Constant Coefficient Linear Differential Equations Research Articles

Digital Film Music Creation Model Based on Inhomogeneous First Order Constant Coefficient Linear Differential Equation

Abstract Computer music production is an essential branch of computer technology in artistic creation. This is an interdisciplinary subject with information science and art. This paper proposes a creation model for digital music with first-order inhomogeneous first-order constant coefficient linear differential equations. This mode includes acoustic modes for waveform envelopes and bands. It has a straight-line mapping of phonetic patterns and rhythmic patterns. Then this paper uses the signal reconstruction principle and iterative extrapolation method to restore the signal in the frequency domain to obtain the simulated music. The experimental results show that this digital music creation model of inhomogeneous first-order constant coefficient linear differentiation can save a lot of labor costs. And this way can improve the audio quality and output effect. Compared with the undetermined factor method and the Laplace transform method in conventional advanced mathematics, the inhomogeneous first-order constant coefficient linear differentiation is more convenient in music composition.

Design of discrete-time matrix all-pass filters using subspace Nevanlinna pick interpolation

Unitary matrix-valued functions of frequency are matrix all-pass systems, since they preserve the norm of the input vector signals. Typically, such systems are represented and analyzed using their unitary-matrix valued frequency domain characteristics, although obtaining rational realizations for matrix all-pass systems enables compact representations and efficient implementations. However, an approach to obtain matrix all-pass filters that satisfy phase constraints at certain frequencies was hitherto unknown. In this paper, we present an interpolation strategy to obtain a rational matrix-valued transfer function from frequency domain constraints for discrete-time matrix all-pass systems. Using an extension of the Subspace Nevanlinna Pick Interpolation Problem (SNIP), we design a construction for discrete-time matrix all-pass systems that satisfy the desired phase characteristics. An innovation that enables this is the extension of the SNIP to the boundary case to obtain efficient time-domain implementations of matrix all-pass filters as matrix linear constant coefficient difference equations, facilitated by a rational (realizable) matrix transfer function. We also show that the derivative of matrix phase constraints, related to the group delay at the interpolating points, can be optimized to control the all-pass transfer matrices at the unspecified frequencies. Simulations show that the proposed technique for unitary matrix filter design performs as well as traditional DFT based interpolation approaches, including Geodesic interpolation and the popular Givens rotation based matrix parameterization.

Differential equations defined by (convergent) Laurent series

The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to (convergent) Laurent series. It is thought that these more general differential equations can be used as an alternative description of some special functions.

Nonuniform Sampling Rate Conversion:An Efficient Approach

We present a discrete-time algorithm for nonuniform sampling rate conversion that presents low computational complexity and memory requirements. It generalizes arbitrary sampling rate conversion by accommodating time-varying conversion ratios, i.e., it can efficiently adapt to instantaneous changes of the input and output sampling rates. This approach is based on appropriately factorizing the time-varying discrete-time filter used for the conversion. Common filters that satisfy this factorization property are those where the underlying continuous-time filter consists of linear combinations of exponentials, e.g., those described by linear constant-coefficient differential equations. This factorization separates the computation into two parts: one consisting of a factor solely depending on the output sampling instants and the other consists of a summation -- that can be computed recursively -- whose terms depend solely on the input sampling instants and its number of terms is given by a relationship between input and output sampling instants. Thus, nonuniform sampling rates can be accommodated by updating the factors involved and adjusting the number of terms added. When the impulse response consists of exponentials, computing the factors can be done recursively in an efficient manner.

Detection of transient signals based on the tricepstrum

In this paper, the problem of detecting transient signals of unknown waveforms and arrival times embedded in white Gaussian noise is addressed. The use of the cepstrum coefficients of the fourth order correlations of the transient signal for forming a detection statistic is demonstrated. It is considered both a batch and adaptive approach for the detection of the signal which is assumed to satisfy a linear constant coefficient difference equation. The adaptive approach is a least squares realization based on Q–R decomposition of the fourth order statistics matrix involved in the computation of the cepstrum coefficients. Their performance is compared in terms of probability of detection and probability of false alarm with the conventional energy detector by means of Monte-Carlo simulations and improved performance is demonstrated. It is shown that the adaptive approach allows for detection of short length transients which are of unknown arrival times using a single data record even before the whole amount of data becomes available. The Q–R decomposition offers good numerical properties when fourth order statistics are involved and in contrast to the energy detector the proposed receiver has false alarm rate independent of the noise if it is of zero mean and independent, identically, distributed (i.i.d).

