Signal approximation with Pascal’s triangle and sampling

Chapter 1 - Continuous-Time Signals

• A unified continuous-time framework is developed for enterprise-wide scheduling and planning optimization • Continuous-time representation eliminates the time-scale inconsistencies between planning and scheduling • Three representative continuous-time formulations are systematically evaluated in terms of optimality and computational efficiency • The unit-slot formulation achieves superior efficiency and solution quality Inconsistent continuous-time and discrete-time representations have been a big challenge in achieving the enterprise-wide optimum for integrated crude scheduling and refinery planning problems. This paper proposes a unified continuous-time representation for solving the integrated enterprise-wide optimization problem. Instead of implementing continuous-time slots for the crude scheduling subproblem and discrete-time slots for the refinery planning subproblem in a hybrid way, continuous-time slots are applied to both subproblems and synchronized via the crude distillation units. Such a proposed modeling framework provides a more flexible and consistent time formulation to the integrated problem. Furthermore, three typical continuous-time representations for crude scheduling, namely, unit-slot time representation, event-based time representation, and multi-operation sequence time representation, are investigated with cases adapted from practical complexes. Compared to the traditional integrated problem based on a hybrid continuous-time and discrete-time representation, the proposed unified model framework reveals a better optimization performance. Higher-quality solutions can be obtained for both crude scheduling and planning operations. Moreover, among the three time representations, the model based on the unit-slot time outperforms the other two models. This work provides a promising alternative modeling strategy for achieving enterprise-wide optimality in the petrochemical industry.

Improved time representation model for the simultaneous energy supply and demand management in microgrids

The choice of a suitable time representation plays a major role in designing a computationally efficient planning/scheduling framework. For the time representation to be effective, it needs to capture the nuances of the actual process, while ensuring that the resultant model is tractable and computationally efficient. In this paper, a novel time representation scheme, titled hybrid time discretization, which combines both continuous- and discrete-time representations in a single framework, is proposed and used to solve a complex multiproduct maritime inventory routing problem (MIRP) with undedicated compartments. Several continuous- and discrete-time-based frameworks have been proposed to solve the MIRP over the years. While single-/multigrid discretized frameworks do efficiently capture the time-sensitive nuances of the problem (such as active/inactive jetty hours and hourly/daily variations in the demands), they struggle to solve larger problem instances to the required optimality gap in a reasonable time. This is primarily due to the large disparities among the timings of different task components in the model. On the other hand, continuous-time representation can significantly reduce the problem size and computational effort; however, modeling the entire problem using the continuous-time-based framework could have problems in representing some time-sensitive intricacies in the demands and jetty/port activities. To allay the above-mentioned drawbacks, in this paper, a novel hybrid time representation is proposed, which combines a continuous-time representation with a multigrid discrete-time framework. The proposed novel framework is cast in a rolling horizon strategy and tested on various scenarios generated for the above-mentioned maritime transportation problem, which are shown to be significantly more complex than those presented in the literature. The proposed MILP framework is able to solve most of the test cases to the required optimality in a reasonable time without the need for heuristic or meta-heuristic assists. The proposed hybrid time representation could find significant utility in other lateral applications such as batch scheduling and inventory routing problems.

The office is a key setting for intervening to reduce sitting, therefore office-specific activity measures are needed to evaluate interventions. We tested whether valid measures of office time and office-specific activities could be obtained using Bluetooth sensing with a variety of sampling intervals, receiver wear positions, and beacon placements. Workers from one building (n = 29, 72% female, age 23–68 years) wore, for one workday, the activPAL3 on the thigh (measured sitting, standing and stepping) and the Bluetooth-enabled ActiGraph Link on the wrist and thigh. Location (office/not) was estimated by Bluetooth signal presence/absence at two beacons in the wearer’s office (desk, wall), with chest-worn video cameras as the criterion. Accuracy in location classification was assessed and compared across 60-s, 30-s, and 10-s sampling intervals. The validity of Bluetooth-derived measures of total time in the office and in office-specific activities was assessed. For both the wrist and thigh-worn Link, with various beacon placements, accurate classification of location (office/not) was obtained, with a significant (p < .05) but trivial difference in accuracy across sampling interval options (F scores all ≈ .98). With the 60-s sampling interval, mean absolute percent error was very small for office time and office sitting time (<5%), but higher for infrequent activities: standing (17%–23%), incidental stepping (30%–49%), and purposeful walking (57%–86%). The ActiGraph Link can be used to validly measure office time and office location of activity with a 60-s Bluetooth setting. Higher resolution improves accuracy but not to a meaningful degree.

