John Earl Gray

John Earl Gray
Institute of Electrical and Electronics Engineers | IEEE · IEEE Aerospace and Electronic Systems Society

MS

About

124
Publications
12,078
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533
Citations
Additional affiliations
May 1985 - June 2022
Naval Surface Warfare Center, Dahlgren Division
Position
  • Naval Surface Warfare Center
Description
  • Retired
Education
June 1973 - August 1980
University of Mississippi
Field of study
  • physics, mathematics

Publications

Publications (124)
Article
We present a follow-on experiment to the recent study from The University of Rochester [ Opt. Lett. 42 , 2479 ( 2017 ) OPLEDP 0146-9592 10.1364/OL.42.002479 ], which reported a new architecture for an inverse weak-value tiltmeter. We recreate the Rochester tiltmeter and specifically investigate mirror oscillations in the low-kilohertz frequency reg...
Preprint
Full-text available
We present an investigation into the tilt sensitivity of the canonical, optical, weak-value amplification device (COWVAD), introduced by Duck, Stevenson, and Sudarshan in 1989, for potential application in a Coriolis vibratory gyroscope (CVG). We model the breakdown of the weak-value amplification effect in this device with respect to angle of inci...
Preprint
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Parks introduced a formulation of time dependent weak values in 2008, which is the formalism we use in this paper. In this paper we extend notions from time dependent weak values to show that Hamiltonians associated with weak value measurements can be shown to exhibit even or odd symmetric properties. They exhibit PT or anti-PT symmetry, respective...
Article
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We are concerning the emotion of aging global village without children or grandchildren taking care of home alone seniors (HAS). We recommended to capture and endow human emotions, especially the loniless by electromechanical robots, humanoids, who can help HAS moving around. Before a final design of Elon-Musk-like space travel robots that can proc...
Article
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The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further gene...
Preprint
The aim of this work is to study generalizations of the notion of mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the Weak Law of Large Numbers in the same way that the ordinary mean relates to the Strong Law. We propose a further generali...
Article
In this paper, we develop the concept of the post-selection operator current and we use it to extend the methodology of quantum mechanics into related fields of science and engineering. We begin by reviewing some results from standard quantum mechanics. We then introduce the post-selection operator current and note that the concept of the operator...
Article
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Recent studies in neurosciences have produced evidence to suggest that Glial cells play a vital role in assisting neurons to form synaptic connections, contrary to a previously held view that they played less significant role in synaptic function. The neurochemical processes underlying the transmission is observed to be both temperature dependent,...
Article
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The original Aharonov–Vaidman operator action theorem relates the action of an Hermitian operator \({\hat{A}} \) upon a state \(|\psi \rangle \) to its associated mean value \(\langle \psi |\hat{{A}}|\psi \rangle \), its uncertainty \(\Delta A\), and an orthogonal companion state \(|{\psi ^{\bot }} \rangle \). Here a slightly more general version o...
Article
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In order to apply thermodynamics to systems in which entropy is not extensive, it has become customary to define generalized entropies. While this approach has been effective, it is not the only possible approach. We suggest that some systems can be investigated by instead generalizing the concept of extensivity. We begin by reexamining the role of...
Presentation
Full-text available
There is a common framework for the measurement problem for sensors such as radars, sonars, and optics in a common language by casting analysis of signals in the language of quantum mechanics (Rigged Hilbert Space). The use of this language can reveal a more detailed understanding of the underlying interactions of a return signal that are not usual...
Conference Paper
We provide a common framework for the measurement problem for radar, sonar, and quantum mechanics by casting them in the common language of quantum mechanics as a Rigged Hilbert Space. This language reveals a more detailed understanding of the underlying interactions of a return signal that are not usually brought out by standard signal processing...
Conference Paper
Since the inception of coherent waveforms, it has been realized that the effect of non-uniform motion of a non-point like object can induce structure in the return spectrum of the waveform that can be exploited. The non-uniform Doppler spectrum has useful information that can be found in the spreading of the Doppler spectrum for the motion models:...
Article
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Calculation of the mean of an observable in quantum mechanics is typically assumed to require that the state vector be in the domain of the corresponding self-adjoint operator or for a mixed state that the operator times the density matrix be in the trace class. We remind the reader that these assumptions are unnecessary. We state what is actually...