DiffEq: A MATLAB-Based Discrete-Time System Simulator [Testing Ourselves

A MATLAB-based discrete-time system simulator (DiffEq) is introduced. This tool investigates linear constant-coefficient difference equations, which represent linear time-invariant finite-dimensional discrete-time systems. Closed-form analytical solutions can be expressed as a superposition of homogeneous and particular solutions, and this distinction gives valuable insights into the transient and steady-state behavior of the system. The transfer function, impulse response, characteristic roots, and simulation diagram of the system are obtained for different input types and initial conditions.

Characterising solution sets of LTI differential equations

We prove that a set of smooth trajectories is LTID (i.e., the solution set of a constant coefficient linear differential equation) if and only if it is the direct sum of a linear time-invariant closed controllable part and a linear time-invariant finite-dimensional part. This characterisation does not directly involve derivatives in its formulation. It solves a problem opened by Willems (1991). We also characterise morphisms between LTID sets as linear time-invariant maps which do not increase support.

Linear Systems and Taylor Complexes

A Taylor complex is a graded collection of polynomials satisfying a few simple axioms. We show that there is a natural one-to-one correspondence between the solution sets of linear constant coefficient differential equations and Taylor complexes. Using this correspondence, we then give a characterization of the solution sets of linear constant coefficient differential equations, which follows closely Willems' characterization of the solution sets of linear constant coefficient difference equations. We also characterize linear differential operators. (The presentation is given for the case of several variables.)

Hardware Approach of a Multipurpose Finite Impulse Response Filter for Real-Time Filtering Applications

Problem statement: Finite Impulse Response (FIR) filters are widely u sed in various DSP applications. The design of digital FIR filters is a very basic problem in digital signal processing. A FIR filter with multiple operation capability is ce rtainly very useful for any real-time filtering applications. This article presents a multipurpose FIR filter design modeled by the hardware description language VHDL for real-time filtering a pplication. Approach: The VHDL has its concept of concurrency to cope with the parallelism of digi tal hardware. The novel feature is the capability o f the design to accomplish up to 127variable filter o rder and an arbitrary filter frequency response. Th e coefficients are calculated by Hamming windowing technique. Basing on selection embedded in the design, the model is able to execute highpass, lowp ass, bandstop and bandpass filtering operations. It is set at 8-bit signed data processing. To filter t he input data in time domain, Linear Constant Coefficient Difference Equation (LCCDE) is used by the filter. Results: The design outputs are validated through simulation and compilation. The o utput results are also compared with the MATLAB implemented calculated output results to test the correctness that proves the effectiveness of the design. Conclusion: With the capability of filtering signal in real ti me mode utilizing arbitrary filter shape, the multipurpose filter proves to be versatile.

Relative completeness and specifiedness properties of continuous linear dynamical systems

The properties of relative completeness and specifiedness are introduced for (continuous) linear dynamical systems. It is shown then that having these two properties and the differentiation-invariance property is necessary and sufficient for a (continuous) linear dynamical system to be represented by means of a linear constant coefficient differential equation.

A note on the Laplace transform method for initial value problems

This note provides a tutorial treatment of the basic (unilateral) Laplace transform method for solving initial value problems for linear constant coefficient differential equations with terms that contain derivatives of the input. The input is allowed to contain jump discontinuities (steps) and so it would first appear that one needs the advanced machinery of generalized functions (distributions) to apply the Laplace transform method here. This is, however, not the case. It suffices to use the basic Laplace transform for ordinary functions, where the integral is interpreted as the most basic integral notion in calculus, i.e. as a Riemann integral. There is thus no need to specify any convention on the lower limit of integration as they will all give the same transform. Furthermore, it suffices to use the derivative rule in the basic form L[f′](s)=sL[f](s)−f(0) for smooth enough differentiable functions. Finally, we point out important limitations of the bilateral Laplace transform method in input–output stability analysis of feedback systems.

