Threshold region performance of maximum likelihood direction of arrival estimators

This paper presents a performance analysis of the maximum likelihood (ML) estimator for finding the directions of arrival (DOAs) with a sensor array. The asymptotic properties of this estimator are well known. In this paper, the performance under conditions of low signal-to-noise ratio (SNR) and a small number of array snapshots is investigated. It is well known that the ML estimator exhibits a threshold effect, i.e., a rapid deterioration of estimation accuracy below a certain SNR or number of snapshots. This effect is caused by outliers and is not captured by standard techniques such as the Crame/spl acute/r-Rao bound and asymptotic analysis. In this paper, approximations to the mean square estimation error and probability of outlier are derived that can be used to predict the threshold region performance of the ML estimator with high accuracy. Both the deterministic ML and stochastic ML estimators are treated for the single-source and multisource estimation problems. These approximations alleviate the need for time-consuming computer simulations when evaluating the threshold region performance. For the special case of a single stochastic source signal and a single snapshot, it is shown that the ML estimator is not statistically efficient as SNR/spl rarr//spl infin/ due to the effect of outliers.     

Estimating particles velocity from laser measurements: maximumlikelihood and Cramer-Rao bounds

We address the problem of estimating particles velocity in the vicinity of an aircraft by means of a laser Doppler system. When a particle passes through the region of interference fringes generated by two coherent laser beams, the signal backscattered is of the form Aexp{-2α²·f_d²t² }cos{2πf_dt}, where the Doppler frequency f_d is related to the aircraft speed. This paper is concerned with the most precise estimation of the parameters A and f_d in the model considered. Cramer-Rao bounds (CRBs) on the accuracy of estimates of A and f_d are derived, and closed-formed expressions are given. Approximated formulas provide quantitative insights into the influence of α and f_d. Additionally, a maximum likelihood estimator (MLE) is presented. Numerical examples illustrate the performance of the MLE and compare it with the CRB. The influence of the SNR, the sample size, the optical parameter α, and the frequency f_d on the estimation performance is emphasized. Finally, an application to real data is presented     

SNR estimation in time-varying fading channels

Signal-to-noise ratio (SNR) estimation is considered for phase-shift keying communication systems in time-varying fading channels. Both data-aided (DA) estimation and nondata-aided (NDA) estimation are addressed. The time-varying fading channel is modeled as a polynomial-in-time. Inherent estimation accuracy limitations are examined via the Cramer-Rao lower bound, where it is shown that the effect of the channel's time variation on SNR estimation is negligible. A novel maximum-likelihood (ML) SNR estimator is derived for the time-varying channel model. In DA scenarios, where the estimator has a simple closed-form solution, the exact performance is evaluated both with correct and incorrect (i.e., mismatched) polynomial order. In NDA estimation, the unknown data symbols are modeled as random, and the marginal likelihood is used. The expectation-maximization algorithm is proposed to iteratively maximize this likelihood function. Simulation results show that the resulting estimator offers statistical efficiency over a wider range of scenarios than previously published methods.     

Estimation of chirp radar signals in compound-Gaussian clutter: acyclostationary approach

Signal detection of known (within a complex scaling) rank one waveforms in non-Gaussian distributed clutter has received considerable attention. We expand on published solutions to consider the case of rank one waveforms that have some unknown parameters, i.e., signal amplitude, initial phase, Doppler shift, and Doppler rate of change. The contribution of this paper is the derivation and performance analysis of two joint estimators of Doppler shift and Doppler rate-the chirp embedded in correlated compound-Gaussian clutter. One solution is based on the maximum likelihood (ML) principle and the other one on target signal second-order cyclostationarity. The hybrid Cramer-Rao lower bounds (HCRLBs) and a large sample closed-form expression for the mean square estimation error (only for the Doppler shift) are also derived. Numerical examples are provided to show the behavior of the proposed estimator under different non-Gaussian clutter scenarios     

Maximum likelihood estimation for angles of arrival of coherentsignals using a coherency profile

This article addresses an ML estimation problem by introducing a coherency profile, which means the sizes of the coherent signal groups. We present the Cramer-Rao (CR) bounds for the estimation. An algorithm for implementing it is proposed based on the alternating maximization. Simulation results show that the performance improvement of the estimator over the conventional ML is significant, even with a small number of samples in low SNR     

Soft Decoding Assisted SNR Estimation Under Block Fading Channels for Orthogonal Modulations

