Analysis of subspace fitting and ML techniques for parameterestimation from sensor array data

It is shown that the multidimensional signal subspace method, termed weighted subspace fitting (WSF), is asymptotically efficient. This results in a novel, compact matrix expression for the Cramer-Rao bound (CRB) on the estimation error variance. The asymptotic analysis of the maximum likelihood (ML) and WSF methods is extended to deterministic emitter signals. The asymptotic properties of the estimates for this case are shown to be identical to the Gaussian emitter signal case, i.e. independent of the actual signal waveforms. Conclusions concerning the modeling aspect of the sensor array problem are drawn     

Localization of multiple sources with moving arrays

We consider the problem of localizing multiple narrowband stationary signals using an arbitrary time-varying array such as an array mounted on a moving platform. We assume a Gaussian stochastic model for the received signals and employ the generalized least squares (GLS) estimator to get an asymptotically efficient estimation of the model parameters. In case the signals are a priori known to be uncorrelated, the estimator allows the exploitation of this prior knowledge to its benefit. For the important case of a translational motion of a rigid array, a computationally efficient spatial-smoothing method is presented. Simulation results confirming the theoretical results are included     

Direction of arrival estimation using parametric signal models

We consider the problem of estimating the directions-of-arrival (DOAs) of narrowband sources with known center frequency. The paper evaluates the potential improvement in estimation accuracy by using spatial-temporal processing for signals obeying a deterministic parametric model. One would expect that prior information about the temporal structure of the signals will yield some gain in performance. By deriving the Cramer-Rao bound (CRB) on the DOA estimates, we quantify this gain and identify the cases for which the gain is significant. We show that for the single-source case, spatial-temporal processing does not yield any gain in performance relative to conventional spatial processing. For multiple noncoherent signals, incorporating temporal processing can achieve the single-source performance, yielding a significant gain for the case of multiple sources with small spatial separation relative to the beamwidth of the array. However, spatial-temporal processing cannot yield any gain in performance for multiple coherent signals     

Source localization using vector sensor array in a multipath environment

Coherent signals from distinct directions is a natural characterization of the multipath propagation effect. This paper addresses the problem of coherent/fully correlated source localization using vector sensor arrays. The maximum likelihood (ML) and minimum-variance distortionless response (MVDR) estimators for source direction-of-arrival (DOA) and signal polarization parameters are derived. These estimators require no search over the polarization parameters. In addition, a novel method for "decorrelating" the incident signals is presented. This method is based on the polarization smoothing algorithm (PSA) and enables the use of eigenstructure-based techniques, which assume uncorrelated or partially correlated signals. The method is implemented as a preprocessing stage before applying eigenstructure-based techniques, such as MUSIC. Unlike other existing preprocessing techniques, such as spatial smoothing and forward-backward (FB) averaging, this method is not limited to any specific array geometry. The performance of the proposed PSA preprocessing combined with MUSIC is evaluated and compared to the Crame/spl acute/r-Rao Bound (CRB) and the ML and MVDR estimators. Simulation results show that the MVDR and PSA-MUSIC asymptotically achieve the CRB for a scenario with two coherent sources with and without an uncorrelated interference source. A sensitivity study of PSA-MUSIC to source polarization was also conducted via simulations.     

Oblique projection for source estimation in a competitive environment: Algorithm and statistical analysis

In this paper, we study the problem where the aim is to estimate the source (complex amplitude) parameter of a single signal contaminated by a structured interference (constituted by the other signals) and by a background Gaussian noise. To solve this problem, we propose an estimator based on a partially estimated oblique projection. We derive closed-form expressions of the variance of this estimator and of the Cramér–Rao bound (CRB) associated with the considered model. In particular, we show that the proposed estimator is (i) asymptotically (for large number of sensors) efficient in the sense that its variance meets the CRB for a single signal in noise and (ii) for a small of moderate number of sensors, the variance remains close to the CRB without structured interference for well separated bearings.     

