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Functions and their curves
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
Engineers use many basic mathematical functions to represent, say, the input/output of systems – linear, quadratic, exponential, sinusoidal and so on, and knowledge of these is needed to determine how these are used to generate some of the more unusual input/output signals such as the square wave, saw-tooth wave and fully rectified sine wave. Periodic functions are used throughout engineering and science to describe oscillations, waves and other phenomena that exhibit periodicity. Graphs and diagrams provide a simple and powerful approach to a variety of problems that are typical to computer science in general, and software engineering in particular; graphical transformations have many applications in software engineering problems. Understanding of continuous and discontinuous functions, odd and even functions and inverse functions are helpful in this – it's all part of the ‘language of engineering’.
Functions and their curves
Published in John Bird, Bird's Engineering Mathematics, 2021
Graphs and diagrams provide a simple and powerful approach to a variety of problems that are typical to computer science in general, and software engineering in particular; graphical transformations have many applications in software engineering problems. Periodic functions are used throughout engineering and science to describe oscillations, waves and other phenomena that exhibit periodicity. Engineers use many basic mathematical functions to represent, say, the input/output of systems – linear, quadratic, exponential, sinusoidal and so on – and knowledge of these are needed to determine how these are used to generate some of the more unusual input/output signals such as the square wave, saw-tooth wave and fully-rectified sine wave. Understanding of continuous and discontinuous functions, odd and even functions and inverse functions are helpful in this – it's all part of the ‘language of engineering’.
Fourier Analysis
Published in L. Prasad, S. S. Iyengar, WAVELET ANALYSIS with Applications to IMAGE PROCESSING, 2020
The most familiar periodic functions are the circular trigonometric functions: sinx, cosx, tanx, etc.. One of the reasons for studying periodic functions on R is the amazing fact that any square integrable function of a real variable with compact support can be expressed as a superposition of sines and cosines of at most countably many frequencies, i.e., if f is a square integrable function with support contained in the closed interval [a, b], then it can be expressed as an infinite series obtained by linear combinations of the functions 1,sin(2πnxb−a),cos(2πnxb−a);n=1,2,3,…,
An Adaptive Algorithm for Battery Charge Monitoring based on Frequency Domain Analysis
Published in IETE Journal of Research, 2021
Poulomi Ganguly, Surajit Chattopadhyay, B.N Biswas
The Fourier transform is a powerful analytical tool in various fields of science. It can help solve cumbersome dynamic response equations. Fourier analysis of a periodic function denotes the extraction of the series of sines and cosines, which when combined will mimic the function. This analysis can be expressed as a Fourier series. The following equation represents the decomposed form of a period function f (t). Here a0, an, and bn are Fourier coefficients and are defined as The Fast Fourier transform (FFT) is an improvement of the Discrete Fourier transform (DFT), which removes duplicated terms in the mathematical algorithm, thereby reducing the number of mathematical operations to be performed. Thus large numbers of samples can be used without compromising the transformation speed. The FFT reduces computation by a factor of N/(log2 (N)). FFT produces the same result as the DFT but at a much faster rate.
Periodic dynamics of a derivative nonlinear Schrödinger equation with variable coefficients
Published in Applicable Analysis, 2020
Qihuai Liu, Wenye Liu, Pedro J. Torres, Wentao Huang
Assume that is a continuous periodic function with the least positive period T and . Then for any constant and any positive integer m, there exists such that, for any positive integer Equation (1) has a sequence of periodic solutions with the form which is periodic with respect to x with the least period mT such that the notation number is given by and The condition that the least positive period of is T is only used to guarantee that the least positive period of on x is also T. The proof of Theorem 3.1 includes two steps. Firstly, for any positive integer m, we shall prove in Section 3.2 the existence of infinitely many mT-periodic solutions for amplitude evolution equation (7). Then in Section 3.3, we shall complete the proof by discussing the rotation numbers.
Modelling of multifilament woven fabric structure using Fourier series
Published in The Journal of The Textile Institute, 2019
Zuhaib Ahmad, Brigita Kolčavová Sirková
The experimental binding waves for the fabric sample (B1) and its approximation using the linear function in Fourier series as in Equation (14) along longitudinal and transverse cross-section can be observed in Figure 18. It can be observed that the approximation done by Fourier series fits well to the experimental binding wave. The difference in amplitude can be analyzed as well, the deformation in longitudinal cross-section (binding wave of warp) is less as compared to deformation in transverse cross-section (binding wave of weft). This is because, the density of weft is low as compared to density of warp and at low pick setting, the weft yarn gets more space to be flat in fabric plane and hence, its binding wave attains more deformation. Moreover, the binding wave of warp yarn can be observed to have slight deformation as in the woven fabrics when one yarn gets more deformation then the other yarn connected to it, deforms less. It is called the balance of crimp between warp and weft. As the Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. In the figure (in red line) is the sum of one term of Fourier series, while (in green line) is the sum of three terms of Fourier series to get better approximation.