The performance of the coded orthogonal modulation (OM) system under slow fading channels heavily depends on the estimation of the signal-to-noise ratio (SNR), including the fading amplitude and the noise spectral density. However, a relatively long packet of pilot symbols is often required to guarantee the accuracy of the SNR estimation, which makes it impractical in some situations. To address this problem, this paper proposes an iterative SNR estimation algorithm using the soft decoding information based on the expectation-maximization algorithm. In the proposed method, a joint iterative loop between the SNR estimator and decoder is performed, where the extrinsic information generated by the soft decoder is employed to enhance the estimation accuracy and the SNR estimated by the estimator is used to generate the soft information to the decoder. Also, no pilot symbols are needed to estimate the SNR in the proposed estimator. The Cramer–Rao lower bound (CRLB) of fully data-aided (FDA) estimation is derived to works as the final benchmark. The performance of the proposed algorithm is evaluated in terms of the normalized mean square errors (NMSEs) and the bit error rates (BERs) under block fading channels. Simulation results indicate that the NMSE of the proposed estimator reaches the CRLB of the FDA estimator and outperforms that of the approximate ML (ML-A) estimator proposed by Hassan et al. by 4.1 dB. The BER performance of coded OM system with the proposed estimation algorithm is close to the ideal case where the channel fading and the noise spectral density are known at the receiver.     

激光多普勒测速参数估计的Cramer-Rao下限

针对固体运动目标高速度、高加速度的特点,研究了高斯白噪声背景下激光多普勒测速同波信号的参数估计问题.通过计算参数矢量的费希尔(Fisher)信息矩阵,分析了实同波信号的多普勒频率和频率变化率估计方差的克拉末-雷奥(Cramer-Rao)下限(CRLB),推导了采样点数较大时同波信号参数方差估计的CRLB计算公式,讨论了各参数的最大似然估计(MLE).指出同波信号参数估计方差的CRLB与采样点数、信噪比及初相有关,采样点数较大时,实信号参数估计方差的CRLB为对应复信号的2倍.在不同的采样点数和信噪比下仿真表明,提高回波信号的信噪比和增加采样点数可以减小各参数估计方差的CRLB,结果与理论分析吻合.     

Performance of coherent DS-SS/QPSK for mobile communications infast-fading multipath and high-frequency offset

We derive a closed-form bit error rate (BER) solution for equal- and nonequal-strength L-path channels considering imperfect channel estimation. The channel model assumes independent paths with Rayleigh fading statistics in a single-cell downlink environment. By using a simple maximum likelihood (ML) estimator, the effects of the channel estimation error due to Doppler shift, residual carrier frequency offset, interference, and additive white Gaussian noise are analyzed. In addition, we present the tradeoff between the noise compression capability and phase tracking capability of the ML estimator with observation length as a parameter. The results indicate that even with channel estimation, the high carrier frequency offset makes an uncoded BER unacceptably high. Also, we present two kinds of modulation techniques such as EC-QPSK and NC-QPSK. Through analysis, we show the performance comparison between these modulation techniques. Finally, we verify the derived BER by using Monte Carlo computer simulation     

Improved Single‐Tone Frequency Estimation by Averaging and Weighted Linear Prediction

This paper addresses estimating the frequency of a cisoid in the presence of white Gaussian noise, which has numerous applications in communications, radar, sonar, and instrumentation and measurement. Due to the nonlinear nature of the frequency estimation problem, there is threshold effect, that is, large error estimates or outliers will occur at sufficiently low signal‐to‐noise ratio (SNR) conditions. Utilizing the ideas of averaging to increase SNR and weighted linear prediction, an optimal frequency estimator with smaller threshold SNR is developed. Computer simulations are included to compare its mean square error performance with that of the maximum likelihood (ML) estimator, improved weighted phase averager, generalized weighted linear predictor, and single weighted sample correlator as well as Cramér‐Rao lower bound. In particular, with smaller computational requirement, the proposed estimator can achieve the same threshold and estimation performance of the ML method.     

Frequency estimation of a single complex sinusoid using a generalized Kay estimator

This letter introduces a generalized version of Kay's estimator for the frequency of a single complex sinusoid in complex additive white Gaussian noise. The Kay estimator is a maximum-likelihood (ML) estimator at high signal-to-noise ratio (SNR) based on differential phase measurements with a delay of one symbol interval. In this letter, the corresponding ML estimator with an arbitrary delay in the differential phase measurements is derived. The proposed estimator reduces the variance at low SNR, compared with Kay's original estimator. For certain delay values, explicit expressions for the window function and the corresponding high SNR variance of the proposed generalized Kay (GK) estimator are presented. Furthermore, for some delay values, the window function is nearly uniform and the implementation complexity is reduced, compared with the original Kay estimator. For a delay value of two, we show that the variance at asymptotically high SNR approaches the Cramer-Rao bound as the sequence length tends to infinity. We also explore the effect of exchanging the order of summation and phase extraction for reduced-complexity reasons. The resulting generalized weighted linear predictor estimator and the GK estimator are compared with both autocorrelation-based and periodogram-based estimators in terms of computational complexity, estimation range, and performance at both low and high SNRs.     

