Recursive Difference Equations Research Articles

Recursive filtering for two-dimensional systems with amplify-and-forward relays: Handling degraded measurements and dynamic biases

This paper is concerned with the recursive filtering issue for a type of two-dimensional shift-varying systems communicated via imperfect networks. The communication network undergoes degraded measurements, stochastic biases, and channel noises, all reflecting real-world constraints. The degraded measurements are governed by three random variable sets with known statistical details, while the stochastic biases follow specific dynamic equations. An amplify-and-forward (AF) relay is implemented to bolster the reliability of transmission from the sensor to the distant filter. The main goal of this study is to formulate an AF relay-based recursive filtering algorithm that can predict system states with the desired performance specification. By employing the inductive technique and meticulous stochastic analysis, an upper bound for the filtering error variance is initially set based on the solutions to recursive difference equations, following which the filter gain is intricately designed to minimize this upper bound at each point in time. Furthermore, a theoretical analysis is conducted on the effect of degradation coefficients on the filtering performance. Finally, the effectiveness of the proposed recursive filtering approach is demonstrated using a numerical example.

RETRACTED: Analysis of governing equations of multibody dynamic systems through kinematic properties of joints and effects of kinematic chains on closed-form mechanism

In this investigation, the fundamental equations dictating the behavior of dynamic systems were ascertained by means of an innovative approach that amalgamated the kinematic properties of joints and the intricate kinematic chains of multibody systems into a series of governing equations. Subsequently, the equations that govern the behavior of multibody systems were converted into ordinary differential equations (ODEs) utilizing the calculus of matrix-valued functions. This computational technique is adept at swiftly acquiring recursive differential equations of motion for multibody systems. Thus, the expenditure of computational costs associated with the simulation was efficaciously minimized. The mechanisms of Andrew's squeezing and carpet scrapping were employed in conjunction with kinematic constraints to authenticate the validity of the proposed methodology. The findings suggest that, notwithstanding the fact that the root mean square error (RMSE) in the generalized coordinates between the two ODE and differential algebraic equation (DAE) techniques is small in the Andrew's squeezing mechanism (at approximately 0.02) and in the carpet scraping mechanism (at approximately 0.7), there are still discrepancies between the two methods. The degree of convergence in the aforementioned methodologies, despite the slight RMSE difference between them, shows a clear disparity. By way of illustration, when employing the conventional DAE technique, the duration required to resolve the Andrew's squeezing and carpet scraping mechanisms is 5 and 11.34 s, respectively. However, by utilizing the ODE methodology in the aforementioned mechanisms, the time required to determine a solution is truncated to 1.19 and 2.1 s, sequentially.

Boundedness Character of the System of Recursive Difference Equations

In this paper, we take into consideration the boundedness character of positive solutions of the difference system $x_{n}=\alpha +\prod \limits_{i=1}^{k}y_{n-i}^{a_{i}} $ $y_{n}=\beta +\prod \limits_{i=1}^{k}x_{n-i}^{b_{i}},$ where $a_{i},b_{i}\in \mathbb{R}$, $i=\overline{1,k},$ $a_{k}\neq 0,$ $b_{k}\neq 0$ and $\alpha $ and $\beta $ are nonnegative real numbers.

Simultaneous State and Unknown Input Estimation for Complex Networks With Redundant Channels Under Dynamic Event-Triggered Mechanisms.

This article addresses the simultaneous state and unknown input estimation problem for a class of discrete time-varying complex networks (CNs) under redundant channels and dynamic event-triggered mechanisms (ETMs). The redundant channels, modeled by an array of mutually independent Bernoulli distributed stochastic variables, are exploited to enhance transmission reliability. For energy-saving purposes, a dynamic event-triggered transmission scheme is enforced to ensure that every sensor node sends its measurement to the corresponding estimator only when a certain condition holds. The primary objective of the investigation carried out is to construct a recursive estimator for both the state and the unknown input such that certain upper bounds on the estimation error covariances are first guaranteed and then minimized at each time instant in the presence of dynamic event-triggered strategies and redundant channels. By solving two series of recursive difference equations, the desired estimator gains are computed. Finally, an illustrative example is presented to show the usefulness of the developed estimator design method.

Set-Membership Filtering Subject to Impulsive Measurement Outliers: A Recursive Algorithm

This paper is concerned with the set-membership filtering problem for a class of linear time-varying systems with norm-bounded noises and impulsive measurement outliers. A new representation is proposed to model the measurement outlier by an impulsive signal whose minimum interval length (i.e., the minimum duration between two adjacent impulsive signals) and minimum norm (i.e., the minimum of the norms of all impulsive signals) are larger than certain thresholds that are adjustable according to engineering practice. In order to guarantee satisfactory filtering performance, a so-called parameter-dependent set-membership filter is put forward that is capable of generating a time-varying ellipsoidal region containing the true system state. First, a novel outlier detection strategy is developed, based on a dedicatedly constructed input-output model, to examine whether the received measurement is corrupted by an outlier. Then, through the outcome of the outlier detection, the gain matrix of the desired filter and the corresponding ellipsoidal region are calculated by solving two recursive difference equations. Furthermore, the ultimate boundedness issue on the time-varying ellipsoidal region is thoroughly investigated. Finally, a simulation example is provided to demonstrate the effectiveness of our proposed parameter-dependent set-membership filtering strategy.