Computationally efficient angle estimation for signals with knownwaveforms

This paper presents a large sample decoupled maximum likelihood (DEML) angle estimator for uncorrelated narrowband plane waves with known waveforms and unknown amplitudes arriving at a sensor array in the presence of unknown and arbitrary spatially colored noise. The DEML estimator decouples the multidimensional problem of the exact ML estimator to a set of 1-D problems and, hence, is computationally efficient. We shall derive the asymptotic statistical performance of the DEML estimator and compare the performance with its Cramer-Rao bound (CRB), i.e., the best possible performance for the class of asymptotically unbiased estimators. We will show that the DEML estimator is asymptotically statistically efficient for uncorrelated signals with known waveforms. We will also show that for moderately correlated signals with known waveforms, the DEML estimator is no longer a large sample maximum likelihood (ML) estimator, but the DEML estimator may still be used for angle estimation, and the performance degradation relative to the CRB is small. We shall show that the DEML estimator can also be used to estimate the arrival angles of desired signals with known waveforms in the presence of interfering or jamming signals by modeling the interfering or jamming signals as random processes with an unknown spatial covariance matrix. Finally, several numerical examples showing the performance of the DEML estimator are presented in this paper     

虚拟子阵平滑算法

主动阵列发射正交信号可以极大地扩展阵列的虚拟孔径,阵列的自由度成几何增长。该文利用主动阵列发射同频带时域正交PCM信号构造的虚拟子阵,提出了一种虚拟子阵平滑(VSS)算法,可解决不同目标回波信号相干的问题。该算法避免了接收阵列物理孔径的损失,当接收阵列具有个阵元时,最大可分辨的相干数目为。仿真实验表明VSS算法的有效性,而且该算法只需要较少的阵元发射信号。     

The achievable accuracy in estimating the instantaneous phase andfrequency of a constant amplitude signal

The approach is based on modeling the signal phase by a polynomial function of time on a finite interval. The phase polynomial is expressed as a linear combination of the Legendre basis polynomials. First, the Cramer-Rao bound (CRB) of the instantaneous phase and frequency of constant-amplitude polynomial-phase signals is derived. Then some properties of the CRBs are used to estimate the order of magnitude of the bounds. The analysis is extended to signals whose phase and frequency are continuous but not polynomial. The CRB can be achieved asymptotically if the estimation of the phase coefficients is done by maximum likelihood. The maximum-likelihood estimates are used to show that the achievable accuracy in phase and frequency estimation is determined by the CRB of the polynomial coefficients and the deviation of true phase and frequency from the polynomial approximations     

Cramer-Rao bounds of DOA estimates for BPSK and QPSK Modulated signals

This paper focuses on the stochastic Cramer-Rao bound (CRB) of direction of arrival (DOA) estimates for binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive circular complex Gaussian noise. Explicit expressions of the CRB for the DOA parameter alone in the case of a single signal waveform are given. These CRBs are compared, on the one hand, with those obtained with different a priori knowledge and, on the other hand, with CRBs under the noncircular and circular complex Gaussian distribution and with different deterministic CRBs. It is shown in particular that the CRBs under the noncircular [respectively, circular] complex Gaussian distribution are tight upper bounds on the CRBs under the BPSK [respectively, QPSK] distribution at very low and very high signal-to-noise ratios (SNRs) only. Finally, these results and comparisons are extended to the case of two independent BPSK or QPSK distributed sources where an explicit expression of the CRB for the DOA parameters alone is given for large SNR.     

Direction-of-Arrival Estimation for Temporally Correlated Narrowband Signals

Signal direction-of-arrival (DOA) estimation using an array of sensors has been the subject of intensive research and development during the last two decades. Efforts have been directed to both, better solutions for the general data model and to develop more realistic models. So far, many authors have assumed the data to be independent and identically distributed (i.i.d.) samples of a multivariate statistical model. Although this assumption reduces the complexity of the model, it may not be true in certain situations where signals show temporal correlation. Some results are available on the temporally correlated signal model in the literature. The temporally correlated stochastic CramÉr–Rao bound (CRB) has been calculated and an instrumental variable-based method called IV-SSF is introduced. Also, it has been shown that temporally correlated CRB is lower bounded by the deterministic CRB. In this paper, we show that temporally correlated CRB is also upper bounded by the stochastic i.i.d. CRB. We investigate the effect of temporal correlation of the signals on the best achievable performance. We also show that the IV-SSF method is not efficient and based on an analysis of the CRB, propose a variation in the method which boosts its performance. Simulation results show the improved performance of the proposed method in terms of lower bias and error variance.      

