Two-Dimensional Polynomial Phase Signals: Parameter Estimation and Bounds

This paper considers the problem of parametric modeling and estimation of nonhomogeneous two-dimensional (2-D) signals. In particular, we focus our study on the class of constant modulus polynomial-phase 2-D nonhomogeneous signals. We present two different phase models and develop computationally efficient estimation algorithms for the parameters of these models. Both algorithms are based on phase differencing operators. The basic properties of the operators are analyzed and used to develop the estimation algorithms. The Cramer-Rao lower bound on the accuracy of jointly estimating the model parameters is derived, for both models. To get further insight on the problem we also derive the asymptotic Cramer-Rao bounds. The performance of the algorithms in the presence of additive white Gaussian noise is illustrated by numerical examples, and compared with the corresponding exact and asymptotic Cramer-Rao bounds. The algorithms are shown to be robust in the presence of noise, and their performance close to the CRB, even at moderate signal to noise ratios.     

Cramer-Rao bounds and maximum likelihood estimation for randomamplitude phase-modulated signals

The problem of estimating the phase parameters of a phase-modulated signal in the presence of colored multiplicative noise (random amplitude modulation) and additive white noise (both Gaussian) is addressed. Closed-form expressions for the exact and large-sample Cramer-Rao Bounds (CRBs) are derived. It is shown that the CRB is significantly affected by the color of the modulating process when the signal-to-noise ratio (SNR) or the intrinsic SNR is small. Maximum likelihood type estimators that ignore the noise color and optimize a criterion with respect to only the phase parameters are proposed. These estimators are shown to be equivalent to the nonlinear least squares estimators, which consist of matching the squared observations with a constant amplitude phase-modulated signal when the mean of the multiplicative noise is forced to zero. Closed-form expressions are derived for the efficiency of these estimators and are verified via simulations     

Fast computation of the exact FIM for deterministic signals incolored noise

A fast algorithm for computing the exact finite-sample Fisher information matrix (FIM) for the parameters of a deterministic signal observed in Gaussian AR noise is derived. In the case of a harmonic signal with random phases, closed-form expressions for the finite-sample posterior Cramer-Rao bound (PCRB) are established. It is shown that the fast algorithm is also useful for computing the conditional CRB when the additive noise is a non-Gaussian AR process. It is seen that the asymptotic CRB may deviate significantly from the exact CRB even when the data length is moderate, whereas the PCRB, which is easy to compute, provides a better approximation. Theoretical results are illustrated via numerical evaluation of the different lower bounds     

Maximum likelihood estimation, analysis, and applications ofexponential polynomial signals

We model complex signals by approximating the phase and the logarithm of the time-varying amplitude of the signal as a finite order polynomial. We refer to a signal that has this form as an exponential polynomial signal (EPS). We derive an iterative maximum-likelihood (ML) estimation algorithm to estimate the unknown parameters of the EPS model. The initialization of the ML algorithm can be performed by using the result of a related paper. A statistical analysis of the ML algorithm is performed using a finite-order Taylor expansion of the mean squared error (MSE) of the estimate about the variance of the additive noise. This perturbation analysis gives a method of predicting the MSE of the estimate for any choice of the signal parameters. The MSE from the perturbation analysis is compared with the MSE from a Monte Carlo simulation and the Cramer-Rao Bound (CRB). The CRB for this model is also derived     

基于特征向量的多谱线自旋回波串信号参数估计

针对存在加性高斯白噪声多参数变量的多谱线自旋回波串(Spin Echo Train,SET)信号参数估计问题,提出基于特征向量的2-D参数估计方法.将SET信号构造成2-D数据矩阵,按照不同的方式构造Hankel块矩阵束,利用子空间转移不变结构解得特征向量,依据特征向量的结构规律获得衰减因子和频率,基于最小二乘方法进一步获得信号幅度估计.该方法具有自动配对的能力,在相对高信噪比以及频率可分辨的情况下能够实现参数的有效估计,且计算复杂度较低.仿真数据结果证明了算法的有效性.     

Cramer-Rao Bounds for parameter estimation of phase-coding signals

In this paper, Cramer-Rao Bound (CRB) is derived from phase-coding signal with additive white noise, where three important parameters are focused on: carrier frequency, chip width and amplitude. Simplified and close form expressions of CRB are obtained through complicated derivation, and then are applied to evaluate the performance of the cyclic estimator. The results are accurate enough and serve well as benchmark for evaluating the performance of parameter estimation method. Numerical simulations illustrate the accuracy and applicability of the derived CRB.     

