér-Rao lower bound (CRLB) analyses.

Localization With Distributed MIMO Using a High-Speed Sigma-Delta-Over-Fiber Testbed

Distributed MIMO (D-MIMO) with synchronized access points (APs) is a promising architecture for both communications and localization in 5G and beyond systems. In this letter, we develop a time-difference-of-arrival (TDOA)-based indoor localization system using a 2.35-GHz high-speed sigma-delta-over-fiber (SDoF) D-MIMO testbed with 40-MHz bandwidth, exploiting the fully synchronized nature of the APs. Experimental results over an area of size 100 m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> demonstrate accuracies below 0.2 m and agree with the theoretical Cramér–Rao bounds (CRBs) at most measurement locations, indicating the localization capability of high-speed SDoF D-MIMO.

On the Accuracy of Wideband Localization in Quantized Uplink Massive MIMO Systems

As an essential technology for 5G and beyond, massive multiple-input multiple-output (MIMO) is well investigated in the aspect of high data-rate communication. However, its potential benefits for accurate positioning are largely undiscovered. In this letter, we derive the Cramér-Rao bound (CRB) on position estimation uncertainty in quantized uplink massive MIMO systems, where orthogonal frequency division multiplexing (OFDM) is considered and the base station (BS) uses few-bit analog-to-digital converters (ADCs) to acquire the incoming signals. With the obtained CRB, one can evaluate the dependence of localization accuracy on the system parameters, such as the number of quantization bits and different combinations of multipaths. Numerical results show that accurate localization is possible in various multipath scenarios, even with low-resolution ADCs.

Joint Frame Detection and Channel Parameter Estimation for OOK Free-Space Optical Communications

We consider a free-space optical (FSO) communication link for packet-based transmissions over a turbulence fading channel. The modulation format is on-off keying (OOK) and a unique-word (UW) composed by a known synch pattern is periodically inserted in the data stream to identify the start of frame. Since an avalanche photo-diode (APD) is used for direct detection of the OOK symbols, the photocurrent signal provided by the APD is plagued by a mixture of thermal and shot noise with signal-dependent power. Our goal is the detection of the UW position in the received stream, along with the estimation of the unknown channel attenuation and noise variances. The aforementioned problem has recently been studied by ignoring any information conveyed by data symbols surrounding the UW. In this work, further investigation is conducted in order to determine the maximum likelihood (ML) solution that, in addition to the UW, exploits all the information-bearing symbols belonging to the observation window. The relevant Cramér-Rao bound (CRB) is also evaluated to establish the ultimate accuracy achievable in the estimation process. Since the true ML estimator leads to a computationally intractable multi-dimensional optimization problem, we develop suitable approximations that enable accurate frame synchronization with affordable complexity. We also present an iterative solution to refine the channel and noise power estimates provided by the UW detection procedure. Numerical simulations demonstrate the superiority of the proposed synchronization and estimation schemes with respect to existing alternatives.

Compressive Wideband Spectrum Sensing and Signal Recovery With Unknown Multipath Channels

We study the problem of joint wideband spectrum sensing and recovery of multi-band signals in a multi-antenna-based sub-Nyquist sampling framework. Specifically, the multi-band signal is composed of a number of uncorrelated narrowband signals spreading over a wide frequency band. Unlike existing works which assume the source signals impinge on the receiver via a line-of-sight (LOS) path, we consider a more practical unknown MIMO channel which results from multipath propagation. A new sub-Nyquist sampling architecture is proposed, where each antenna output passes through two channels, namely, a direct path and a delayed path with a controlled amount of time delay. The signal at each channel is then sampled by a synchronized low-rate analog-to-digital converter (ADC). We utilize the collected data samples to build a set of cross-correlation matrices with different time lags and develop a CANDECOMP/PARAFAC (CP) decomposition-based method to recover the carrier frequencies, power spectra as well as the source signals themselves. Recovery conditions of the proposed method are analyzed, and Cramér-Rao bound (CRB) results for our estimation problem are derived. Simulation results are presented to illustrate the effectiveness of the proposed method.

