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Fourier Series and Integrals
Published in Paolo Di Sia, Mathematics and Physics for Nanotechnology, 2019
Parseval’s identity shows a relation between the average of the square of f(x) and the coefficients in the Fourier series for f(x). It holds: The average of [f(x)]2 is (1/2)∫−LL[f(x)]2dx;The average of (a0/2) is (a0/2)2;The average of [an cos(nx)] is an2/2;The average of [bn sin(nx)] is bn2/2.
Basics of communications using chaos
Published in Marcio Eisencraft, Romis Attux, Ricardo Suyama, Chaotic Signals in Digital Communications, 2018
Géza Kolumbán, Tamás Krébesz, Chi K. Tse, Francis C. M. Lau
The signals r˜TC,mR(t+TC) and r˜TC,mI(t) are periodic signals with a period time of TC. The Parseval’s identity expresses the relationship between the average power of a periodic signal and its Fourier series coefficients. The received signal is an RF bandpass signal; consequently, it has a zero DC component. Exploiting the Parseval’s identity, Equation (4.37) can be interpreted as follows: RHS of Equation (4.37) shows that the sum of two RF bandpass periodic signals with the same periodicity TC has to be considered in the time domain;the two bandpass signals are the delayed reference and the information bearing chips of received signal as depicted in Figure 4.8(b).
Fourier Waveform Analysis
Published in Jerry C. Whitaker, Electronic Systems Maintenance Handbook, 2017
Jerry C. Hamann, John W. Pierre
In Eq. (24.1), the average value or DC term of the waveform is given by A/2, whereas the fundamental frequency is described by Ω0 = 2π/T. Because the waveform is an even function only cosine terms are present. Because the waveform, less its average value, is characterized by half-wave symmetry, only the odd harmonics are present. To examine the reconstruction of the function x(t) from the given decomposition, the middle plot of Fig. 24.2 displays the DC term, and the first and third harmonics. The sum of these terms is overlayed with the ideal square wave at the bottom of Fig. 24.2. This progressive reconstruction is continued in Fig. 24.3 with successive odd harmonics added as per Eq. (24.1). The ultimate goal of reconstructing the ideal square wave theoretically demands completion of the infinite sum, an unrealistic task of computation or analog signal summation. By truncating the summation at a large enough value of the harmonic number k, we are assured of having a best approximation of the ideal square wave in a least-square-error sense: that is, the truncated Fourier series is the best fit at any given truncation level k = kmax < ∞. The ringing or overshoot, which forms near the discontinuity of the ideal square wave, is described by the Gibbs phenomenon, in tribute to an early investigator of the effects of truncating Fourier series summations.
Analysis of real-time jitter in cyber-physical applications using frequency domain perturbation
Published in International Journal of General Systems, 2023
Ricardo Cayssials, Edgardo Ferro
The Fourier series can be expressed as a Fourier Transform but for periodic functions. A time function g(t) is a periodic function with a fundamental period T if: It can be noted that a periodic function with a fundamental period of T is also periodic with periods of 2.T, 3.T, and so on. Therefore, the fundamental period T of a periodic function is the smallest time interval T, greater than 0, that makes the equation (1) true. Then, using the Fourier Series, a periodic function g(t) may be expressed as an infinite sum of sinusoidal functions with frequencies that are an integer multiple of 1/T, as follows: where the coefficients am and bn of the Fourier Series determine the weight of each sinusoid in the periodic function. Consequently, the coefficients ai and bi determine the amplitude and phase of the component of frequency i/T in the periodic function g(t).
A multimodal differential privacy framework based on fusion representation learning
Published in Connection Science, 2022
Chaoxin Cai, Yingpeng Sang, Hui Tian
A signal can be represented in terms of the linear combination of orthogonal functions. It is known as the trigonometric Fourier series when these orthogonal functions are trigonometric functions. As the limit form of Fourier series, the Fourier transform (or FT) decomposes functions from time domain to frequency domain in the form of linear transformation. The Fourier transform of a function in n-dimensional space is , defined as follows (Equation (10)): where . Equation (11) is the inverse Fourier transform (or IFT) of a function . Note that the n-fold multiple integral goes from to ∞ and the output shape of FT and IFT is the same as input.
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.