Asymptotic properties of 2-D MUSIC estimator and comparison to 2-D MP estimator

The MUSIC estimator of two-dimensional frequencies (2-D MUSIC) is studied assuming a one-measurement data model with deterministic phases and additive complex white Gaussian noise. The large sample estimation covariance of the 2-D MUSIC is derived and compared to that of the 2-D matrix pencil (MP) estimator. The theoretical estimation variances for both the MP and MUSIC estimators are compared with the simulated MP and MUSIC estimation variances and the Cramer-Rao Bound (CRB). In the single 2-D sinusoid case, the most revealing form of the estimation covariance for both estimators are provided. The results shown in this paper are valid for a median range of SNR.     

The true Crame/spl acute/r-Rao lower bound for data-aided carrier-phase-independent time-delay estimation from linearly Modulated waveforms

In this paper, we present the derivation and analysis of the true Crame/spl acute/r-Rao lower bound (CRB) for the variance of unbiased, data-aided (DA) symbol-timing estimates, obtained from a block of K samples of a linearly modulated information signal, transmitted through an additive white Gaussian noise channel with random carrier phase. We consider a carrier-phase-independent time-delay estimation scenario wherein the carrier phase is viewed as an unwanted or nuisance parameter. The new bounds require only a moderate computational effort and are tighter than the CRB for the variance of unbiased time-delay estimates obtained under the assumption that the carrier phase is known. These bounds are particularly useful to assess the ultimate accuracy that can be achieved by pilot-assisted symbol synchronizers. Conversely, they may be used to evaluate data sequence suitability for the purpose of time-delay estimation. Comparison of the actual variance of a DA feedforward timing estimator with the new bounds show that these are attainable by practical synchronizers.     

Pilot-Assisted and Blind Joint Data Detection and Channel Estimation in Partial-Time Jamming

We consider a communication scenario in which a message is received in the presence of partial-time Gaussian jamming and additive white Gaussian noise. We consider a quasi-static channel, in which the amplitude and phase are constant over each packet transmission. The receiver does not know the amplitude and phase of the incoming signal, which symbols are jammed, or even the statistics of the jammer, such as the jamming power and jamming probability. In this scenario, the receiver must accurately estimate the parameters of the channel and the jamming to achieve good performance. We apply the expectation-maximization (EM) algorithm to iteratively approximate the maximum-likelihood (ML) estimator for all of the parameters. We find that the overall performance of the EM algorithm is very sensitive to the initial estimates, so we propose a new initial estimator that offers good performance. The EM algorithm approach requires pilot symbols to resolve a phase ambiguity. Thus, we also present a blind estimation algorithm to avoid the reduction in overall code rate from the use of pilot symbols     

On the performance of cyclic moments-based parameter estimators ofamplitude modulated polynomial phase signals

Parameter estimation of amplitude-modulated polynomial phase signals embedded in additive white Gaussian noise is considered. The amplitude modulation is modeled as the sum of a real-valued deterministic function and a zero mean correlated stationary random process. It is shown that cyclic moments-based estimators, previously proposed for parameter estimation of polynomial phase signals modulated by stationary random processes, can be adapted to the more general signal model considered here. The covariance matrices of the cyclic moments-based amplitude and phase parameter estimators are derived for large sample lengths. Using this result, it is shown how the lags can be chosen to minimize the large-sample variances of the cyclic moments-based phase parameter estimators. Comparisons with the Cramer-Rao bounds are performed under the assumption of a Gaussian modulating process. The theoretical derivations are confirmed by simulation results     

On the Miller-Chang lower bound for NDA carrier phase estimation

In this letter, we derive a new Miller-Chang lower bound (MCB) for the variance of unbiased non-data-aided (NDA) carrier phase estimates obtained from a block of N samples of a linearly modulated pulse-amplitude modulation or quadrature-amplitude modulation information signal, transmitted through an additive white Gaussian noise channel. This bound is tighter than the corresponding Crame/spl acute/r-Rao lower bound (CRB) for data-aided or continuous-wave carrier phase estimation (CW-CRB), particularly when N is small. For some given N and sufficiently high signal-to-noise ratios, the MCB is tighter than the corresponding true NDA CRB. The main limitation of this new MCB is that its application is restricted to carrier phase estimators which are unbiased for all possible values of the received symbol sequence.     