Target Sensing With Intelligent Reflecting Surface: Architecture and Performance

Intelligent reflecting surface (IRS) has emerged as a promising technology to reconfigure the radio propagation environment by dynamically controlling wireless signal’s amplitude and/or phase via a large number of reflecting elements. In contrast to the vast literature on studying IRS’s performance gains in wireless communications, we study in this paper a new application of IRS for sensing/localizing targets in wireless networks. Specifically, we propose a new <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">self-sensing IRS</i> architecture where the IRS controller is capable of transmitting probing signals that are not only directly reflected by the target (referred to as the direct echo link), but also consecutively reflected by the IRS and then the target (referred to as the IRS-reflected echo link). Moreover, dedicated sensors are installed at the IRS for receiving both the direct and IRS-reflected echo signals from the target, such that the IRS can sense the direction of its nearby target by applying a customized multiple signal classification (MUSIC) algorithm. However, since the angle estimation mean square error (MSE) by the MUSIC algorithm is intractable, we propose to optimize the IRS passive reflection for maximizing the average echo signals’ total power at the IRS sensors and derive the resultant Cramer-Rao bound (CRB) of the angle estimation MSE. Last, numerical results are presented to show the effectiveness of the proposed new IRS sensing architecture and algorithm, as compared to other benchmark sensing systems/algorithms.

Ant Colony Pheromone Mechanism-Based Passive Localization Using UAV Swarm

The problem of passive localization using an unmanned aerial vehicle (UAV) swarm is studied. For multi-UAV localization systems with limited communication and observation range, the challenge is how to obtain accurate target state consistency estimates through local UAV communication. In this paper, an ant colony pheromone mechanism-based passive localization method using a UAV swarm is proposed. Different from traditional distributed fusion localization algorithms, the proposed method makes use of local interactions among individuals to process the observation data with UAVs, which greatly reduces the cost of the system. First, the UAVs that have detected the radiation source target estimate the rough target position based on the pseudo-linear estimation (PLE). Then, the ant colony pheromone mechanism is introduced to further improve localization accuracy. The ant colony pheromone mechanism consists of two stages: pheromone injection and pheromone transmission. In the pheromone injection mechanism, each UAV uses the maximum likelihood (ML) algorithm with the current observed target bearing information to correct the initial target position estimate. Then, the UAV swarm weights and fuses the target position information between individuals based on the pheromone transmission mechanism. Numerical results demonstrate that the accuracy of the proposed method is better than that of traditional localization algorithms and close to the Cramer–Rao lower bound (CRLB) for small measurement noise.

Computationally attractive and statistically efficient estimator for noise resilient TOA localization

Localization technologies have received considerable attentions and been well-studied in past few decades. However, the performance under limited number of sensors and large noise/error environments still needs to be improved for balancing the performance and complexity. Taking these factors into consideration, this paper focuses on the time of arrival (TOA) localization problem where the measurement noise level can be high, the sensor position errors are present and the number of sensors is limited. We propose a location estimator that consists of a coarse solution and a refined estimate to enhance the quality of the coarse solution. The proposed estimator is closed-form and simple to compute. It is asymptotically unbiased, and having the estimation covariance matrix analytically matching the Cramér–Rao lower bound (CRLB). Simulations validate the theoretical results and illustrate the proposed algorithm works well with small number of sensors, has more noise resilient behavior in the large error region than the existing methods, and performs close to the Maximum Likelihood Estimator with ideal initialization.

