Cramer-Rao bound for Gaussian random processes and applications toradar processing of atmospheric signals

Calculations of the exact Cramer-Rao bound (CRB) for unbiased estimates of the mean frequency, signal power, and spectral width of Doppler radar/lidar signals (a Gaussian random process) are presented. Approximate CRBs are derived using the discrete Fourier transform (DFT). These approximate results are equal to the exact CRB when the DFT coefficients are mutually uncorrelated. Previous high SNR limits for CRBs are shown to be inaccurate because the discrete summations cannot be approximated with integration. The performance of an approximate maximum likelihood estimator for mean frequency approaches the exact CRB for moderate SNR and moderate spectral width     

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     

Maximum likelihood estimation and Cramer-Rao bounds for directionof arrival parameters of a large sensor array

A maximum likelihood (ML) method is developed for estimation of direction of arrival (DOA) and associated parameters of narrowband signals based on the Taylor's series expansion of the inverse of the data covariance matrix R for large M, M specifying number of sensors in the array. The stochastic ML criterion function can thus be simplified resulting in a computationally efficient algorithm for DOA estimation. The more important result is the derivation of asymptotic (large M) expressions for the Cramer-Rao lower bound (CRB) on the covariance matrix of all unknown DOA angles for the general D source case. The derived bound is expressed explicitly as a function of snapshots, signal-to-noise ratio (SNR), sensors, separation, and correlation between signal sources. Using the condition of positive definiteness of the Fisher information matrix a resolution criterion is proposed which gives a tight lower limit on the minimum resolvable angle     

Fast algorithms for single frequency estimation

In this paper, some new estimators of the frequency of a single complex sinusoid are presented. The rotate-add-decimate (RAD) method of Crozier is first refined to more closely approach the Cramer-Rao Bound (CRB). An additional modification yields an unbiased estimator (ERAD) that essentially achieves the CRB above a signal-to-noise ratio (SNR) threshold comparable to that of RAD. In addition, this estimator is proven to achieve the CRB for high SNR. The ERAD method requires approximately 2N complex multiply-adds and log/sub 2/N arctangents. A modified ERAD (MERAD) is proposed that matches the SNR threshold and computational complexity of the RAD method (approximately 3N complex multiply-adds and log/sub 2/N arctangents) but achieves the CRB for high SNR.     

一种基于窄带信号多普勒频率测量的运动目标直接定位方法

相比于常规的两步定位方法,目标直接定位方法具有更高的定位精度,但现有的直接定位方法主要是针对静止目标所提出的,并具有较大的运算量.本文提出了一种针对匀速运动目标的多普勒频率直接定位方法.文中首先基于最大似然准则推导了直接估计目标初始位置和运动速度的优化模型,针对该优化模型是以矩阵特征值的形式给出而难以数值求解的问题,提出了一种基于矩阵特征值扰动定理的Newton型迭代算法,该算法可以避免多维参数网格搜索所导致的庞大运算量.此外,文中还推导了关于目标初始位置、运动速度以及运动航迹估计的克拉美罗界的闭式表达式.数值实验表明新方法的目标位置估计方差可以达到克拉美罗界,并且具有较少的运算量.     

A New Barankin Bound Approximation for the Prediction of the Threshold Region Performance of Maximum Likelihood Estimators

It is well known that the ML estimator exhibits a threshold effect, i.e., a rapid deterioration of estimation accuracy below a certain signal-to-noise ratio (SNR) or number of snapshots. This effect is caused by outliers and is not captured by standard tools such as the CramÉr-Rao bound (CRB). The search of the SNR threshold value (where the CRB becomes unreliable for prediction of maximum likelihood estimator variance) can be achieved with the help of the Barankin bound (BB), as proposed by many authors. The major drawback of the BB, in comparison with the CRB, is the absence of a general analytical formula, which compels one to resort to a discrete form, usually the Mcaulay–Seidman bound (MSB), requesting the search of an optimum over a set of test points. In this paper, we propose a new practical BB discrete form that provides, for a given set of test points, an improved SNR threshold prediction in comparison with existing approximations (MSB, Abel bound, Mcaulay–Hofstetter bound) at the expense of the computational complexity increased by a factor $leq(P+1)^{3}$ , where $P$ is the number of unknown parameters. We have derived its expression for the general Gaussian observation model to be used in place of existing approximations.      

