On the Sensitivity of a Fixed-Point Smoothing Algorithm

This paper presents a simple approach to the derivation of sensitivity measures of smoothing algorithms. It is proposed that the known sensitivity results of the Kalman filtering algorithm be utilized along with the state augmentation approach for this purpose. The sensitivity measures so obtained are easier to compute than the ones available in the literature. It is then shown that the fixed-point smoothing algorithm, derived recently by Biswas and Mahalanabis [1], [2], is less sensitive to model parameter variations than the algorithm studied by Griffin and Sage [7], [8]. The case of a satellite tracking problem is presented by way of illustrating the results.     

A New Approach to Estimation of Initial Conditions and Smoothing Problems

A unified approach utilizing the Kalman-Bucy filtering technique istaken to solve the estimation problem of initial conditions and thesmoothing problem in linear dynamic systems. The equivalencebetween the forward integration method and the backward integrationmethod of the smoothing solution is proved. Complete analyticalsolutions of filtering and smoothing problems of rectilinear motion ofa randomly accelerated spacecraft are derived when the vehicle istracked by the ranging system at the ground station.     

A novel variable-lag probability hypothesis density smoother for multi-target tracking

It is understood that the forward-backward probability hypothesis density (PHD) smoothing algorithms proposed recently can significantly improve state estimation of targets. However, our analyses in this paper show that they cannot give a good cardinality (i.e., the number of targets) estimate. This is because backward smoothing ignores the effect of temporary track drop- ping caused by forward filtering and/or anomalous smoothing resulted from deaths of targets. To cope with such a problem, a novel PHD smoothing algorithm, called the variable-lag PHD smoother, in which a detection process used to identify whether the filtered cardinality varies within the smooth lag is added before backward smoothing, is developed here. The analytical results show that the proposed smoother can almost eliminate the influences of temporary track dropping and anomalous smoothing, while both the cardinality and the state estimations can significantly be improved. Simulation results on two multi-target tracking scenarios verify the effectiveness of the proposed smoother.     

Interacting multiple model fixed-lag smoothing algorithm forMarkovian switching systems

We investigate a suboptimal approach to the fixed-lag smoothing problem for Markovian switching systems. A fixed-lag smoothing algorithm is developed by applying the basic Interacting Multiple Model (IMM) approach to a state-augmented system. The computational load is roughly d (the fixed lag) times beyond that of filtering for the original system. In addition, an algorithm that approximates the “fixed-lag” mode probabilities given measurements up to current time is proposed. The algorithm is illustrated via a target tracking simulation example where a significant improvement over the filtering algorithm is achieved at the cost of a time delay (i.e., data up to time k are used to produce the smoothed state estimate at time k-d where the fixed large d>0). the IMM fixed-lag smoothing performance for the given example is comparable to that of an existing IMM fixed-interval smoother. Compared with fixed-interval smoothers, the fixed-lag smoothers can be implemented in real-time with a small delay     

GNSS最优载波相位平滑伪距研究

平滑常数是影响载波相位平滑伪距精度的关键参数,实际数据处理时主要依据经验设定平滑常数。这种主观设定过程缺乏理论依据,无法达到最优平滑效果。针对此问题,以适用于实时GNSS载波相位平滑伪距的经典Hatch递推滤波算法为基础,在连续时间域上分析了载波相位平滑伪距误差的主要构成,给出了总误差估算公式,分析了平滑常数对平滑精度的影响传导机制。进一步,采用令平滑总误差最小为目标的极值法推导给出了最优载波相位平滑常数的计算公式,给出了最优载波相位平滑伪距的完整处理步骤。最优平滑常数算法在数学意义上最优,大幅压缩了伪距测量误差,又不会引入过大的电离层发散误差。通过两个实际算例,证明了算法有效性。     

A simple smoother: closed-form solution and analytical results

A closed-form solid-state solution for a simple one-dimensional smoother is presented. The target position-velocity dynamics are modeled as a linear discrete Gauss-Markov process: the discrete position measurements are assumed to be independent and identically normally distributed. Analytical results derived from estimating optimum steady-state position and velocity are given as well as the improvement provided by smoothing relative to pure filtering estimates. Analytical results, including appropriate limiting cases, show that the ratios of the RMS smoothing to the RMS filtering error are from 1/2 to 1 in position and from 1/2 to 1/√2 in velocity. As a numerical example a highly maneuvering target characterized by an acceleration width of 30 m/s² is considered     

Sensitivity Analysis of Finite-Lag Receivers

This paper presents the sensitivity analysis of a class of receivers called finite-lag receivers, introduced by the authors in [1] through [3]. Since these receivers are based on the use of fixed-lag smoothing techniques, algorithms for the calculation of large-scale and small-scale sensitivities of fixed-lag smoothing are derived using a state augmentation approach. Steady-state analysis of these algorithms shows that an explicit relation can be obtained between sensitivity coefficients of fixed-lag smoothing and filtering. The specific case of sensitivity to variations in the measurement (channel) noise is considered as an example. These results are applied to study the sensitivity performance of the finite-lag receivers for analog communication. It is shown, for example, that finite-lag receivers for AM signals, besides being superior in performance [1]-[3], [11], in terms of output SNR or error variance, are also much less sensitive to the additive noise power level, compared to zero-lag receivers.     