Several studies have been venturing into developing a model for controlling blood glucose among diabetes patients. It is because diabetes mellitus is a severe and common chronic disease affecting almost all populations in many countries. This study collected retrospective clinical data from five patients receiving insulin therapy in the ICU of HUSM. The auto-regressive with exogenous (ARX) and auto-regressive moving average with exogenous (ARMAX) model structure techniques were used to generate a model converter that best describes the glucose and insulin relationship of the subject. The simulation of ARX were started from model order (1,1,1) to model order (5,5,10) while, for ARMAX the simulation was started from model order (1,1,1,1) until model order (5,5,5,10). The three best model orders from ARX and ARMAX models were chosen. The best model fits for ARX and ARMAX were compared to identify the best model order in predicting the glucose-insulin system. The finding indicated that the ARX model recorded the best model fit for all patients in the 5 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> model order. Meanwhile, the ARMAX model recorded patients with different medical backgrounds and produced a different model order. Besides, the ARMAX model was considered the best option for most of the patients in this study due to the highest model fit, time-delay and lowest percentage of peak error. A more extensive data set may be required to ensure the structure of the model precisely describe the glucose-insulin interaction of the patient.Clinical Relevance– This study establishes a prediction model of the glucose-insulin system that can assist clinicians in providing appropriate insulin value and consequently reduce the incidence of hypoglycemia and hyperglycemia.

Performance degradation prediction method of PEM fuel cells using bidirectional long short-term memory neural network based on Bayesian optimization

Fundamental frequency is one of the most important characteristics of speech and audio signals. Harmonic model-based fundamental frequency estimators offer a higher estimation accuracy and robustness against noise than the widely used autocorrelation-based methods. However, the traditional harmonic model-based estimators do not take the temporal smoothness of the fundamental frequency, the model order, and the voicing into account as they process each data segment independently. In this paper, a fully Bayesian fundamental frequency tracking algorithm based on the harmonic model and a first-order Markov process model is proposed. Smoothness priors are imposed on the fundamental frequencies, model orders, and voicing using first-order Markov process models. Using these Markov models, fundamental frequency estimation and voicing detection errors can be reduced. Using the harmonic model, the proposed fundamental frequency tracker has an improved robustness to noise. An analytical form of the likelihood function, which can be computed efficiently, is derived. Compared to the state-of-the-art neural network and nonparametric approaches, the proposed fundamental frequency tracking algorithm has superior performance in almost all investigated scenarios, especially in noisy conditions. For example, under 0 dB white Gaussian noise, the proposed algorithm reduces the mean absolute errors and gross errors by 15% and 20% on the Keele pitch database and 36% and 26% on sustained /a/ sounds from a database of Parkinson's disease voices. A MATLAB version of the proposed algorithm is made freely available for reproduction of the results.11An implementation of the proposed algorithm using MATLAB may be found in https://tinyurl.com/yxn4a543.

The reconstruction of laser reflection tomography target reconstruction is an important part of the laser reflection tomography radar target detection technology.But currently the acquisition of imaging data for laser reflection tomography mainly stays in the laboratory test acquisition phase, for the complete slave laser pulse. There are still gaps in the comprehensive simulation of the complete process of launch propagation, laser and target effects, acceptance of echo data, and target reconstruction. In this paper, based on the above problems, studying the problem of large computational complexity firstly when the sub-rays and the bins are successively intersected. Proposed a fast intersection algorithm based on the boundary of the laser beam, which improves the computational efficiency. Secondly, based on the spatio-temporal distribution characteristics of the lidar signal, the response function of the detection target to the signal is deduced, and the action process model of the two is established. Based on the above key technologies, a target reconstruction simulation system based on laser reflection tomography was constructed. Finally, according to the effect of different sampling intervals and different detection angles on the reconstructed image quality, a contrast simulation experiment was conducted. The image back-propagation algorithm was used to complete the reconstruction of the target 2D contour under different conditions.