Notes on the Tightness of the Hybrid CramÉr–Rao Lower Bound

In this paper, we study the properties of the hybrid CramÉr-Rao bound (HCRB). We first address the problem of estimating unknown deterministic parameters in the presence of nuisance random parameters. We specify a necessary and sufficient condition under which the HCRB of the nonrandom parameters is equal to the CramÉr-Rao bound (CRB). In this case, the HCRB is asymptotically tight [in high signal-to-noise ratio (SNR) or in large sample scenarios], and, therefore, useful. This condition can be evaluated even when the CRB cannot be evaluated analytically. If this condition is not satisfied, we show that the HCRB on the nonrandom parameters is always looser than the CRB. We then address the problem in which the random parameters are not nuisance. In this case, both random and nonrandom parameters need to be estimated. We provide a necessary and sufficient condition for the HCRB to be tight. Furthermore, we show that if the HCRB is tight, it is obtained by the maximum likelihood/maximum a posteriori probability (ML/MAP) estimator, which is shown to be an unbiased estimator which estimates both random and nonrandom parameters simultaneously optimally (in the minimum mean-square-error sense).      

Cisoid parameter estimation in the colored noise case: asymptoticCramer-Rao bound, maximum likelihood, and nonlinear least-squares

The problem of estimating the parameters of complex-valued sinusoidal signals (cisoids, for short) from data corrupted by colored noise occurs in many signal processing applications. We present a simple formula for the asymptotic (large-sample) Cramer-Rao bound (CRB) matrix associated with this problem. The maximum likelihood method (MLM), which estimates both the signal and noise parameters, attains the performance corresponding to the asymptotic CRB, as the sample length increases. More interestingly, we show that a computationally much simpler nonlinear least-squares method (NLSM), which estimates the signal parameters only, achieves the same performance in large samples     

The Large Sample Performance of a Maximum Likelihood Method for OFDM Carrier Frequency Offset Estimation

This paper is concerned with the performance analysis of a blind method for carrier frequency offset (CFO) estimation in OFDM systems. The method generates a maximum likelihood CFO estimate when message-carrying symbols and channel are assumed nonrandom. It has been observed in simulation that, when a large number of OFDM blocks are in use under noisy scenarios, the performance of the method does not achieve the corresponding Cramer-Rao lower bound (CRB) and is lower bounded by another (larger) CRB when symbols are assumed random. This phenomenon is caused by the inconsistency of nonrandom symbol estimates and is further verified by using first-order perturbation analysis and comparison with the two CRBs.     

Stochastic Crame/spl acute/r-Rao bound for noncircular signals with application to DOA estimation

After providing an extension of the Slepian-Bangs formula for general noncircular complex Gaussian distributions, this paper focuses on the stochastic Crame/spl acute/r-Rao bound (CRB) on direction-of-arrival (DOA) estimation accuracy for noncircular sources. We derive an explicit expression of the CRB for DOA parameters alone in the case of noncircular complex Gaussian sources by two different methods. One of them consists of computing the asymptotic covariance matrix of the maximum likelihood (ML) estimator, and the other is obtained directly from our extended Slepian-Bangs formula. Some properties of this CRB are proved, and finally, it is numerically compared with the CRBs under circular complex Gaussian and complex discrete distributions of sources.     

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     

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     

Design of the Periodic Training Sequence for Joint Channel and Frequency Estimation in MIMO Frequency-Selective Channels

This paper considers the design of a periodic training sequence (TS) for joint channel and frequency estimation in multiple-input, multiple-output (MIMO) frequency-selective channels. The design criterion for the periodic TS is to jointly minimize the mean-squared error (MSE) of the maximum likelihood channel estimation and the asymptotic Cramer-Rao bound (CRB) for the frequency estimation. This paper shows that all the TSs that minimize the MSE of the channel estimation have the same asymptotic CRB for the frequency estimation. Furthermore, they also minimize the asymptotic CRB as long as the channel is i.i.d. Rayleigh fading. The design of low-complexity frequency estimators based on the proposed periodic TS is also investigated. Finally, the performance of the proposed periodic TS is evaluated by simulation results.     

Simultaneous Blind Separation of Instantaneous Mixtures With Arbitrary Rank

This paper presents a gradient-based method for simultaneous blind separation of arbitrarily linearly mixed source signals. We consider the regular case (i.e., the mixing matrix has full column rank) as well as the ill-conditioned case (i.e., the mixing matrix does not have full column rank). We provide one necessary and sufficient condition for the identifiability of simultaneous blind separation. According to our identifiability condition and the existing general identifiability condition, all source signals are separated into two categories: separable single sources and inseparable mixtures of several single sources. A sufficient condition is also derived for the existence of optimal partition of the mixing matrix which leads to a unique maximum set of separations. One sufficient condition is proved to show that each maximum partition of the mixing matrix corresponds to a unique class of separated signals and as a result we can determine the number of maximum partitions from the classes of outputs under different separation matrices. For sub-Gaussian or super-Gaussian source signals, a cost function based on fourth-order cumulants is introduced to simultaneously separate all separable single sources and all inseparable mixtures. By minimizing the cost function, a gradient-based method is developed. Finally, simulation results show the effectiveness of the present method.     

