Capacitor-input filter

Abstract: This paper discusses a family of filters that have been designed for Quadrature Mirror Filter (QMF) Banks. These filters provide a significant improvement over conventional optimal equiripple and window designs when used in QMF banks. The performance criterion for these filters differ from those usually used for filter design in a way which makes the usual filter design techniques difficult to apply. Two filters are actually designed simultaneously, with constraints on the stop band rejection, transition band width, and pass and transition band performance of the QMF filter structure made from those filters. Unlike most filter design problems, the behavior of the transition band is constrained, which places unusual requirements on the design algorithm. The requirement that the overall passband behavior of the QMF bank be constrained (which is a function of the passband and stop band behavior of the filter) also places very unusual requirements on the filter design. The filters were designed using a Hooke and Jeaves optimization routine with a Hanning window prototype. Theoretical results suggest that exactly flat frequency designs cannot be created for filter lengths greater than 2, however, using the discussed procedure, one can obtain QMF banks with as little as ±.0015dB ripple in their frequency response. Due to the nature of QMF filter applications, a small set of filters can be derived which will fit most applications.

Abstract: In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion of the extended Kalman filter, and a relatively simple (tangible) example with real numbers & results. 1. welch@cs.unc.edu, http://www.cs.unc.edu/~welch 2. gb@cs.unc.edu, http://www.cs.unc.edu/~gb Welch & Bishop, An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discretedata linear filtering problem [Kalman60]. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. A very “friendly” introduction to the general idea of the Kalman filter can be found in Chapter 1 of [Maybeck79], while a more complete introductory discussion can be found in [Sorenson70], which also contains some interesting historical narrative. More extensive references include [Gelb74; Grewal93; Maybeck79; Lewis86; Brown92; Jacobs93]. The Process to be Estimated The Kalman filter addresses the general problem of trying to estimate the state of a discrete-time controlled process that is governed by the linear stochastic difference equation , (1.1) with a measurement that is . (1.2) The random variables and represent the process and measurement noise (respectively). They are assumed to be independent (of each other), white, and with normal probability distributions , (1.3) . (1.4) In practice, the process noise covariance and measurement noise covariance matrices might change with each time step or measurement, however here we assume they are constant. The matrix in the difference equation (1.1) relates the state at the previous time step to the state at the current step , in the absence of either a driving function or process noise. Note that in practice might change with each time step, but here we assume it is constant. The matrix B relates the optional control input to the state x. The matrix in the measurement equation (1.2) relates the state to the measurement zk. In practice might change with each time step or measurement, but here we assume it is constant. The Computational Origins of the Filter We define (note the “super minus”) to be our a priori state estimate at step k given knowledge of the process prior to step k, and to be our a posteriori state estimate at step k given measurement . We can then define a priori and a posteriori estimate errors as x R ∈ xk Axk 1 – Buk 1 – wk 1 – + + = z R ∈ zk H xk vk + = wk vk p w ( ) N 0 Q , ( ) ∼ p v ( ) N 0 R , ( ) ∼ Q R n n × A k 1 – k A n l × u R ∈ m n × H H x̂k R ∈ x̂k R n ∈ zk ek xk x̂k , and – ≡ ek xk x̂k. – ≡ UNC-Chapel Hill, TR 95-041, March 1, 2004 Welch & Bishop, An Introduction to the Kalman Filter 3 The a priori estimate error covariance is then , (1.5) and the a posteriori estimate error covariance is . (1.6) In deriving the equations for the Kalman filter, we begin with the goal of finding an equation that computes an a posteriori state estimate as a linear combination of an a priori estimate and a weighted difference between an actual measurement and a measurement prediction as shown below in (1.7). Some justification for (1.7) is given in “The Probabilistic Origins of the Filter” found below. (1.7) The difference in (1.7) is called the measurement innovation, or the residual. The residual reflects the discrepancy between the predicted measurement and the actual measurement . A residual of zero means that the two are in complete agreement. The matrix K in (1.7) is chosen to be the gain or blending factor that minimizes the a posteriori error covariance (1.6). This minimization can be accomplished by first substituting (1.7) into the above definition for , substituting that into (1.6), performing the indicated expectations, taking the derivative of the trace of the result with respect to K, setting that result equal to zero, and then solving for K. For more details see [Maybeck79; Brown92; Jacobs93]. One form of the resulting K that minimizes (1.6) is given by1

Abstract: Under certain conditions, the orbit estimated by a Kalman filter has errors that are much greater than predicted by theory. This phenomenon is called divergence, and renders the operation of the Kalman filter unsatisfactory. This paper investigates the control of divergence in a Kalman filter used for autonomous navigation in a low earth orbit. The system studied utilizes stellar-referenced angle sightings to a sequence of known terrestrial landmarks. A Kalman filter is used to compute differential corrections to spacecraft position, spacecraft velocity, and landmark location. A variety of filter modifications for the control of divergence was investigated. These included the Schmidt-Pines analytical modification and an "empirical" modification based upon Pines' machine noise treatment. Several simplified approximations to the theoretically optimum analytical modifications were also investigated. The principal numerical results are presented in graphs of the magnitude of the error in estimated position and velocity vs time for sixteen orbits. These graphs compare actual position and velocity errors with the theoretical estimates furnished by the trace of the position and velocity covariance matrices. Numerical results indicate that a properly modified filter achieves a steady-state operating level.