Distributed Fusion Estimator for Multisensor Multirate Systems With Correlated Noises

A distributed fusion estimation algorithm is studied for multisensor multirate systems with correlated noises, where the state update rate is positive integer multiples of the measurement sampling rates and different sensors sample uniformly with different sampling rates. The measurement noises from different sensors are correlated with each other and are also correlated with the process noise. First, the state space model is established at the measurement sampling points (MSPs). Then, the optimal local filter at the MSPs and optimal local estimators (LEs) at the state update points are presented by an innovation analysis approach, respectively. Moreover, the cross-covariance matrices of estimation errors between any two LEs are derived, which involves three jointly recursive difference equations. At last, a distributed fusion estimator is proposed by applying the matrix-weighted fusion estimation algorithm in the linear minimum variance sense. The stability of the proposed algorithms is analyzed. Simulation results illustrate the effectiveness of the algorithms.

Implementation of a cusp-like for real-time digital pulse shaper in nuclear spectrometry

Pulse shaping, which improves signal-to-noise ratio excellently, has been extensively used in nuclear signal processing. This paper presents a cusp-like pulse-shaping technique developed through the recursive difference equation in time domain. It can be implemented in field programmable gate array hardware system. Another flat-topped cusp-like shaper is developed to optimize the time constant of pulse shaping and reduce the influence of ballistic deficit. The methods of both baseline restoration and pile-up rejection are described. The 137Cs energy spectra measured with the digital cusp-like shaper are 6.6% energy resolution, while those by traditional analog pulse shaper are 7.2% energy resolution, under the same conditions. This technique offers flexibility, too, in adjusting the pulse shaper parameters.

Poisson-Type Processes Governed by Fractional and Higher-Order Recursive Differential Equations

We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. $\partial_tp_k(t)=-\lambda(p_k(t)-p_{k-1}(t))$, $k\geq0$, $t>0$ by introducing fractional time-derivatives of order $\nu,2\nu,\ldots,n\nu$. We show that the so-called "Generalized Mittag-Leffler functions" $E_{\alpha,\beta^k}(x)$, $x\in\mathbb{R}$ (introduced by Prabhakar [24] )arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for $t\to\infty$. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter $\nu$ varying in $(0,1]$. For integer values of $\nu$, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.

SU‐FF‐T‐38: A Recursive Differential Equation Approach Towards the Problem of Estimating the Probability of M Surviving Clonogens After the Ith Irradiation

Purpose: A model is developed that describes the tumor dynamics of a repopulating tumor during a fractionated external radio treatment. Method and Materials: An iterative algorithm is constructed for the calculation of the probability distribution of the surviving number of tumor cells for any point in time during the treatment and for post treatment times. This is accomplished through the assumption, that cell repopulation takes place only during the intervals between the fractions of irradiation. Thus, the processes of cell repopulation and of radiation‐induced cell kill are considered decoupled and are therefore modeled separately. Binomial statistics is applied to describe the radiation induced cell damage. Natural death is ignored. Tumor repopulation is modeled through the construction of differential equations for the probability that a certain number of cells exist at a certain point in time. The differential equations are solved in a straightforward manner. Results: The probability of zero survivals is compared numerically to the Zaider‐Minerbo tumor control probability (TCP) model for an arbitrarily chosen regime of treatment. An excellent agreement is found to exist between the two expressions. The Poisson approximation of the probability distribution of the number of cells is also studied and it is compared to the proposed method for several different combinations of the model parameter values. Conclusion: It is concluded that the Poisson approximation works well only for slowly repopulating tumors and for rather high cell radiosensitivity (i.e. low cell survival probability after a fraction of radiation). Therefore, the use of the exact method for the description of the radiation response of repopulating tumors is recommended.

On a (2,2)-rational recursive sequence

We investigate the asymptotic behavior of the recursive difference equation yn+1 = (α+βy n )/(1+yn-1) when the parameters α < 0 and β ∈ ℝ. In particular, we establish the boundedness and the global stability of solutions for different ranges of the parameters α and β. We also give a summary of results and open questions on the more general recursive sequences yn+1 = (a + by n )/(A + Byn-1), when the parameters a, b, A, B ∈ ℝ and abAB ≠ 0.