U-D分解的固定点平滑新算法及其在飞行试验中的应用

 本文根据Rauch固定点平滑公式,提出了一种U-D分解的固定点平滑新算法。这一算法不仅具有良好的数值稳定性和可靠性,而且计算量较少;计算效率是Bryson-Ho Y C固定点平滑计算效率的1.5倍以上。将这种新算法用于飞机运动状态初值的确定,提高了飞机气动参数辨识精度。     

Coordinate Conversion and Tracking for Very Long Range Radars

The problem of tracking with very long range radars is studied in this paper. First, the measurement conversion from a radar's r-u-v coordinate system to the Cartesian coordinate system is discussed. Although the nonlinearity of this coordinate transformation appears insignificant based on the evaluation of the bias of the converted measurements, it is shown that this nonlinearity can cause significant covariance inconsistency in the conventionally converted measurements (CM1). Since data association depends critically on filter consistency, this issue is very important. Following this, it is shown that a suitably corrected conversion (CM2) eliminates the inconsistency. Then, initialized with the converted measurements (using CM2), four Cartesian filters are evaluated. It is shown that, among these filters, the converted measurement Kalman filter with second order Taylor expansion (CM2KF) is the only one that is consistent for very long range tracking scenarios. Another two approaches, the range-direction-cosine extended Kalman filter (ruvEKF) and the unscented Kalman filter (UKF) are also evaluated and shown to suffer from consistency problems. However, the CM2KF has the disadvantage of reduced accuracy in the range direction. To fix this problem, a consistency-based modification for the standard extended Kalman filter (E1KF) is proposed. This leads to a new filtering approach, designated as measurement covariance adaptive extended Kalman filter (MCAEKF). For very long range tracking scenarios, the MCAEKF is shown to produce consistent filtering results and be able to avoid the loss of accuracy in the range direction. It is also shown that the MCAEKF meets the posterior Carmer-Rao lower bound for the scenarios considered.     

Nonlinear Estimation with Radar Observations

The nonlinearities of the radar measurement model equation are examined and their influence upon the accuracy of filtering and smoothing is determined. A simple algorithm by which the effects from these nonlinearities can be significantly reduced is derived. It consists of processing the radar observations in this order: azimuth, elevation, and range, using the azimuth residual to evaluate the elevation residual, and then using this combined result to evaluate the range residual. The accuracy of estimates obtained via the algorithm is compared with and shown to be superior to that of the extended Kalman filter (EKF). Furthermore, use of the algorithm does not increase the computational complexity of estimation beyond that of the EKF.     

R-T-S平滑算法与抗差Kalman在数据后处理中的应用

为了提高组合导航系统后处理精度和数据稳定性,将R-T-S最优固定区间平滑算法引入数据后处理中,在前向Kalman滤波的基础上,进行了后向R-T-S最优固定区间平滑处理,并针对GPS观测值中存在异常的问题,将抗差Kalman滤波算法引入数据后处理中,并对该算法进行实物仿真。结果表明,与传统Kalman滤波相比,R-T-S平滑算法不仅可以提高位置、姿态精度,而且在卫星信号失锁的情况下精度也得到显著改善,并且在不丢星的时刻,抗差Kalman滤波可以有效处理GPS信号中的异常观测值,遏制滤波发散,是一种有效的数据处理方法。     

基于数字滤波的低频时码秒脉冲抖动平滑方法研究

低频时码授时信号在接收时,由于环境等因素的影响会导致解调得到的低频时码秒脉冲出现抖动。为减小该抖动现象对低频时码定时精度的影响,基于我国低频时码授时系统(BPC),研究并设计基于数字滤波的秒脉冲抖动平滑方法,通过仿真实验,验证分析了最小均方误差(LMS)自适应滤波算法和卡尔曼滤波算法的有效性和抖动平滑性能。结果表明,低频时码秒脉冲的抖动现象可以通过BPC 1PPS (one pulse per second)和本地1PPS之间的相位差波动情况反映;LMS自适应滤波算法和卡尔曼滤波算法对BPC 1PPS的抖动平滑处理均有明显效果,卡尔曼滤波算法的抖动平滑效果更优,但存在一定的收敛时间,LMS自适应滤波算法的滤波结果响应速度快,但抖动平滑性能受滤波器阶数的影响。所以,在实际应用中,应根据实际需求选择合适的滤波器及相关参数。     

New Autocorrelation and Prediction Techniques for the Demodulation of Binary Signals

The classic problem of detecting a narrowband signal received in a wideband noise is treated via new techniques, namely, autocorrelation computation and linear adaptive filtering. It is shown by analysis and simulation that the proposed techniques yield probility of bit errors cimparable to those provided by coding techniques. Expressions are derived for the detectability parameter of the new test statistics, and it is shown that these statistics are very close to the Gaussian approximation. The improvements obtained in the signal-to-noise ratio and the Gaussian nature of the new statistic implies an improvement in the capacity of the channel by the nonlinear and storage processes involved in the new techniques, something not achievable by linear filtering and/or memoryless detection techniques.     