Cross-Track Motion of Satellite Formations in the Presence of J2 Disturbances

Introduction T HIS Note augments and completes a previous work1 by the authors in which a set of constant coefficient linear differential equations were developed that describe the relative motion between two satellites in the presence of the Earth’s J2 geopotential. In the previous work, the motion was separated into in-plane and cross-track motion. The analysis of the in-plane motion was relatively straightforward and remains untouched in this Note. The cross-track motion was more difficult to derive; linearizations with respect to time were used to develop analytical solutions. In this Note, the equations are instead simplified using small angle approximations. These simplifications are valid due to the very small angles that describe the cross-track motion. Although no longer linear, the equations can still be solved analytically. The result is a set of differential equations that are more accurate and more intuitive.

Formulas for the distribution of sums of independent exponential random variables

This paper derives a new type of formula for the probability that, among a collection of items with s-independent exponential times to failure, a certain subset of them fails in a given order before a certain time, and all the remaining items survive beyond that time. This formula is in the form of a power series that satisfies a certain constant coefficient linear differential equation with specified initial conditions. This provides an alternative to existing closed-form formulas of the exponomial variety, viz., a nonlinear combination of exponential terms, where the coefficients of the exponential terms are polynomials in the mission time. Some results are given which quantify the computation effort required to achieve a specified accuracy using partial sums of the infinite series; a simple example illustrates these results. This approach can be very efficient for system reliability analysis where the product of the mission time and the sum of the failure rates down any path leading to system failure is small. Further work is needed to expand the practical applicability of this approach to cases where some rates are large and/or the mission time is long.

Hamiltonian and Variational Linear Distributed Systems

We use the formalism of bilinear- and quadratic differential forms in order to study Hamiltonian and variational linear distributed systems. It was shown in [1] that a system described by ordinary linear constant-coefficient differential equations is Hamiltonian if and only if it is variational. In this paper we extend this result to systems described by linear, constant-coefficient partial differential equations. It is shown that any variational system is Hamiltonian, and that any scalar Hamiltonian system is contained (in general, properly) in a particular variational system.

A natural vector space of two-sided discrete-time signals

The vector space of two-sided sequences expressible as finite sums of unit sample sequences, complex exponential sequences, the unit step sequence, and their elementwise products forms a natural space for the theory of discrete-time linear shift-invariant operators. This space is (algebraically) closed under translation, elementwise multiplication, and solution of linear constant-coefficient difference equations. Every sequence in this space can be uniquely decomposed into a sum of elementary sequences. An unexpected consequence of this decomposition is that the impulse response and frequency response of an LSI operator can be specified independently.

Nonlinear control of mechatronic systems with permanent-magnet DC motors

The current trends in development and deployment of advanced electromechanical systems have facilitated the unified activities in the analysis and design of state-of-the-art motion devices, electric motors, power electronics, and digital controllers. This paper attacks the motion control problem (stabilization, tracking, and disturbance attenuation) for mechatronic systems which include permanent-magnet DC motors, power circuity, and motion controllers. By using an explicit representation of nonlinear dynamics of motors and switching converters, we approach and solve analysis and control problems to ensure a spectrum of performance objectives imposed on advanced mechatronic systems. The maximum allowable magnitude of the applied armature voltage is rated, the currents are limited, and there exist the lower and upper limits of the duty ratio of converters. To approach design tradeoffs and analyze performance (accuracy, settling time, overshoot, stability margins, and other quantities), the imposed constraints, model nonlinearities, and parameter variations are thoroughly studied in this paper. Our goal is to attain the specified characteristics and avoid deficiencies associated with linear formulation. To solve these problems, an innovative controller is synthesized to ensure performance improvements, robust tracking, and disturbance rejection. One cannot neglect constraints, and a bounded control law is designed to improve performance and guarantee robust stability. The offered approach uses a complete nonlinear mechatronic system dynamics with parameter variations, and this avenue allows one to avoid the conservative results associated with linear concept when mechatronic system dynamics is mapped by a linear constant-coefficient differential equation. To illustrate the reported framework and to validate the controller, analytical and experimental results are presented and discussed. In particular, comprehensive analysis and design with experimental verification are performed for an electric drive. A nonlinear bounded controller is designed, implemented, and experimentally tested.

