Cramer-Rao bound analysis for active array with antenna switching

An active radar array with antenna switching provides information on the direction, velocity and range of multiple-targets, in which the receiving antennas are switched to the single receiving channel periodically. The Cramer-Rao bound expressions for estimating the movement parameters using the radar system are derived.     

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     

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     

Posterior Cramer-Rao bounds for discrete-time nonlinear filtering

A mean-square error lower bound for the discrete-time nonlinear filtering problem is derived based on the van Trees (1968) (posterior) version of the Cramer-Rao inequality. This lower bound is applicable to multidimensional nonlinear, possibly non-Gaussian, dynamical systems and is more general than the previous bounds in the literature. The case of singular conditional distribution of the one-step-ahead state vector given the present state is considered. The bound is evaluated for three important examples: the recursive estimation of slowly varying parameters of an autoregressive process, tracking a slowly varying frequency of a single cisoid in noise, and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation     

Linearization method for finding Cramer-Rao bounds in signalprocessing

A new approach for finding and interpreting Cramer-Rao bounds in signal processing is presented in this correspondence. Using the linearization method in nonlinear models and the decoupling technique in linear models, the new method simplifies the burdensome derivations in finding Cramer-Rao bounds and offers insight to their interpretations     

Cramer-Rao bounds for wavelet transform-based instantaneous frequency estimates

We use an asymptotic integral approximation of a wavelet transform as a model for the estimation of instantaneous frequency (IF). Our approach allows the calculation of the Cramer-Rao bound for the IF variance at each time directly, without the need for explicit phase parameterization. This is in contrast to other approaches where the Cramer-Rao bounds rely on a preliminary decomposition of the IF with respect to a (usually polynomial) basis. Attention is confined to the Morlet wavelet transform of single-component signals corrupted with additive Gaussian noise. Potential computationally and statistically efficient IF extraction algorithms suggested by the analysis are also discussed.     

Cramer-Rao lower bounds for low-rank decomposition ofmultidimensional arrays

Unlike low-rank matrix decomposition, which is generically nonunique for rank greater than one, low-rank three-and higher dimensional array decomposition is unique, provided that the array rank is lower than a certain bound, and the correct number of components (equal to array rank) is sought in the decomposition. Parallel factor (PARAFAC) analysis is a common name for low-rank decomposition of higher dimensional arrays. This paper develops Cramer-Rao bound (CRB) results for low-rank decomposition of three- and four-dimensional (3-D and 4-D) arrays, illustrates the behavior of the resulting bounds, and compares alternating least squares algorithms that are commonly used to compute such decompositions with the respective CRBs. Simple-to-check necessary conditions for a unique low-rank decomposition are also provided     

Cramer-Rao lower bounds for QAM phase and frequency estimation

In this paper, we present the true Cramer-Rao lower bounds (CRLBs) for the estimation of phase offset for common quadrature amplitude modulation (QAM), PSK, and PAM signals in AWGN channels. It is shown that the same analysis also applies to the QAM, FSK, and PAM CRLBs for frequency offset estimation. The ratio of the modulated to the unmodulated CRLBs is derived for all QAM, PSK, and PAM signals and calculated for specific cases of interest. This is useful to determine the limiting performance of synchronization circuits for coherent receivers without the need to simulate particular algorithms. The hounds are compared to the existing true CRLBs for an unmodulated carrier wave (CW), BPSK, and QPSK. We investigated new and existing QAM phase estimation algorithms in order to verify the new phase CRLB. This showed that new minimum distance estimator performs close to the QAM bound and provides a large improvement over the power law estimator at moderate to high signal-to-noise ratios     

Cramer-Rao lower bounds for a damped sinusoidal process

The Cramer-Rao lower bounds (CRLBs) for the parameter estimators of a damped sinusoidal process are derived in this paper. Succinct matrix expressions for CRLB's of frequency, damping factor, amplitude, and initial phase are given for both scalar and vector processes. The relationships between the CRLBs of the characteristic parameters are established in the general multimode case. In particular, explicit, closed-form expressions for the single mode scalar/vector-damped/undamped cases are provided     

Cramer-Rao bounds of SNR estimates for BPSK and QPSK modulatedsignals

We derive Cramer-Rao lower bounds (CRLBs) for the estimation of signal-to-noise ratio (SNR) of binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals. The received signal is corrupted by additive white Gaussian noise (AWGN). The lower bounds are derived for non-data-aided estimation where the transmitted symbols are unknown at the receiver. The bounds are compared to those for data-aided estimations (known symbols at the receiver). It is shown that at low SNR there is a significant difference between the bounds for non-data-aided and data-aided estimations     

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     

Cramer-Rao bounds for parametric shape estimation in inverse problems

We address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems, when the boundaries are parameterized by a finite number of unknown variables. Our model applies to multiple unknown objects, each with its own unknown gray level, or color, and boundary parameterization, on an arbitrary known background. While such fundamental bounds on the performance of shape estimation algorithms can in principle be derived from the Cramer-Rao lower bounds, very few results have been reported due to the difficulty of computing the derivatives of a functional with respect to shape deformation. We provide a general formula for computing Cramer-Rao lower bounds in inverse problems where the observations are related to the object by a general linear transform, followed by a possibly nonlinear and noisy measurement system. As an illustration, we derive explicit formulas for computed tomography, Fourier imaging, and deconvolution problems. The bounds reveal that highly accurate parametric reconstructions are possible in these examples, using severely limited and noisy data.     