Image Algebra and Automatic Shape Recognition

Shape recognition can be carried out with set theory operations. In addition to the usual set theory operations of union, complement, etc., the operation of erosion between two sets (or images) is incorporated. Then it is shown that these operations between sets serve as a basis for representation theorems for all, generally nonlinear, translation invariant transformations, (i.e., transformations which commute with translations). By treating images and shapes as point sets in n-dimensional Cartesian space (n = 2 for binary images, n = 3 for gray scale images, and larger n for images that incorporate color, polarization, and the like), the problem of shape or pattern recognition is converted to the problem of detecting the occurrences of specific sets within an image. This problem is closely related to the operation of erosion. By introducing complement images and complement shapes, a generic computer for automatic shape recognition is found which provides a constructive proof of the representation theorem and related results.     

基于固定区间最优平滑的传递对准精度评估

惯导系统传递对准是战术导弹发射前必须完成的任务,其对准精度直接影响导弹系统的制导精度。为评估传递对准精度性能,在介绍传递对准精度评估原理的基础上,研究了Kalman固定区间最优平滑算法;并从理论上分析了顺序滤波与逆序平滑的关系,得出平滑依赖于滤波的结论。通过实测数据半物理仿真试验证明了理论分析的正确性,表明固定区间最优平滑算法能有效地评估传递对准精度,并与间接法对比得出间接法与最优平滑算法具有一致性。     

卡尔曼平滑的自校准技术

介绍了卡尔曼平滑技术的固定区间平滑方法，推导了非线性时变模型的固定区间平滑公式，并将之应用到轨道约束自校准技术中，且导出了相应公式。     

高速采样测量数据处理方法研究

探讨在新一代高精度测速体制下,外测数据原有处理方法应用到高采样率数据时的缺陷,提出将原始采样数据分段、分层进行多次处理,结合二步最小二乘、等距样条拟舍和区间长度自适应的时间多项式中心平滑技术,可降低每次处理的数据量,并逐步滤除周期噪声与色噪声,提高了处理结果的精度。     

Dynamics of the Phase-Locked Loop without a Limiter

The general (nth order) phase-locked loop is analyzed, of which the amplitude is not constant. The input carrier signal is amplitude-modulated by wide-band stationary Gaussian noise, and the signal, superposed with the additive white stationary Gaussian noise, enters the nonlimited phase-locked loop. Under the above assumptions the loop can be shown to constitute an n-dimensional vector Markov process, so that the process satisfies the n-dimensional Fokker-Plank equation. The probability density function depends on the effective loop signal-to-noise ratio and the effective modulation power.     

Mixed H/sub 2//H/sub /spl infin// filtering for uncertain systems with regional pole assignment

The mixed H/sub 2//H/sub /spl infin// filtering problem for uncertain linear continuous-time systems with regional pole assignment is considered. The purpose of the problem is to design an uncertainty-independent filter such that, for all admissible parameter uncertainties, the following filtering requirements are simultaneously satisfied: 1) the filtering process is asymptotically stable; 2) the poles of the filtering matrix are located inside a prescribed region that compasses the vertical strips, horizontal strips, disks, or conic sectors; 3) both the H/sub 2/ norm and the H/sub /spl infin// norm on the respective transfer functions are not more than the specified upper bound constraints. We establish a general framework to solve the addressed multiobjective filtering problem completely. In particular, we derive necessary and sufficient conditions for the solvability of the problem in terms of a set of feasible linear matrix inequalities (LMIs). An illustrative example is given to illustrate the design procedures and performances of the proposed method.     

Improved Phase-Locked Loop Performance with Adaptive Phase Comparators

A major problem in phase-locked loop (PLL) design is to meet the requirements of both fast signal acquisition and good synchronous mode performance. This relation is reviewed for different types of phase comparators. As a result a new phase-and-frequency comparator is proposed. This comparator is based on an up-down counter principle and can be considered as an adaptive acquisition control circuit. The analysis of a PLL with the proposed phase comparator is based on an exact calculation method for the pull-in time. It is shown that fast signal acquisition is possible without affecting the filtering properties of the loop. Experimental results are given of the acquisition behavior of a second-order type-2 loop which show a good correspondence with the theoretical analysis.