Optimal Utilization of Signal-Free Samples for Array Processing in Unknown Colored Noise Fields

Many algorithms for direction-of-arrival (DOA) estimation require the noise covariance matrix to be known or to possess a known structure. In many cases, the noise covariance is, in fact, estimated from separate measurements. This paper addresses the combined effects of finite sample sizes, both in the estimated noise covariance matrix and in the data with signals present. It is assumed that a batch of signal-free samples is available in addition to the signal-containing samples. No assumption is made on the structure of the noise covariance. In this paper, the asymptotic covariance of the weighted subspace fitting (WSF) algorithm is derived for the case in which the data are whitened using an estimated noise covariance. The expression obtained suggests an optimal weighting that improves performance compared to the standard choice. In addition, a new method based on covariance matching is proposed. Both methods are asymptotically statistically efficient. The CramÉr–Rao lower bound (CRB) on the covariance of the estimate for the data model is also derived. Monte Carlo simulations show promising small sample performance for the two new methods and confirm the asymptotic results.     

Maximum likelihood array processing for stochastic coherent sources

Maximum likelihood (ML) estimation in array signal processing for the stochastic noncoherent signal case is well documented in the literature. We focus on the equally relevant case of stochastic coherent signals. Explicit large-sample realizations are derived for the ML estimates of the noise power and the (singular) signal covariance matrix. The asymptotic properties of the estimates are examined, and some numerical examples are provided. In addition, we show the surprising fact that the ML estimates of the signal parameters obtained by ignoring the information that the sources are coherent coincide in large samples with the ML estimates obtained by exploiting the coherent source information. Thus, the ML signal parameter estimator derived for the noncoherent case (or its large-sample realizations) asymptotically achieves the lowest possible estimation error variance (corresponding to the coherent Cramer-Rao bound)     

虚拟空间平滑算法

根据MASK、BPSK和AM等信号的实值特性,利用阵列接收数据及其共轭信息,构造了虚拟子阵,提出了虚拟空间平滑算法,较好地解决了信号高度相关问题.虚拟子阵具有M个阵元,避免了阵列孔径的损失,使该算法最大可分辨的相干信号数目为M-1.仿真实验证明,与前后空间平滑算法相比,该算法提高了空间分辨率,适应小样本,在一定程度上可以降低了计算量.     

Preprocessing for direction finding with minimal variancedegradation

Numerous authors have advocated the use of preprocessing in high-resolution direction of arrival (DOA) algorithms. The benefits cited include reduced computation, improved performance in spatially colored noise, and enhanced resolution. The authors identify the preprocessing matrices that provide minimum variance estimates of DOA for a number of models and algorithms. They examine the Cramer-Rao bound (CRB) for Gaussian signals, the CRB for deterministic signals, and the asymptotic variance of the MUSIC estimator for preprocessed data. They also study the effect of array manifold errors on the direction estimates. As expected, the optimal preprocessor requires knowledge of the source directions. However, they show that performance that is close to optimal can be obtained with only approximate knowledge of the source directions (with an error not exceeding the array beamwidth) if the design rules outlined in this paper are used     

Target Velocity Estimation and Antenna Placement for MIMO Radar With Widely Separated Antennas

This paper studies the velocity estimation performance for multiple-input multiple-output (MIMO) radar with widely spaced antennas. We derive the Cramer–Rao bound (CRB) for velocity estimation and study the optimized system/configuration design based on CRB. General results are presented for an extended target with reflectivity varying with look angle. Then detailed analysis is provided for a simplified case, assuming an isotropic scatterer. For given transmitted signals, optimal antenna placement is analyzed in the sense of minimizing the CRB of the velocity estimation error. We show that when all antennas are located at approximately the same distance from the target, symmetrical placement is optimal and the relative position of transmitters and receivers can be arbitrary under the orthogonal received signal assumption. In this case, it is also shown that for MIMO radar with optimal placement, velocity estimation accuracy can be improved by increasing either the signal time duration or the product of the number of transmit and receive antennas.      

