We address a p -shift finite impulse response (FIR) unbiased estimator (UE) for linear discrete time-varying filtering (p=0), p-step prediction (p >; 0), and p-lag smoothing (p <; 0) in state space with no requirements for initial conditions and zero mean noise. A solution is found in a batch form and represented in a computationally efficient iterative Kalman-like one. It is shown that the Kalman-like FIR UE is able to outperform the Kalman filter if the noise covariances and initial conditions are not known exactly, noise is not white, and both the system and measurement noise components need to be filtered out. Otherwise, the errors are similar. Extensive numerical studies of the FIR UE are provided in Gaussian and non-Gaussian environments with outliers and temporary uncertainties.

An Iterative Kalman-Like Algorithm Ignoring Noise and Initial Conditions

This paper addresses a general p-shift linear optimal finite impulse response (FIR) estimator intended for solving universally the problems of filtering (p = 0), smoothing (p 0) of discrete time-invariant models in state space. An optimal solution is found in the batch form with the initial mean square state function self-determined by solving the discrete algebraic Riccati equation. An unbiased solution represented both in the batch and recursive forms does not involve any knowledge about noise and initial state. The mean square errors in both the optimal and unbiased estimates are found via the noise power gain (NPG) and a recursive algorithm for fast computation of the NPG is supplied. Applications are given for FIR filtering with fixed, receding, and full averaging horizons.

Linear Optimal FIR Estimation of Discrete Time-Invariant State-Space Models

In this correspondence, we propose an efficient estimator of optimal memory (averaging interval) for discrete-time finite impulse response (FIR) filters in state-space. Its crucial property is that only real measurements and the filter output are involved with no reference and noise statistics. Testing by the two-state polynomial model has shown a very good correspondence with predicted values. Even in the worst case of the harmonic model, the estimator demonstrates practical applicability.

Optimal Memory for Discrete-Time FIR Filters in State-Space

The first- and second-order extended finite impulse response (EFIR1 and EFIR2, respectively) filters are addressed for suboptimal estimation of nonlinear discrete-time state-space models with additive white Gaussian noise. It is shown that, unlike the extended Kalman filter (EKF) and EFIR2 filter, the EFIR1 one does not require noise statistics and initial errors. Only within a narrow region around actual noise covariances, EFIR filters fall a bit short of EKF and they demonstrate better performance otherwise. It is shown that the optimal averaging interval for EFIR filters can be determined via measurement without a reference model in a learning cycle. We also notice that the second-order approximation can improve the local performance, but it can also deteriorate it. We thus have no recommendations about its use, at least for tracking considered as an example of applications.

Suboptimal FIR Filtering of Nonlinear Models in Additive White Gaussian Noise

We address an unbiased finite impulse response (FIR) filter for discrete-time state-space models with polynomial representation of the states. The unique l-degree polynomial FIR filter gain and the estimate variance are found for a general case. The noise power gain (NG) is derived for white Gaussian noises in the model and in the measurement. The filter does not involve any knowledge about noise in the algorithm. It is unstable at short horizons, 2 les N les l, and inefficient (NG exceeds unity) in the narrow range l 1, the estimate noise becomes negligible and the filter thus optimal in the sense of zero bias and zero noise. Having such properties, the proposed unbiased FIR filter fits well slowly changing with time models. An example is given for a two-state system.

Unbiased FIR Filtering of Discrete-Time Polynomial State-Space Models

In this paper, we discuss optimal synchronization of local clocks using Global Positioning System (GPS) one-pulse-per-second (1PPS) timing signals. To eliminate sawtooth errors that are peculiar to the 1PPS signals and optimally steer the clock errors each M seconds, we propose the use of a ramp-predictive finite-impulse-response (FIR) filter that is known to be optimal for clock models on large averaging horizons. A low-pass filter is used to smooth the hold filter output between the optimally predicted points. A GPS-locked crystal clock has been investigated in detail in terms of the time interval error, Allan deviation, and precision time protocol (PTP) variance. A high-efficiency implementation of the proposed synchronization algorithm is experimentally demonstrated.

Optimal Synchronization of Local Clocks by GPS 1PPS Signals Using Predictive FIR Filters

Optimal horizons for a one-parameter family of unbiased FIR filters

A thinning algorithm for GPS-based unbiased FIR estimation of a clock TIE model

This paper addresses a probabilistic analysis of sawtooth noise in the one pulse per second (1PPS) output of the timing SynPaQ III GPS Sensor. We show that sawtooth noise is uniformly distributed within the bounds caused by period of the Local Time Clock of the sensor and that the probability density function (pdf) of this noise is formed with 1ns sampling interval used in the sensor to calculate the negative sawtooth. We also show that the pdf has at zero a spike of 1ns width caused by roll-off. It is demonstrated that an unbiased finite impulse response filter is an excellent suppresser of such a noise in the estimates of the time interval errors of local clocks. Copyright © 2007 IFSA.

An Analysis of Sawtooth Noise in the Timing SynPaQ III GPS Sensor

In this work, we present a novel technique for ultrasound image processing (2D signals). The moving average hybrid FIR filter (MAH-FIR) which it work with two different FIR filter, the moving average (MA) and median hybrid (FMH) filters. We experimented with continuous linear regression function and Lagrange multiplier to obtain the approximation of the MA-FIR and FMH-FIR filters, respectively. Furthermore, we showed both filters composed into a block diagram. Finally, we present evaluations qualitative and quantitative of ultrasound image processing.

Luis Arceo-Miquel

Papers

Optimal Synchronization of Local Clocks by GPS 1PPS Signals Using Predictive FIR Filters

Optimal horizons for a one-parameter family of unbiased FIR filters

A thinning algorithm for GPS-based unbiased FIR estimation of a clock TIE model

An Analysis of Sawtooth Noise in the Timing SynPaQ III GPS Sensor

Moving Average Hybrid FIR Filter in Ultrasound Image Processing