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Sampling Quantization and Discrete Fourier
Published in Francis F. Li, Trevor J. Cox, Digital Signal Processing in Audio and Acoustical Engineering, 2019
is periodic with period L, since it is the sum of harmonically related components, the fundamental (k = 1), which has a period equal to L. The function sn(x) in Equation 2.9 is called a trigonometric polynomial, order n, period L. Fortunately for us, many periodic signals found in audio and acoustics may be usefully approximated as high order trigonometric polynomials (or accurately described by infinite order series). In describing a signal in terms of an equivalent trigonometric polynomial, we are stating that combination of harmonic components that represents the signal, i.e. we are performing frequency analysis.
Difference equations in the complex plane: quasiclassical asymptotics and Berry phase
Published in Applicable Analysis, 2022
Alexander Fedotov, Ekaterina Shchetka
As is a trigonometric polynomial, it is natural to consider w as a function of the variable . Then the Riemann surface Γ of w turns out to be a hyperelliptic curve. In particular, in the case where is a first-order trigonometric polynomial, relation (6) implies that where , and are constants, and . Therefore w is single-valued on the Riemann surface of the function , which is a hyperelliptic curve of genus one, see [21].
Periodic motion planning and control for underactuated mechanical systems
Published in International Journal of Control, 2018
Zeguo Wang, Leonid B. Freidovich, Honghua Zhang
A systematic periodic motion planning and periodic motion control design method is proposed in this paper. It could be applied to underactuated mechanical systems with arbitrary underactuation degree. The idea originates from the fact that any continuous periodic function could be decomposed into an infinite uniformly converging Fourier series and, therefore, it can be arbitrarily accurately approximated by a trigonometric polynomial. Hence, the reference trajectory of each state of the system is assumed to be a truncated Fourier series. Moreover, this reference trajectory should satisfy differential constraints imposed by passive dynamics equations. Therefore, a numerical optimisation search is implemented to find the parameters to minimise the error, which is given by passive dynamics equations. An almost feasible periodic motion is defined in this way. The obtained reference trajectory does not satisfy the equations precisely but a feedback controller can be used to force the closed-loop system's feasible trajectories into a small neighbourhood of the desired approximately feasible motion.
Ordinary differential equations defined by a trigonometric polynomial field: behaviour of the solutions
Published in Dynamical Systems, 2023
A function is called a trigonometric polynomial if there exists a finite sequence such that where is the usual scalar product on . A function is a trigonometric polynomial if each component is a trigonometric polynomial function.