Vector Miss Distance Parameter Estimation Based on Direct Position Determination

The vector miss distance measurement (VMDM) plays a key role in the scoring system. In the traditional two-step method, the estimation accuracy of intermediate measurements is limited by the nonstationary characteristics of echoes present in the rendezvous process. To address this problem, an innovative vector miss distance parameter estimation method based on direct position determination (DPD) is proposed in this study. The proposed method can directly estimate vector miss distance parameters with higher accuracy from the echoes received, in which the intermediate measurements need not be estimated in advance. The closed-form expressions of the Cramér-Rao bound (CRB) for vector miss distance parameters are also derived. The simulation results show that the proposed method outperforms the traditional two-step method in terms of the estimation accuracy of vector miss distance parameters. Moreover, the proposed method is more robust to the high noise levels and inconsistency of the receiving channels. The experimental data further validated the efficiency of the proposed method.

Efficient solutions for target localization in asynchronous MIMO networks

Multiple-input multiple-output (MIMO) technique has been drawing extensive attention for its spatial diversity to enhance the performance in the target localization. Considering a fully asynchronous MIMO network, we propose a differential delay time (DDT) method for target localization. Semidefinite programming (SDP) and closed-form solutions (CFS) always converge to global optimum. To efficiently solve the DDT-based localization problem, we propose a global SDP (GSDP) solution to determine the target position. The complexity of the GSDP solution is high. Hence, we also develop a two-stage CFS (TSCFS) to reduce the complexity. We also prove that the performance of the TSCFS and GSDP solutions is able to attain the Cramér–Rao Lower Bound (CRLB) accuracy. The simulated results show that the GSDP solution provides comparable performance with the TSCFS at the low noise levels. Compared with the TSCFS, the GSDP shows its superiority in the estimation performance at the high noise levels.

Metabolite-cycled echo-planar spectroscopic imaging ofthehuman heart.

PurposeSpectroscopic imaging could provide insights into regional cardiac triglyceride variations, but is hampered by relatively long scan times. It is proposed to synergistically combine echo‐planar spectroscopic imaging (EPSI) with motion‐adapted gating, weighted acquisition and metabolite cycling to reduce scan times to less than 10 min while preserving spatial‐spectral quality. The method is compared to single‐voxel measurements and to metabolite‐cycled EPSI with conventional acquisition for assessing triglyceride‐to‐water (TG/W) ratios in the human heart.MethodsMeasurements were performed on 10 healthy volunteers using a clinical 1.5T system. EPSI data was acquired both without and with motion‐adapted gating in combination with weighted acquisition to assess TG/W ratios and relative Cramér‐Rao lower bounds (CRLB) of TG. For comparison, single‐voxel (PRESS) spectra were acquired in the interventricular septum.ResultsBland–Altman analyses did not show a significant bias in TG/W when comparing both metabolite‐cycled EPSI methods to PRESS for any of the cardiac segments. Scan time was 8.05 ± 2.06 min and 17.91 ± 3.93 min for metabolite‐cycled EPSI with and without motion‐adapted gating and weighted acquisition, respectively, while relative CRLB of TG did not differ significantly between the two methods for any of the cardiac segments.ConclusionsMetabolite‐cycled EPSI with motion‐adapted gating and weighted acquisition allows detecting TG/W ratios in different regions of the in vivo human heart. Scan time is reduced by more than 2‐fold to less than 10 min as compared to conventional acquisition, while keeping the quality of TG fitting constant.

Indoor scenario-based UWB anchor placement optimization method for indoor localization

Currently, ultra-wideband (UWB)-based localization systems are attracting much attention due to their high ranging accuracy. It is well known that the relative sensor–source geometry and non-line-of-sight (NLOS) measurement can severely impact the accuracy of the location estimate. To improve the localization accuracy, we propose an anchor placement optimization method that is based on the corresponding indoor scenario. Specifically, concrete walls affect the ranging error and constrain the placement of UWB sensors. An indoor scenario modeling method is presented for generating NLOS ranging paths and dividing the search space. Furthermore, a stochastic/deterministic combination ranging model is proposed based on the through-the-wall path and Gaussian process regression technique. We propose a heuristic differential evolution algorithm for searching for an optimal anchor placement that is based on minimizing the Cramer–Rao lower bound (CRLB) and predicting the ranging error. In addition, a region division method and a region combination method are proposed for reducing the search space, optimizing anchor placement in parallel in different regions, and providing full line-of-sight (LOS) coverage regions to the greatest extent possible. Finally, field experiments demonstrate the effectiveness of the proposed indoor scenario modeling method and ranging model. The proposed indoor scenario-based UWB anchor placement method achieves decimeter-level localization accuracy. A thorough comparison confirms the localization performance improvement attributed to the proposed anchor placement method.