Chandrasekhar-type recursive Wiener filter using covariance information in linear discrete-time wide-sense stationary stochastic systems

This paper designs the Chandrasekhar-type recursive Wiener filter for the white observation noise in linear discrete-time wide-sense stationary stochastic systems. The system matrix in the state-space model of the signal, the crossvariance function of the state variable of the signal with the observed value, the observation matrix for the signal, the variance of the white observation noise and the observed value are assumed to be known. In particular, this paper extends the Chandrasekhar-type recursive Wiener filter for a scalar observation equation to the case of a vector observation equation. The characteristic of the Chandrasekhar-type filter is to calculate the filter gain directly by solving the recursive difference equations.

Global asymptotic stability of a nonlinear recursive sequence

A sufficient condition is obtained to guarantee the global asymptotic stability of the following nonlinear recursive difference equation: xn+1=xnxn−1b+xn−2b+axn−1b+xnxn−2b+a,n=0,1,2,…, where a, b [0, ∞] and the initial values x−2, x−1, x0 (0, ∞). Some known results are included and improved.

Leaky Delayed LMS Algorithm: Stochastic Analysis for Gaussian Data and Delay Modeling Error

This paper presents a stochastic analysis of the delayed least-mean-square (DLMS) adaptive algorithm with leakage. This analysis is obtained taking into account that mismatches between the system delay and its estimate may occur. Such an approach is not considered in previous models. In addition, it is shown that the introduction of a leakage factor into the adaptive algorithm keeps the adaptive algorithm stable under an imperfect delay estimate condition. Recursive difference equations for the first and second moments of the adaptive filter weights are derived. An expression for the critical value of the step size is also determined. Results of Monte Carlo simulations present excellent agreement with the proposed model for both white and colored Gaussian inputs.

Global asymptotic stability for two recursive difference equations

Two sufficient conditions are obtained for the global asymptotic stability of the following two recursive difference equations x n+1= x nx n−1+x n−2+a x n+x n−1x n−2+a , n=0,1,2,…, and x n+1= x n−1+x nx n−2+a x nx n−1+x n−2+a , n=0,1,2,…, where a∈[0,∞) and the initial values x −2, x −1, x 0∈(0,∞). Some known results are included.

A novel method for designing allpass digital filters

A novel method for designing recursive allpass digital filters is presented. This method uses cepstral coefficients to design the allpass filter. For a stable allpass filter, the denominator polynomial must be of minimum phase. For a minimum phase polynomial the magnitude function and the group delay function are related through the cepstral coefficients. Hence, given the group delay function, the cepstral coefficients corresponding to the denominator polynomial part are first determined. From the cepstral coefficients the minimum phase denominator polynomial coefficients are determined through a nonlinear recursive difference equation.

Optimum phase-acquisition technique for charge-pump PLL

In this paper, we propose a new optimum phase-acquisition algorithm controlling the loop gain of a charge-pump PLL (CP-PLL) in the sense of the MMSE criterion. A set of recursive difference equations minimizing RMS jitter of output phase is derived to obtain an optimum gear-shifting sequence with a zero-phase start (ZPS) assumption. It is shown that the optimum gear-shifting sequence is independent of the variance of the input phase jitter. A procedure for applying this sequence to the design of CP-PLL circuits is described, Both behavoral simulation and HSPICE circuit-level simulation demonstrate that the proposed design leads to an efficient CP-PLL having both fast acquisition and significant jitter reduction characteristics. The optimal gear-shifting CP-PLL outperforms the conventional CP-PLLs. These methods can be used for clock recovery applications such as data communication receivers, disk drive read/write channels, and local area networks, as well as for other applications requiring very short initial preamble periods.

Recursive Estimation and Difference Equations

Recursive Estimation and Difference Equations

Limit cycle bounds for floating point implementations of second-order recursive digital filters

It is shown that floating point realizations of linearly stable systems can exhibit four fundamental types of free responses. Sufficient conditions for the existence or nonexistence of some of these periodic response types in a given system are presented. Explicit closed form conditions on the mantissa length to guarantee certain limit cycle bounds are provided. The effect of various floating point arithmetic reformatting schemes on the convergence of recursive difference equations is also addressed. Truncation and rounding quantization schemes as well as double- and single-length product mantissa schemes are analyzed and compared. Although the method introduced is applicable to any digital filter realization implemented in floating point format, the analysis focuses on the zero-input behavior of second-order direct-form digital filters.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Filtering of a semi-Markov queueing system with retrials

A recursive differential equation for the conditional probability of the number of waiting customers is derived for the MwrM|G|1 queue with continuous monitoring of the server state.

Periodically time-varying two-dimensional systems

A two-dimensional (2-D) single-input/single-output (SISO) system represented by a recursive linear difference equation with periodically varying coefficients is considered. An equivalent multiple-input/multiple-output (MIMO) time-invariant realisation is then derived, whose structure is suitable for VLSI realisation.