Bounds for estimation of complex exponentials in unknown colorednoise

We consider the problem of estimating the parameters of complex exponentials in the presence of complex additive Gaussian noise with unknown covariance. Bounds are derived for the accuracy of jointly estimating the parameters of the exponentials and the noise. We first present an exact Cramer-Rao bound (CRB) for this problem and specialize it for the cases of circular Gaussian processes and autoregressive processes. We also derive an approximate expression for the CRB, which is related to the conditional likelihood function. Numerical evaluation of these bounds provides some insights on the effect of various signal and noise parameters on the achievable estimation accuracy     

Harmonics in Gaussian multiplicative and additive noise: Cramer-Raobounds

The concern here is retrieval of single and multiple tone harmonics observed in white Gaussian multiplicative and additive noise. Computable Cramer-Rao bound (CRB) expressions are derived on the frequency and phase estimates as well as on the sample mean or variance of the multiplicative noise processes. The zero- and nonzero-mean multiplicative noise cases are addressed separately and are shown to yield distinct CRBs on the frequency and phase estimates. Tight lower and upper bounds on the CRBs themselves are developed, which, relative to the CRBs, are intuitively more appealing and easier to implement. Well-established formulas on the achievable accuracy for estimates of constant amplitude harmonics turn put to be special cases of our results. Numerical studies support our claims     

一种单自旋回波串信号参数估计方法

针对存在加性高斯白噪声多参数变量的单自旋回波串信号参数估计问题,提出一种参数分离化的2-D参数估计方法.利用2-D数据矩阵秩为1的特性,依照迭代加权最小二乘方法,从左、右主奇异值向量中以参数分离的方式分别估计出衰减因子和频率,基于最小二乘方法进一步获得信号幅度估计.该方法在相对高信噪比和/或大数据样本下可达到克拉美罗下界,且计算复杂度较低.仿真数据结果证明了算法的有效性.     

复加性白高斯噪声中随机幅值多项式相位信号的Cramer—Rao下界

本文考虑了在复的白高斯加性噪声中观测的多个随机幅值多项式相位信号的参数估计问题，导出了多项式相位系数估计以及幅值样本均值的Ｆｉｓｈｅｒ信息矩阵，因而利用数值计算可得到相应的Ｃｒａｍｅｒ－Ｒａｏ界（ＣＲＢ），模拟结果验证了所得结果的正确性。     

Bounds for estimation of multicomponent signals with randomamplitude and deterministic phase

We study a class of nonstationary multicomponent signals, where each component has the form a(t) exp jφ(t), where a(t) is a random amplitude function, and φ(t) is a deterministic phase function. The amplitude function consists of a stationary Gaussian process and a time varying mean. The phase and the amplitude mean are characterized by a linear parametric model, while the covariance of the amplitude function is parameterized in some general manner. This model encompasses signals that are commonly used in communications, radar, sonar, and other engineering systems. We derive the Cramer-Rao bound (CRB) for the estimates of the amplitude and phase parameters, and of functions of these parameters, such as the instantaneous frequencies of the signal components     

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     

非最小相位ARMA模型的一种自适应辨识算法

本文提出了一种加性有色高斯噪声中因果非最小相位ARMA模型的自适应辨识算法。模型输入假定为非高斯独立同分布随机过程。算法只利用了观测信号的高阶统计量。在每次迭代中,先估计AR参数,再估计MA参数,但不用计算残差序列。在参数递推中采用了随机梯度法。仿真实验证实了本文算法的有效性。     

Algorithm of Robust Frequency Estimation in a Channel with White Gaussian Noise and Pulse Interferences

An algorithm for robust frequency estimation in a channel with additive white Gaussian noise and pulse interference is proposed. The algorithm involves iterative elimination of pulse interferences and subsequent calculation of the maximum likelihood estimate. It is demonstrated that the proposed filter-cleaner differs from the well-known Martin–Thomson filter in the functional dependence of adaptive coefficients on the estimates of parameters of distributions of the linear prediction error and the amplitude envelope of the signal. The results of mathematical modeling and numerical estimation of the threshold point of the method (0.49) are presented.     

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