Article
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Joining previous authors, we propose axiomatic properties that yield the mean as the unique measure of center of a data set. In addition to familiar properties such as symmetry, homogeneity, and translativity, we make use of a condensation principle, first considered by Bemporad, that ties together means for different sample sizes and identifies an...
Article
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Target tracking in radar and sonar is often done in either spherical or rectangular coordinates. The objective of the work presented here is to provide a means to analytically characterize the probability density function (pdf) of these transformations. Such pdf can be used to accurately characterize the measurement statistics needed for filter des...
Article
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Exact expressions for the mean pointer position, the mean pointer momentum and their variances are obtained for projectio operator measurements performed upon ensembles of pre- and postselected (PPS) and preselected-only (PSO) quantum systems. These expressions are valid for any interaction strength which couples a measurement pointer to a quantum...
Article
Full-text available
We present an axiomatic approach to the mean and discuss generalizations of the mean, including one due to Kolmogorov based on the Weak Law of Large Numbers. We offer examples and counterexamples, describe conventional and unconventional uses of the mean in statistical mechanics, and resolve an anomaly in quantum theory concerning apparent simultan...
Article
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It is well known that the pointer state |{\Phi}> resulting from the von Neumann measurement of a projection operator A performed upon an ensemble of quantum systems in the preselected state |{\psi}> depends upon A|{\psi}>. Here it is shown that the pointer state |{\Psi}> obtained from such a measurement performed upon an ensemble that is also posts...
Article
Full-text available
In previous work, we have argued that measurement using a radar can be viewed as taking the expected value of an operator. The operator usually represents some aspect of the characteristics of the object being tracked (such as Doppler, distance, shape, polarization, etc.) that is measured by the radar while the expectation is taken with respect to...
Article
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The variance of an arbitrary pointer observable is considered for the general case that a complex weak value is measured using a complex valued pointer state. For the typical cases where the pointer observable is either its position or momentum, the associated expressions for the pointer's variance after the measurement contain a term proportional...
Article
Full-text available
We introduce the concept of a random gauge. We propose two distinct types of random gauge that can be defined based on the concept of phase noise in scattering theory. In the context of quantum physics, we discuss a variety of possible realizations of this concept that can be connected to various Aharonov-other effects and make some connections wit...
Article
Full-text available
Exact pointer states are obtained for projection operator measurements performed upon pre-selected (PS) and upon pre- and post-selected (PPS) quantum systems. These states are used to provide simple exact expressions for both the pointer spatial probability distribution profiles and the mean values of arbitrary pointer observables associated with P...
Article
Full-text available
In order to apply thermodynamics to systems in which entropy is not extensive, it has become customary to define generalized entropies. While this approach has been effective, it is not the only possible approach. We suggest that some systems, including nanosystems, can be investigated by instead generalizing the concept of extensivity. We begin by...
Conference Paper
Full-text available
We suggest a new interpretation of Woodward's ambiguity function as the expected value of an operator. The operator represents the physics of the interaction of the waveform with the object. This approach provides a new approach to understanding the return signal at the receiver and can reveal more detailed understanding of the underlying interacti...
Conference Paper
Full-text available
In this work, we discuss a variety of problems that can be cast as specific instances of the superposition of random amplitudes times random phase functions, the Rayleigh problem. A wide number of problems that occur in sensor domain are equivalent to specific instances of this problem. Using characteristic functions, it is possible to determine ho...
Article
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We discuss the principle of superposition relative to the definition of extensivity in thermodynamics and discuss how it can be extended to systems that do not obey the linear principle of superposition. This leads to the concept of generalized superposition, of which, there are multiple types defined in the paper. Generalized superposition can be...
Article
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Ideas from classical signal analysis and Khinchin's theorem suggest a different way to think about signals and quantum mechanics so that probability and noise are both treated in the same manner. This suggest connections between ideas of receivers and the Von Neuman theory of measurement. Weak values are connected to these ideas
Article
Full-text available
We introduce a notation for probability current and operator current in a normal quantum mechanics setting. We then extend these concepts to the concept of post-selection that was introduced by Schordinger and has found wide application by Aharonov and his colleagues. We introduce the concept of a post-selection operator current and then use it as...