Cramer-Rao Lower Bounds for Semi-Blind ML Channel Estimation in UWB Systems

This paper derives a unified representation of the Cramer-Rao lower bounds (CRLBs) for semi-blind maximum-likelihood (ML) channel estimation in time-hopping (TH) ultra-wideband systems with pulse amplitude and position modulation, pulse amplitude modulation and pulse position modulation (PPM). These bounds subsume the existing results for both pilot-based and blind ML channel estimation in TH-PPM as special cases. A numerical example is used to illustrate the effects of different data modulation formats and system parameters on the CRLBs derived     

Approximate maximum likelihood estimators for array processing inmultiplicative noise environments

We consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise. This scenario is encountered, for example, in communications, owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media. Since the exact maximum likelihood (ML) estimator is computationally intensive, two approximate solutions are proposed, originating from the analysis of the high and low signal to-noise ratio (SNR) cases, respectively. First, starting with the no additive noise case, a very simple approximate ML (AML₁) estimator is derived. The performance of the AML₁ estimator in the presence of additive noise is studied, and a theoretical expression for its asymptotic variance is derived. Its performance is shown to be close to the Cramer-Rao bound (CRB) for moderate to high SNR. Next, the low SNR case is considered, and the corresponding AML₂ solution is derived. It is shown that the approximate ML criterion can be concentrated with respect to both the multiplicative and additive noise powers, leaving out a two-dimensional (2-D) minimization problem instead of a four-dimensional (4-D) problem required by the exact ML. Numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis     

Joint maximum-likelihood parameter estimation for burst DSspread-spectrum transmission

We first derive the joint optimal maximum-likelihood (ML) estimator of the carrier phase, both Doppler shift and Doppler rate, and the spreading code delay for a short burst direct sequence/spread spectrum (DS/SS) transmission in the absence of the data modulation. The typical burst duration is three data bit periods (60 ms). The performance of the joint estimator is analytically derived separately for high and low carrier-to-noise ratio (CNR) values. A suboptimal ML estimator based on the method of averaged periodogram is proposed for the data modulation present case, allowing the joint estimation of the data bit values. Then the above parameters are assumed correctly estimated and a segmentation approach is adopted, deriving the optimal joint ML estimator for the bit synchronization epoch and data. Simulations show the joint estimators perform reliable down to a CNR of approximately 30 dB·Hz     

基于正交训练序列的MIMO系统联合最大似然时频同步和信道估计

该文推导了多输入多输出(MIMO)系统中的符号定时、频偏和信道参数的联合最大似然(ML)估计。针对联合ML估计没有闭合的表达式、数值计算复杂度高的问题,该文提出了一种基于重复结构的正交训练序列的简化估计算法。该估计算法形式简单、复杂度低,且仍为最大似然估计。最后仿真分析了最大似然参数估计的均方误差与接收信噪比和天线数目的关系,并与Cramer-Rao界作了比较,表明了该算法的有效性。     

Initial synchronization of DS-CDMA via bursty pilot signals

We consider initial timing acquisition in discrete-sequence code-division multiple-access (DS-CDMA) when propagation is affected by multipath and fading, and where the base-station broadcasts a synchronization pilot signal in the form of bursts of modulated chips transmitted periodically and separated by long silent intervals. Subject to certain simplifying assumptions, we derive the maximum-likelihood (ML) estimator by solving a constrained maximization problem. Our ML timing estimator has constant complexity per observation sample. The relation to other estimation methods is addressed, and performance comparisons are provided by simulation. The proposed estimator yields good performance independently of the multipath-intensity profile of the channel, provided that the delay spread is not larger than a given maximum spread. Moreover, our estimator is fairly robust to the mismatch in the fading Doppler spectrum and provides good performance for both fast and slow fading     

Pilot-based estimation of time-varying multipath channels forcoherent CDMA receivers