An Information Appraisal Procedure: Endows Reliable Online Parameter Identification to Lithium-Ion Battery Model

Online parameter identification is vital for boosting the accuracy of the battery equivalent circuit model (ECM) under dynamic profiles. However, traditional recursive least squares (RLS) method easily decays with the noise corruption from sensors or insufficient exciting signal in reality, which further limits the performance of ECM in battery modeling and states estimation. This article thus proposes a reliable online parameter identification method for battery ECM, which utilizes a well-designed information appraisal procedure based on the Fisher-information-based Cramer–Rao lower bound (CRLB). Without increasing much computing complexity, a comprehensive appraisal indicator, derived recursively from CRLB, enables a new mechanism for online parameter updating. Simulation and experimental results prove the validity of the proposed method under different driving cycles, temperatures, and aging conditions. The results show that the identification accuracy of the proposed method has been significantly improved comparing with a typical RLS and a multiple adaptive forgetting factors RLS method.

Impact of Low-Resolution ADC on DOA Estimation Performance for Massive MIMO Receive Array

In this paper, we present a new scenario of direction of arrival (DOA) estimation using massive multiple-input multiple-output (MIMO) receive array with low-resolution analog-to-digital convertors (ADCs), which can strike a good balance between performance and circuit cost. Based on the linear additive quantization noise model (AQNM), the effect of low-resolution ADCs on the methods, such as Root-MUSIC method, is analyzed. Also, the closed-form expression of Cramer-Rao lower bound (CRLB) is derived to evaluate the performance loss caused by the low-resolution ADCs. The simulation results show that the Root-MUSIC methods can achieve the corresponding CRLB. Furthermore, 2-3 bits are acceptable for most applications if the 1dB performance loss.

Deep Learning-Based Frequency-Selective Channel Estimation for Hybrid mmWave MIMO Systems

Millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) systems typically employ hybrid mixed signal processing to avoid expensive hardware and high training overheads. However, the lack of fully digital beamforming at mmWave bands imposes additional challenges in channel estimation. Prior art on hybrid architectures has mainly focused on greedy optimization algorithms to estimate frequency-flat narrowband mmWave channels, despite the fact that in practice, the large bandwidth associated with mmWave channels results in frequency-selective channels. In this paper, we consider a frequency-selective wideband mmWave system and propose two deep learning (DL) compressive sensing (CS) based algorithms for channel estimation. The proposed algorithms learn critical apriori information from training data to provide highly accurate channel estimates with low training overhead. In the first approach, a DL-CS based algorithm simultaneously estimates the channel supports in the frequency domain, which are then used for channel reconstruction. The second approach exploits the estimated supports to apply a low-complexity multi-resolution fine-tuning method to further enhance the estimation performance. Simulation results demonstrate that the proposed DL-based schemes significantly outperform conventional orthogonal matching pursuit (OMP) techniques in terms of the normalized mean-squared error (NMSE), computational complexity, and spectral efficiency, particularly in the low signal-to-noise ratio regime. When compared to OMP approaches that achieve an NMSE gap of <inline-formula> <tex-math notation="LaTeX">$\mathrm {\{4-10\}\,\,dB}$ </tex-math></inline-formula> with respect to the Cramer Rao Lower Bound (CRLB), the proposed algorithms reduce the CRLB gap to only <inline-formula> <tex-math notation="LaTeX">$\mathrm {\{1-1.5\}\,\,dB}$ </tex-math></inline-formula>, while reducing complexity by two orders of magnitude.