Article
Full-text available
There is a connection between radar theory and the formalism of quantum mechanics that has not been explored by many radar engineers. We illustrate this for a number of other radar observables that are also equivalent to taking the expected value of certain operators. We then compare this to an approach to waveform design based on the ambiguity fun...
Article
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Perception is the apparatus of many of the potential new sciences of the twenty-first century. We can imagine science as a symbolic language where there are concrete words which are taken from measurements while abstract words are defined by theory to construct new ways to think about more fuzzy concepts that occur in more human observation based s...
Article
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In this paper, we investigate two topics related to mitigating the effect of radar bias in ballistic missile tracking applications. We determine the absolute bias between two radars in polar coordinates when their relative bias is given in rectangular coordinates. Using this result, we then obtain the optimized steady-state filter to handle the ran...
Conference Paper
Full-text available
Statistical mechanics has proven to be a useful model for drawing inferences about the collective behavior of individual objects that interact according to a known force law (which for a more general usage is referred to as interacting units.). Collective behavior is determined not by computing F = ma for each interacting unit because the problem i...
Article
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We introduce the concept of memory of an instrument into quantum mechanics. The connection between quantum memory and contextuality arises via the lack of the Markov property of instrumentation. Also, a further connection is made between memory and gauge transformations that arises in the definition of distance relative to measurement.
Article
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We have establish a more rigorous formulation of the AV-formula by illustrating how it can be derived from the Projection Theorem. This connection with the Projection Theorem enables us to establish an additional formula as well as show how to calculate the higher powers of the observable in terms of the formulas for A vertical bar Psi > and A vert...
Conference Paper
Full-text available
Statistical Mechanics, which is due primarily to Maxwell, Gibbs, and Boltzmann in the ninetieth century, has proven to be useful model for drawing inferences about the collective behavior of individual objects that interact according to a known force law (which for more general usage is referred to as interacting units.). Collective behavior is det...
Article
Full-text available
Target tracking with radar and sonar is done in either spherical or rectangular coordinates. Often tracking is done in one reference frame while filtering, usually Kalman, is done in another reference frame. It is commonly assumed that the probability density functions can be treated the same in both reference frames. An extended Kalman filter is u...
Article
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A study is performed of several multiple model tracking filter architectures that do not employ a Markov Switching Matrix in its weighting mathematics. The Markov Switching Matrix which is common to multiple model tracking filters does not have an "optimum" rule for defining its constituent probabilities. The only real constraint on the probabiliti...
Conference Paper
Full-text available
For many applications of radar and sensor based filtering, simulations can not represent the sole estimate of performance, provide points where threats become engagable, or determine when to use weapons' platform based sensors effectively in an engagement, etc... No significant advances have been proposed to analytically characterize performance or...
Article
Full-text available
Our work on the solution to the Rayleigh problem has concentrated on developing the theoretical and mathematical aspects to determining the probability density function or characteristic function for superpositions of random phases. In this paper, we apply the methodology to solve a number of practical problems of interest to radar engineers. The d...
Article
This paper provides a simplified solution to the general asynchronous track fusion problem of the authors. The original solution solves a practical sensor to sensor track fusion problem when the sensors used are asynchronous, communication delays exist between sensor platforms and track fusion center, and tracks may arrive out-of-sequence. The new...
Conference Paper
Full-text available
In this note we develop the methodology for computing the moments of the characteristic function for the superposition of sinusoidal transformed random variables. Thus we solve an important case of the Rayleigh problem, and point how to use this technique to completely solve it
Conference Paper
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In this paper, we discuss the symbolic and visual approach to combinations of noise plus noise. Just as in other application chaos is used to disguise noise, we suggest that their is a dual process where chaos may be used to amplify signals in noise. An example of this is presented and some of the implications are discussed
Conference Paper
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A brief introduction to the history of chaos and chaotic encryption is provided. Simple circuits that oscillate chaotically are reviewed. Methods of removing the masking produced by a single chaotic oscillator are reviewed. Methods that have been proposed to produce a secure encryption scheme using chaotic oscillators are described and evaluated. B...
Conference Paper
The construction and testing of a class of chaotic circuits corresponding to the differential equation xumldot = -Axuml - xdot + D(x) - alpha are described. In this equation D(x) represents the non-linear components that cause the circuit to oscillate chaotically. The suitability of circuits containing different non-linear elements as components in...