相干激光探测中微动参数估计的克拉美-罗界

为选择最佳参数估计方法估计目标微多普勒特征,需要研究参数估计的克拉美-罗界,来评价各估计方法的性能。以相干激光探测为背景,考虑噪声方差未知的影响,严格推导了高斯白噪声环境下微动目标回波信号各参数估计的克拉美-罗界的闭合表达式,仿真分析了目标相对于雷达的位置信息、数据处理长度以及回波信噪比与参数估计方差下界的关系。结果表明,克拉美-罗界与噪声方差无关,目标相对于雷达的方位角、俯仰角越小,数据长度和信噪比越大,参数估计的方差下界越小。对目前常用的两种微动参数估计方法方差进行了计算,并与推导克拉美-罗界进行了对比。最后,与通过近似处理方法得到的克拉美-罗界进行了对比,指出了精确推导方差下界的意义。     

双星定位系统定位精度的Cramer-Rao下界分析

介绍了双星时频差定位原理,在此基础上研究了在地球表面约束条件下的双星定位系统的定 位精度理论下界。利用推导的定位精度Cramer-Rao下界,仿真分析了定位参数估计精度、卫 星星历、目标辐射源位置以及辐射源载波频率等因素对定位精度的影响,并给出了物理解释 。定位仿真试验结果表明,定位精度变化趋势与Cramer-Rao下界一致,证明了理论推导的正 确性。     

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.     

Improved Cramer-Rao lower bounds for phase and frequency estimationwith M-PSK signals

We derive new Cramer-Rao lower bounds (CRBs) for the estimation of carrier phase and frequency offset from an unmodulated carrier or from a M-PSK data signal. The new CRBs are obtained formally from the exact likelihood function for the carrier phase and frequency offset, build from a block of received phase samples, and are applicable for a general alphabet size M. The bounds are compared with other previously obtained bounds (which also take into account the effect of random phase modulation) and with the performance of some popular feedforward algorithms for carrier phase and frequency offset estimation     

Worst case Cramer-Rao bounds for parametric estimation ofsuperimposed signals with applications

The problem of parameter estimation of superimposed signals in white Gaussian noise is considered. The effect of the correlation structure of the signals on the Cramer-Rao bounds is studied for both the single and multiple experiment cases. The best and worst conditions are found using various criteria. The results are applied to the example of parameter estimation of superimposed sinusoids, or plane-wave direction finding in white Gaussian noise, and best and worst conditions on the correlation structure and relative phase of the sinusoids are found. This provides useful information on the limits of the resolvability of sinusoid signals in time series analysis or of plane waves in array processing. The conditions are also useful for designing worst-case simulation studies of estimation algorithms, and for the design of minimax signal acquisition and estimation procedures, as demonstrated by an example     

Performance bounds for estimating vector systems

We propose a unified framework for the analysis of estimators of geometrical vector quantities and vector systems through a collection of performance measures. Unlike standard performance indicators, these measures have intuitive geometrical and physical interpretations, are independent of the coordinate reference frame, and are applicable to arbitrary parameterizations of the unknown vector or system of vectors. For each measure, we derive both finite-sample and asymptotic lower bounds that hold for large classes of estimators and serve as benchmarks for the assessment of estimation algorithms. Like the performance measures themselves, these bounds are independent of the reference coordinate frame, and we discuss their use as system design criteria     

Statistical AM-FM models, extended Kalman filter demodulation,Cramer-Rao bounds, and speech analysis

A stochastic dynamical system model for describing time signals that are jointly amplitude (AM) and frequency (FM) modulated is presented. The signal is assumed to be bandpass, perhaps originating from a filter bank applied to a broadband signal, and includes the constraint that the magnitude of the complex baseband signal is positive. Motivated by speech processing and the desire for narrowband modulating signals, time is divided into frames, and the modulating signals are smoothly interpolated across each frame. The model allows a detailed characterization of the bandwidth of the modulating signals and the statistical character of the measurement noise. An adaptive estimation algorithm based on extended Kalman filtering ideas for extracting the modulating signals from the measured signal is described and demonstrated on both voiced and unvoiced speech signals. The Cramer-Rao bound on the performance of any estimator is computed