Effect of Signal-Dependent Shot Noise on Visible Light Positioning

This paper investigates the error bounds for position estimation in downlink visible light communication (VLC) systems. More precisely, we use Cramér-Rao lower bounds (CRLBs) to study the effect of signal-dependent shot noise (SDSN) on the error performance in visible light positioning. We consider synchronous, quasi-synchronous, and asynchronous visible light positioning (VLP) systems. The system performance is assessed based on the position estimation, and uses information from parameters relating to the time of arrival (TOA), time difference of arrival (TDOA), and received signal strength (RSS). The results demonstrate that SDSN has a negative impact on the error bounds in all considered scenarios. Moreover, the level of degradation observed is not uniform among all cases at higher SDSN levels.

Low-Complexity CFO Estimator for Spectrally Efficient Frequency Division Multiplexing System

In this paper, pilot-based carrier frequency offset (CFO) estimation for spectrally efficient frequency division multiplexing (SEFDM) system is focused in the downlink. Firstly, the designed objective function for CFO estimation is proved to fit parabolic curve approximately. Then, a low-complexity CFO estimator is proposed by computing only four trials. Moreover, the Cramer Rao bound (CRB) of CFO estimation for SEFDM is derived. Considering the non-orthogonality of SEFDM, the novel pilot sequence is proposed for SEFDM via minimizing the derived CRB. Under the effects of timing offset and multi-path, simulation results demonstrate that the proposed pilot sequence for SEFDM outperforms traditional sequences.

Target Localization in Asynchronous Distributed MIMO Radar Systems With a Cooperative Target

In general, target localization in distributed multiple-input multiple-output (MIMO) radar systems requires synchronization among the sensors, which may not be feasible when the sensors are widely separated. In this paper, the target localization problem is addressed under asynchronous distributed MIMO radar systems in the presence of clock and frequency offsets with the aid of a cooperative target, whose position and velocity are known with measurement errors. Specifically, the influence of the cooperative target position and velocity errors on the unknown target localization accuracy is analyzed through the Cramer-Rao lower bound (CRLB) derivation firstly. Then two methods, namely, the offset estimation based localization method and the offset elimination based localization method, are proposed to estimate the unknown target location. The former estimates the clock and frequency offsets firstly and then locates the unknown target based on the offset estimates, and the latter achieves the unknown target location estimate based on the differential measurements of the two targets. Through the theoretical derivation and numerical simulations, the two proposed methods are demonstrated to be approximately unbiased and can attain the CRLB under mild noise. Moreover, the simulation results show that the proposed methods enjoy higher localization accuracy than the state-of-the-art algorithms.

Energy Efficiency of Angle of Arrival Estimation in Massive MIMO Systems

In the next generation of wireless systems, massive Multiple Input Multiple Output (MIMO) offers high angular resolution for localization. By virtue of large number of antennas, Angle of Arrival (AoA) of User Terminals (UTs) can be estimated accurately. Measurements of Dense Multipath Channel (DMC) indicate multipath signals contribute up to 95% of total link power. This fact urges the necessity of studying the contribution of multipath signals accompanying dominant path in AoA estimation. We obtain a deterministic form for Cramer-Rao Lower Bound ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$CRLB$ </tex-math></inline-formula> ) in multi-user scenario when contribution of multipath signals is considered. We do this when the multipath signals are independent and identically distributed (i.i.d) with arbitrary distribution. Then, we redefine a localization efficiency function for multi-user scenario and optimize it with respect to (w.r.t) the number of antennas. When only a subset of available antennas is used, we prove that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$CRLB$ </tex-math></inline-formula> can be minimized w.r.t which set is used. An antenna selection strategy that minimizes <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$CRLB$ </tex-math></inline-formula> is proposed. As a benchmark, we apply the proposed antenna selection to MUltiple SIgnal Classification (MUSIC) algorithm and study its efficiency. Numerical results validate the accuracy of our analysis and show significant improvement in efficiency when the proposed antenna selection is employed.