Conference Paper
When integrating track data across multiple platforms, one tries to create a super-platform that has performance capability which is greater than an individual platform. The performance of the super-platform ideally would be the superposition of the individual performances of the single platforms. From an estimation perspective, one of the key prob...
Conference Paper
Full-text available
In order to understand the problems intractable problems nature presents us, we are forced to make simplifications till we arrive at simple analytical models that we are capable of understanding. Such canonical models are rare, but useful as tools for constructing more realistic models that we can use analyze nature. The class of analytical models...
Article
Full-text available
This paper solves a general track fusion problem with feedback when the sensors used are asynchronous, communication delays exist between sensor platforms and track fusion center, and tracks may arrive out- of-sequence. For the proposed linear fusion rule, the solution is shown to be optimal in the minimum mean square sense. The batch processing of...
Conference Paper
Full-text available
In this note we show how to arrive at the characteristic function for sum of N sinusoidal random variables under the conditions that a characteristic function for the individual component can be calculated. Using this formalism, it is possible to generalize Huygen's principle to random variables with each point of space to be an antenna with a gain...
Article
Full-text available
This work solves a practical sensor to sensor track fusion problem when the sensors used are asynchronous, communication delays exist between sensor platforms and track fusion center, and tracks may arrive out-of-sequence. For the proposed linear fusion rule, the solution is shown to be optimal in the minimum mean square sense. The new fusion rule...
Article
Full-text available
In the Mandelstam–Tamm version of the time-energy uncertainty principle Δt denotes the infimum of time intervals that elapse before the change in the mean of any observable has the same magnitude as its standard deviation. We clarify this interpretation, and show that the infimum is achieved for certain observables and thus that this famous inequal...
Conference Paper
We introduce a notation of a random Fourier series based on a communication theory model of phase noise and discuss the relation between this model and those discussed in Marcus. Two notions of representation of a random variable in terms of one another are introduced and a ''uniqueness" theorem is introduced. Applications to different engineering...
Conference Paper
Research is becoming an integral part of the educational experiences of undergraduates. The University of Central Arkansas (UCA) has developed programs that provide authentic research experiences for undergraduate majors in its College of Natural Sciences in Mathematics. An investigation of chaotic encryption involves mentors from UCA and NSWCDD. C...
Article
The problem of prediction of target location and the role that this problem plays in the design and evolution of a modern combat system is analyzed. Connected with this problem is the task of choosing one set of hypotheses from among many viable sets. The functional dependence of the ability to make a prediction is discussed with particular attenti...
Article
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We discuss the concept of symmetry and its importance to understanding evolutionary processes that arise in dynamics. Using Curie’s principle that the cause must be at least as symmetric as the expect, along with Rosen’s generalizations, we arrive at a number of interesting results. These principles prove very useful in classifying quantum machine...
Conference Paper
Full-text available
To take advantage of recent and future advances in communications and information processing technology, military systems of the 21st century will create systems that enable the elements to operate as nodes in a geographically distributed information collection and processing network. Each network node is immersed in an information environment. An...
Conference Paper
Full-text available
The problem of modulated noise first arose in Lord Rayleigh's investigations of acoustical backscatter off of rough sea surfaces. The same problem occurs in radar when the electromagnetic waveform takes an indirect return or transmit path to a scatterer (target) and then is received as a noise corrupted signal at the radar receiver. The effect is t...
Article
Tracking maneuvering targets has always been a significant challenge to the tracking community, so new approaches to this problem are always being pursued. One approach is to use multiple model filters as an attractive design logic for both maneuver detection and filter re-initialization. Common current practice in multiple model tracking uses a sw...
Article
Full-text available
In this note we introduce the idea of adaptive scheduling based on a cost function methodology. As the warfare environment becomes more complex, individual sensor resources are stretched, and the usage of the sensors has grown. In a multi-ship multi-platform environment, one has the potential to share information across platforms. This would dramat...
Article
Full-text available
Tracking maneuvering target using multiple models is an attractive approach that is an alternative to a design that needs logic for both maneuver detection and filter re-initialization. Common current practice in multiple model tracking uses a switching Markov model. Recently the authors developed a multiple model approach to tracking maneuvering t...
