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Complex Aperture Theory – Volume Apertures – General Results
Published in Lawrence J. Ziomek, An Introduction to Sonar Systems Engineering, 2023
It is well known from Fourier transform theory that the time-domain Fourier transform of the product of two functions of time is equal to a convolution integral in the frequency domain. Similarly, the three-dimensional spatial Fourier transform of the product of two functions of three spatial variables is equal to a three-dimensional convolution integral in the spatial-frequency domain. Therefore, (1.2-47) can be expressed as follows: I=χM(f,r,α)=X(f,α)*αDT(f,r,α),
Complex Aperture Theory – Volume Apertures – General Results
Published in Lawrence J. Ziomek, An Introduction to Sonar Systems Engineering, 2017
It is well known from Fourier transform theory that the time-domain Fourier transform of the product of two functions of time is equal to a convolution integral in the frequency domain. Similarly, the three-dimensional spatial Fourier transform of the product of two functions of three spatial variables is equal to a three-dimensional convolution integral in the spatial-frequency domain. Therefore, (1.2-47) can be expressed as follows: ()
Analytic pavement modelling with a fragmented layer
Published in International Journal of Pavement Engineering, 2022
Vertical loading representing tire-pavement contact interaction is applied at the surface (i.e. top of Layer 1), just above the coordinate system origin. It entails a uniform stress with intensity operating over a circular area with radius . Such loading is mathematically expressed as: where is the signum function. Based on the Hankel transform theory, a mathematically equivalent expression for is where is a dimensionless loading radius, is a dimensionless radial coordinate, and are Bessel functions of the first kind of order zero and one (respectively), and is a unitless positive integration parameter (Hankel parameter or wave number).
A new DFT-based dynamic detection framework for polygonal wear state of railway wheel
Published in Vehicle System Dynamics, 2023
Qiushi Wang, Zhongmin Xiao, Jinsong Zhou, Dao Gong, Zegen Wang, Zhanfei Zhang, Tengfei Wang, Yanling He
Traditional analysis method is based on discrete Fourier transform theory. The frequencies, amplitudes, and phases of harmonic contained in the signal can be identified easily: Where:z(n) is the time series of axle box vertical acceleration, n = 0,1, … , N-1. N is the total number of time series. f0 is fundamental frequency, f0 = N−1. Z(k) is the discrete Fourier transform of z(n), and |Z(k)| represents the amplitude of harmonic with frequency fk, fk = k·f0.
Teaching transfer functions without the Laplace transform
Published in European Journal of Engineering Education, 2022
Imad Abou-Hayt, Bettina Dahl, Camilla Østerberg Rump
In the Laplace transform theory, the differentiation theorem states that the Laplace transform of is equal to , and the Laplace transform of is and so on for higher derivatives. Now, since all initial conditions , and are assumed to be zero when finding a transfer function of a system, we conclude that multiplying by the kth power of the Laplace variable s in the Laplace domain corresponds to the kth derivative in the time domain. We can therefore simplify the analysis of the governing mathematical model by defining the differential operator or ‘D operator’ as Using this notation, the time derivatives in a differential equation can be written as powers of the operator D: for example, , . We can then replace the differential operator D in the differential equation with s to obtain the transfer function and vice versa. As an example, if a system is modelled by the differential equation, where u is the input and y is the output, then we can get the output-to-input ratio by applying the D operator Finally, replacing the operator D with s yields the transfer function of the system