Dead-beat tracking of complex signals

The dead-beat response problem of discrete-time systems was thoroughly studied in the past within the framework of the control systems theory, but has been traditionally restricted to prototype time-domain inputs only. This article presents an extension to the tracking of complex signals that can be generated by systems described by homogeneous linear constant-coefficient difference equations. The proposed approach takes advantage of a general theory for the computation of the response of a linear system to time-domain inputs and the moment-and-cumulant theory for a discrete-time signal. >

Response of blood flow entering the heart to dissynchronous ventricular recoil

To account for mechanical suction at the initiation of the cardiac filling process, we have previously modeled transmitral blood flow driven by synchronous ventricular recoil in terms of a one-dimensional simple harmonic oscillator (SHO) in the Parametrized Diastolic Filling (PDF) formalism. PDF model predictions of blood flow velocity have shown excellent agreement with the diastolic Doppler velocity profile (DVP), the echocardiographically measured blood flow velocity of filling. However, dissynchrony of the cardiac recoil process, defined by the early or late recoil of one region relative to the rest of the ventricle, is known to exist in-vitro and has been observed in-vivo. To model dissynchronous recoil and mathematically explore the range of model predicted blood flow velocity responses, we have developed a new model. Our two-dimensional model utilizes two second order constant coefficient linear differential equations, whose solutions combine to form v( t), the new model's prediction of blood flow velocity. The differential equations are solved as initial value problems with zero initial velocities, unknown initial displacements and unknown damping and spring constants. For a given data set (DVP) we solve the ‘inverse problem’ and determine initial oscillator displacements and parameters iteratively by graphical comparison of v( t) to the DVP contour. We discuss the kinematic features of this refined model, its testable predictions, and provide evidence for its validity by comparison of v( t) to the clinical DVP in a case of known dissynchrony. We comment on the benefits and limitations of this modeling approach.

A link between the Fourier and Heaviside representations of the square wave

By Laplace transforming the Fourier expansion of the odd periodic extension of the square wave and then taking inverse transforms on the result, we arrive at a representation of this periodic function as an infinite combination of delayed step functions. This helps to clarify the connection between solutions obtained for constant coefficient linear differential equations with periodic step forcing functions, when using, on the one hand, the Fourier Series approach and on the other, the Laplace transform [1], [2].

High-Speed Motion Control of Mechanisms under Average Heat Generation Restriction. 3rd Report. Trajectory Planning Method of Three-Degrees-of-Freedom SCARA-Type Robotic Manipulator with Short Calculation Time.

The objective of this paper is to solve the minimum-time trajectory planning problem of a robotic manipulator under both the average heat generation restriction and the intermediate state variable restriction. We have already proposed a new numerical method to solve this problem in which the trajectory is approximated by the fifth-order Hermite polynomial expression. In analyzing a many-degrees-of-freedom manipulator, however, this method cannot be carried out within a short calculation time because of its duplicated iterative calculation procedure. To reduce the calculation time, we investigated the properties of the method and found two useful characteristics. The first characteristic is that the duplicated iterative calculation is not necessary when the robotic manipulator is a conservative system in a mechanical sense. The second one is that the polynomial solution can be derived when the robotic manipulator is governed by the constant coefficient linear differential equations. Using these characteristics, we constructed the trajectory planning method, which can generate the trajectory of the three-degrees-of-freedom SCARA-type robotic manipulator within a short calculation time. Numerical examples were shown to prove that the proposed method was practical enough to be applied to industrial robotic manipulators.