Conference Paper
Full-text available
Presented in this paper is a solution to the Lyapunov equation in constant gain filtering. A specific method for deriving the noise reduction ratios for the alpha-beta and alpha-beta-gamma filters are explored using the Lyapunov equation. This enables us to simplify the computation drastically. For a 2 × 2 matrix, one has to solve for one unknown i...
Conference Paper
Full-text available
There are three commonly used relationships between alpha and beta that are reported in the literature: Kalata, Benedict-Bordner, and continuous white noise. The Kalata relation is obtained from steady state Kalman filter theory assuming zero mean white noise in the position and velocity state equations. The Benedict-Bordner relation is derived bas...
Conference Paper
The treatment of propagation in a linear dispersive medium is a problem that is outlined in many electromagnetic texts which consider the continuous time case. These texts don't deal with initial value problems, but instead refer to Stratton for a complete treatment of the problem. Stratton uses both the Fourier transform and the Laplace transform...
Conference Paper
To take advantage of recent and future advances in communications and information processing technology, military systems of the 21st century will create systems that enable the elements to operate as nodes in a geographically distributed information collection and processing network. Each network node is immersed in its local information environme...
Article
Full-text available
Discussed is a number of mechanisms for complex systems that can lead to unexpected behavior. All are related to graph theoretic models of interactions. In addition, the theory of random interval graphs can be applied to the characterization of simultaneously occurring variable finite length events. The theory is sufficiently general to provide sim...
Article
Full-text available
A particular method of detecting unresolved targets using simulated generic monopulse radar data is examined in detail. The system is assumed to be incorrectly calibrated i.e. the decision boundary is calculated based on erroneous values governing the hypothesis that only a single target is present in the range cell. The system performance is analy...
Article
Full-text available
The effect of nonuniform motion on radar waveforms is discussed. By considering the physical model of a perfectly reflecting mirror with an arbitrary law of motion r(t), it is possible to determine the functional form of the scattered wave at the receiver for any waveform in general. The particular example of an interrupted continuous wave waveform...
Article
Recently the authors developed a new filter that uses data generated by asynchronous sensors to produce a state estimate that is optimal in the minimum mean square sense. The solution accounts for communications delay between sensors platform and fusion center. It also deals with out of sequence data as well as latent data by processing the informa...
Article
Tracking maneuvering target using multiple models is an attractive approach that is an alternative to a design that needs logic for both maneuver detection and filter re-initialization. Common current practice in multiple model tracking uses a switching Markov model. This paper also uses a multiple model approach to tracking maneuvering targets, bu...
Article
Full-text available
In this report, we consider two classes of threats: Anti-Ship Cruise Missiles (ASCMs) and Tactical Ballistic Missiles (TBMs). ASCMs are treated as being capable of maneuver, while the trajectory of a TBM is determined solely by its initial state. A threat is evaluated according to two basic criteria: the amount of damage it can potentially cause, a...
Article
Full-text available
The Doppler effect is a widely treated phenomena in both radar and sonar for objects undergoing uniform motion. There are many different models (Censor has written a history of the subject) one can use to derive the Doppler effect. The treatment of non-uniform motion is not widely discussed in the literature of radar and sonar. Some authors argue i...
Article
What underpins this vision as axiomatic is the mantra information is power. Besides the necessary requirement of information exchange networks with sufficient bandwidth and computational power to treat the data being passed around the network; algorithms are required to make sense of the data. It is estimation algorithms that turn the straw (data)...
Article
Sensor data fusion has long been recognized as a means to improve target tracking. Common practice assumes that the sensors used are synchronous, i.e., they have identical data rate, measurements are taken at the same time, and have no communication delays between sensors platform and central processing center. Such assumptions are invalid in pract...
Conference Paper
Full-text available
For pt.I see National Radar Conference, Atlanta Ga, May 2001). In this paper, the future problems of kill assessment in the exo-atmosphere is discussed. The concentration is on characterizing the Doppler spectrum of objects that are undergoing non-uniform motion. Then a data collection experiment is discussed for gathering exo-atmospheric Doppler d...
Conference Paper
Full-text available
The purpose of this investigation was to determine what approach is best to deal with the question of how to choose the model transition matrix for the interactive multiple model filter. The desire is to determine an approach to choosing the model transition matrix that is "best" in some sense. In the study, many target tracking simulations were ru...