空间步进频雷达多参数联合稀疏估计方法

在复杂环境下,由于少快拍、低信噪比和相干信号导致目标距离-角度二维超分辨估计性能下降,误差增大,本文提出了一种基于空间步进频雷达的距离-角度二维稀疏估计方法.该方法根据空间步进频雷达的回波特性得到距离和角度联合的等效接收信号模型;然后利用迭代的加权lq-范数的稀疏渐进最小化方差方法获得距离-角度二维功率谱估计.仿真结果...     

Estimation of amplitude and phase parameters of multicomponentsignals

This paper considers the problem of estimating signals consisting of one or more components of the form a(t)e/sup jφ(t/), where the amplitude and phase functions are represented by a linear parametric model. The Cramer-Rao bound (CRB) on the accuracy of estimating the phase and amplitude parameters is derived. By analyzing the CRB for the single-component case, if is shown that the estimation of the amplitude and the phase are decoupled. Numerical evaluation of the CRB provides further insight into the dependence of estimation accuracy on signal-to-noise ratio (SNR) and the frequency separation of the signal components. A maximum likelihood algorithm for estimating the phase and amplitude parameters is also presented. Its performance is illustrated by Monte-Carlo simulations, and its statistical efficiency is verified     

The Wald test and Crame/spl acute/r-Rao bound for misspecified models in electromagnetic source analysis

By using signal processing techniques, an estimate of activity in the brain from the electro- or magneto-encephalogram (EEG or MEG) can be obtained. For a proper analysis, a test is required to indicate whether the model for brain activity fits. A problem in using such tests is that often, not all assumptions are satisfied, like the assumption of the number of shells in an EEG. In such a case, a test on the number of sources (model order) might still be of interest. A detailed analysis is presented of the Wald test for these cases. One of the advantages of the Wald test is that it can be used when not all assumptions are satisfied. Two different, previously suggested, Wald tests in electromagnetic source analysis (EMSA) are examined: a test on source amplitudes and a test on the closeness of source pairs. The Wald test is analytically studied in terms of alternative hypotheses that are close to the hypothesis (local alternatives). It is shown that the Wald test is asymptotically unbiased, that it has the correct level and power, which makes it appropriate to use in EMSA. An accurate estimate of the Crame/spl acute/r-Rao bound (CRB) is required for the use of the Wald test when not all assumptions are satisfied. The sandwich CRB is used for this purpose. It is defined for nonseparable least squares with constraints required for the Wald test on amplitudes. Simulations with EEG show that when the sensor positions are incorrect, or the number of shells is incorrect, or the conductivity parameter is incorrect, then the CRB and Wald test are still good, with a moderate number of trials. Additionally, the CRB and Wald test appear robust against an incorrect assumption on the noise covariance. A combination of incorrect sensor positions and noise covariance affects the possibility of detecting a source with small amplitude.     

Cramer-Rao bounds for estimating range, velocity, and directionwith an active array

We derive Cramer-Rao bound (CRB) expressions for the range (time delay), velocity (Doppler shift), and direction of a point target using an active radar or sonar array. First, general CRB expressions are derived for a narrowband signal and array model and a space-time separable noise model that allows both spatial and temporal correlation. We discuss the relationship between the CRB and ambiguity function for this model. Then, we specialize our CRB results to the case of temporally white noise and the practically important signal shape of a linear frequency modulated (chirp) pulse sequence. We compute the CRB for a three-dimensional (3-D) array with isotropic sensors in spatially white noise and show that it is a function of the array geometry only through the “moments of inertia” of the array. The volume of the confidence region for the target's location is proposed as a measure of accuracy. For this measure, we show that the highest (and lowest) target location accuracy is achieved if the target lies along one of the principal axes of inertia of the array. Finally, we compare the location accuracies of several array geometries