Reliable coherent wireless communication requires accurate estimation of the time-varying multipath channel. This paper addresses two issues in the context of direct-sequence code-division multiple access (CDMA) systems: (i) linear minimum-mean-squared-error (MMSE) channel estimation based on a pilot transmission and (ii) impact of channel estimation errors on coherent receiver performance. A simple characterization of the MMSE estimator in terms of a bank of filters is derived. A key channel characteristic controlling system performance is the normalized coherence time, which is approximately the number of symbols over which the channel remains strongly correlated. It is shown that the estimator performance is characterized by an effective signal-to-noise ratio (SNR)-the product of the pilot SNR and the normalized coherence time. A simple uniform averaging estimator is also proposed that is easy to implement and delivers near-optimal performance if properly designed. The receivers analyzed in this paper are based on a time-frequency RAKE structure that exploits joint multipath-Doppler diversity. It is shown that the overall receiver performance is controlled by two competing effects: shorter coherence times lead to degraded channel estimation but improved inherent receiver performance due to Doppler diversity, with opposite effects for longer coherence times. Our results demonstrate that exploiting Doppler diversity can significantly mitigate the error probability floors that plague conventional CDMA receivers under fast fading due to errors in channel estimation     

广义Pareto分布海杂波模型参数的组合双分位点估计方法

广义Pareto分布的复合高斯模型可以很好地描述高分辨低擦地角对海探测场景中海杂波的重拖尾特性,实现该杂波模型下双参数的有效估计对雷达检测性能具有重要意义。对此,该文提出一种双参数的组合双分位点(CBiP)估计方法。该估计方法基于低阶多项式方程的显式求根表达式,充分组合利用回波中的样本信息,旨在实现高精度的双参数估计过程。此外,考虑到实际雷达工作中存在岛礁、渔船等造成的功率异常大的野点样本时,不同于传统的矩估计、最大似然(ML)估计等方法,组合双分位点估计方法仍可保持估计性能的鲁棒性。仿真及实测数据实验表明,在纯杂波环境中,组合双分位点估计方法可以实现与最大似然估计方法近似的估计精度,若存在异常样本,组合双分位点估计方法的估计性能优于上述几种传统估计方法。     

Cramer-Rao bounds in the parametric estimation of fadingradiotransmission channels

The context of this paper is parameter estimation for linearly modulated digital data signals observed on a frequency-flat time-selective fading channel affected by additive white Gaussian noise. The aim is the derivation of Cramer-Rao lower bounds for the joint estimation of all those channel parameters that impact signal detection, namely, carrier phase, carrier frequency offset (Doppler shift), frequency rate of change (Doppler rate), signal amplitude, fading power, and Gaussian noise power. Time-selective frequency-flat fading is modeled as a low-pass autoregressive multiplicative distortion process. In particular, the important case of “slow” fading, with the multiplicative process remaining constant over the whole data burst, is specifically discussed. Asymptotic expressions of the bounds, valid for a large observed sample or for high signal-to-noise ratio (SNR), are also derived in closed form. A few charts with numerical results are finally reported to highlight the dependence of the bounds on channel status (SNR, fading bandwidth, etc.)     

利用频谱特性的速度自适应信道估计方法

本文利用信道在移动环境下频谱带限和对称的特征,提出一种新的对移动速度自适应的信道估计方法。该方法通过寻找频谱自相关的峰值来确定频域滤波器的长度。相对传统速度自适应方法,该方法不需要估计多普勒频率或移动速度,降低了系统实现的复杂度;也不需要信道的长时统计量或信噪比信息,实时性较高。信号在频域的处理必然会有频谱泄漏问题,传统方法滤除噪声的同时也去掉了部分信号,对此本文同时还提出一种恢复泄漏的频域信号的方法,使估计性能有所提高。仿真结果显示,从步行到高速移动的环境下,本文提出的信道估计方法的均方误差(MSE)都保持在较低的水平,具有良好的对速度自适应的估计性能。     

Maximum Likelihood Carrier Phase Recovery for Linear Suppressed-Carrier Digital Data Modulations

The problem of ML estimation of the Phase of a general data-modulated carrier is considered. The shortcomings of current iterative approaches to the problem are pointed out, and the correct conceptual approach is proposed. The true ML estimator is then obtained and found to be nonimplementable. However, by specializing to limits of high and low SNR, the general ML estimator is shown to reduce to implementable DA and NDA ML estimators, respectively. The DA receiver's performance in terms of phase tracking and symbol error probability can be analyzed, and even the effects of past decision errors on current system performance can be assessed. For circular signal constellations, the DA receiver has a simple and totally linear structure which is easy to implement. The NDA ML estimator is shown to be equivalent to the common carrier loops. Our emphasis here on explicit computation of the ML phase estimate from the past received signal leads to detection strategies which do not require a carrier loop and a VCO for coherent detection.