Timing Synchronization for Cooperative Communications with Detect and Forward Relaying

A non-data-aided near maximum likelihood (NDA-NML) symbol timing estimator is presented, which is applied to a cooperative communication system with a source, relay and destination. A Cramer rao bound (CRB) for the estimator for asymptotically low signal-to-noise (SNR) ratio case is derived. The timing complexity of the NDA-NML estimator is derived and compared with the correlation based data-aided maximum likelihood (DA-ML) estimator. It is demonstrated that the complexity of the NDA-NML estimator is much less than that of correlation based DA-ML estimator. The bit-error-rate (BER) performance of this system operating in a detect-and-forward (DAF) mode is studied where the channel state information (CSI) is available at the receiver and the symbol timings are estimated independently for each channel. SNR combining (SNRC) and equal ratio combining (ERC) methods are considered. It is found that timing estimation error has a significant effect on BER performance. It is also found that for large timing error the benefit of cooperative diversity could vanish. It is demonstrated that significant gains can be made with both combining methods with cooperation and timing estimation, where the gains are the same for both estimators.     

一种基于方差融合的双均匀线阵阵列测向算法

本文提出了一种基于双均匀线阵的快速高精度测向算法。针对双均匀线阵不同轴上的接收噪声互不相关的特点,可以得到理论上不含噪声的互相关矩阵。并且由于在两个轴上的阵列流型均具有移不变特性,因此本文所提的快速算法首先利用x轴上最大不重叠的两个子阵列对y轴上最大不重叠的子阵列做互相关,利用y轴上最大不重叠的两个子阵列对x轴上最大不重叠的子阵列做互相关,接着对上述两个相关矩阵采用旋转不变子空间算法,分别计算出目标的角度,再利用方差融合的方法,将两次求得的角度信息进行融合,最后得到较高精度的角度信息,理论分析表明该算法是无偏的,并且在较高信噪比下趋近于克拉莫罗界。最后利用蒙特卡洛仿真验证了本算法的有效性。      

基于信号重构和最小二乘的线性调频信号估计

通过多延时离散多项相位变换重构出具有较高信噪比的信号,随后解出瞬时相位曲线,利用最小二乘法估计出线性调频信号的一阶、二阶相位参数.在低信噪比时,估计量均方误差仍达到CRB.另外,该方法信噪比门限低且可调,运算量小,易于实现.仿真验证了其有效性.     

Maximum likelihood array processing in spatially correlated noisefields using parameterized signals

This paper deals with the problem of estimating signal parameters using an array of sensors. This problem is of interest in a variety of applications, such as radar and sonar source localization. A vast number of estimation techniques have been proposed in the literature during the past two decades. Most of these can deliver consistent estimates only if the covariance matrix of the background noise is known. In many applications, the aforementioned assumption is unrealistic. Recently, a number of contributions have addressed the problem of signal parameter estimation in unknown noise environments based on various assumptions on the noise. Herein, a different approach is taken. We assume instead that the signals are partially known. The received signals are modeled as linear combinations of certain known basis functions. The exact maximum likelihood (ML) estimator for the problem at hand is derived, as well as computationally more attractive approximation. The Cramer-Rao lower bound (CRB) on the estimation error variance is also derived and found to coincide with the CRB, assuming an arbitrary deterministic model and known noise covariance     

用于正弦波频率估计的修正Kay算法

Kay算法能够估计出采样点较少的正弦波频率,但低信噪比下估计性能不佳.针对此问题,提出了修正Kay算法.首先基于最大似然估计准则,推导了观测信号模值与相位的条件概率密度函数,进而重建了Kay算法的相位差噪声矢量协方差矩阵与权值矩阵.实验结果表明,修正算法能够有效估计正弦波信号频率,与Kay算法相比,抗噪性更强.     

MIMO-OFDM系统中导频最优设计准则及性能分析

信道的准确估计是提高MIMO-OFDM系统性能的关键,利用输入序列矩阵特征值,提出了MIMO-OFDM系统中基于最大似然估计的最优导频设计准则,给出了接收端信噪比损失的闭解.此准则可用于指导不同信道条件下的导频设计.