Basics of Multirate Digital Signal Processing

The objective of this paper is the development and solution of nonlinear and mixed-integer (MIP) optimization models for real-world planning and scheduling problems in petroleum refineries. Firstly, we present a nonlinear planning model that represents a general refinery topology and allows implementation of nonlinear process models as well as blending relations. The optimization model is able to define new operating points, thus increasing the production of the more valuable products and simultaneously satisfying all specification constraints. The second part addresses scheduling problems in oil refineries, which are formulated as MIP optimization models and rely on both continuous and discrete time representations. Three practical applications closely related to the current refinery scenario are presented. The first one addresses the problem of crude oil inventory management of a refinery that receives several types of crude oil delivered exclusively by a single oil pipeline. Subsequently, two optimization models intended to define the optimal production policy, inventory control and distribution are proposed and solved for the fuel oil and asphalt plant. Finally, the planning model of Moro et al. (1998) is extended in order to sequence decisions at the scheduling level in the liquefied petroleum gas (LPG) area for maximization of the production of petrochemical-grade propane and product delivery.

Simultaneous re-design and scheduling of multiple effect evaporator systems

Results of the analysis of the measuring device error of integrated characteristics of the harmonious signals, using formation of additional signals with the help of the phase-shifting block are given. The received analytical expressions and diagrams of the error allow to choose optimum parameters of measuring process and a phase corner shift of the phase-shifting block providing reduction of the given error kind .

This chapter deals with the minimum sampling interval \(T_s\) needed to correctly represent an analog signal by samples extracted periodically from it, so as to be able to reconstruct the continuous-time signal from its discrete-time version. The sampling theorem prescribes this lower limit and highlights the fact that a representative sampling is possible if, and only if, the analog signal does not contain frequencies higher than the Nyquist frequency \(1/(2 T_s)\): no finite-rate sampling can capture the variations of an analog signal which is not bandlimited. Other issues related to analog signals, such as the signal’s concentrations in the time and frequency domains and their mutual inverse dependence (uncertainty principle), as well as the definition of bounded support in both domains, are also discussed. An appendix provides a summary of the relations among the variables used to express the concept of frequency in the continuous-time and discrete-time cases.

When building autonomous real-time systems (RTS), it is necessary to solve the problem of optimal multitasking loading of a number of parallel functioning digital signal processors. One of the reserves for achieving the desired result is the implementation of samples from the sensor signals of information about the magnitude of the signal most rarely in time. In this case, it is necessary to provide a linear or stepwise approximation of the signal by samples with an acceptable reconstruction error. One of the system tasks of these processors is filtering signals or limiting the spectrum to the cutoff frequency. A distinctive feature of the approach proposed in the article is the fulfillment of the condition: if the measurement of this frequency is difficult (for example, in the electromechanical means of the RTS), then for such signals it is proposed to match the maximum values of the harmonic half-wave parameters: approximation error, speed and acceleration. The study opens up the prospect of applying new approaches to sampling the time of signals in the amplitude-time domain and determining the equivalent cutoff frequency of the signal spectrum for such signals. In this article, the dependences of the value of the unit of system time for input-output of data on the degree of agreement between the maximum values of the signal parameters are obtained. A mathematical model of the extreme behavior of a signal between two adjacent samples is given in the form of a harmonic half-wave. The study is also extended to convex composite harmonic functions, according to which the signal can deviate from the results of a linear or stepwise approximation of the signal for these samples. The comparison of the models by the value of the relative time sampling intervals, depending on the degree of matching of the maximum parameters of the harmonic half-wave, is carried out. When comparing, in addition to these maximum parameters, the relationship of the maximum signal speed with the error of approximating the samples by steps and the relationship of the maximum acceleration of the signal with the maximum error of the linear approximation was taken into account. The results make it possible to determine the duration of the intervals of uniform sampling of the signal time based on the results of the inspection of the control object, substantiate a significant increase in the sampling interval of time or a similar increase in the number of tasks to be solved per unit of system time.