Conference Paper
We often encounter problems that involve either nonlinear or multi-dimensional combinations of random variables. One would like to be able to determine the probability density function of these combinations. The individual components that contribute to the combination are assumed known, but the combination is not known. Computations to determine th...
Conference Paper
Typically, the tracking and estimation community has been concerned with the direct problem of tracking a threat in a near optimal fashion. The usage of secondary quantities or derived quantities from state estimates that can be used for other purposes than tracking have been ignored. An example of a derived quantity that is useful directly for tra...
Article
Full-text available
The role of linearity in the definition of entropy is examined. While discussions of entropy often treat extensivity as one of its fundamental properties, the extensivity of entropy is not axiomatic in thermodynamics. It is shown that systems in which entropy is an extensive quantity are systems in which a entropy obeys a generalized principle of l...
Article
Sensor data fusion has long been recognized as a means to improve target tracking. Common practice assumes that the sensors used are synchronous, i.e., they have identical data rate, measurements are taken at the same time, and have no communication delays between sensors platform and central processing center. Such assumptions are invalid in pract...
Article
Full-text available
Sensor data fusion has long been recognized as a means to improve target tracking. Common practice assumes that the sensors used an synchronous, i.e., they have identical data rate, measurements are taken at the same time, and have no communication delays between sensors platform and central processing center. Such assumptions are invalid in practi...
Article
Full-text available
This report presents a general solution to Constant Gain Tracking Filters, which are nothing more than matrix difference equations. These equations can be solved regardless of the measurement model provided the model is deterministic. The general solution can then be used to compute the covariance matrix under very general assumptions about the noi...
Conference Paper
This paper examines the propensities of distributed control systems to lapse into temporally and topologically decoupled states and recommends a generally viable control methodology for distributed systems which reduces this propensity
Conference Paper
There are three problems that arise from attempts to characterize the Rayleigh problem that are of interest to computer scientists: characterizing noise distributions that occur in real time scheduling of computer resources, distinguishing between pseudo-random and random signals that occur on networks (the Internet for example) and testing the goo...
Conference Paper
Both current and future problems of kill assessment are discussed from a surface navy perspective. The basic terminology of kill assessment is discussed in the introduction, which provides the pedestrian background to the common vocabulary. Some of the underlying sources of potential kill assessment information and aspects of current decision takin...
Conference Paper
In this paper, we develop a methodology for comparing filter gain selection for several different filter coefficient relationships. When designing tracking filters, an important issue is the selection of the filter gains. The steady state solution of the Kalman filter leads to different filter coefficient relationships depending on what one assumes...
Conference Paper
A formalism to determine the probability density function (PDF) resulting from a coordinate transformation applied to an arbitrary PDF is developed. The results are applied to the coordinate transformation used in target tracking, specifically the transformation from spherical to Cartesian coordinates. This result is then applied to the specific ex...
Article
Full-text available
Two explanations of how to obtain electromagnetic waves with E//B are presented. The nature of these solutions is discussed, with a caveat on what constitutes a general solution. Finally, an example of an electromagnetic wave with the electric field parallel to magnetic field is presented.
Conference Paper
In this note, we determine several different ways of relating the tracking index of Kalata (1984) to a cost function optimization method of calculating the filter coefficients. The cost function method can be used to define a tracking index. Thus, we demonstrate the relationship between the two methods
Conference Paper
The general solution of the equations that arise in constant coefficient filter theory are solved for arbitrary number of dimensions. Namely, we solve the first order matrix equation for the special case of scalar measurements. We also determine the covariance matrix for the general case. We then illustrate these results for the specific case of an...
Conference Paper
The notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al. (Phys. Rev A 41,11 (1990)), hereafter referred to as AAV. We have performed an experiment similar to one proposed by Knight et al. (1990) and demonstrated that weak values can be observed and that they exhibit the properties described by...
Conference Paper
Presents a method of gain adjustment for an alpha-beta filter when data points are lost or when the tracking interval changes. The steady-state position and velocity lags are first derived for a step acceleration input. The standard predictor-corrector forms of the filter equations are algebraically rearranged into two uncoupled difference equation...