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Convolutional Neural Networks
Published in Robert H. Chen, Chelsea Chen, Artificial Intelligence, 2022
A convolution is simply a mathematical operation of two functions to produce a third function that represents how the functions are conjoined. In two dimensions, a mapped function f(x, y), is convolved by figuratively placing a convolving filter h(x, y) over f(x, y), spatially stepping through it and integrating the result, Convolution=∬f(x,y)h(x+1,y+1)dxdy.
Analytical Techniques for Ultra-Wideband Signals
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Muriladhar Rangaswamy, Tapan K. Sarkar
The convolution is useful in obtaining the response of a linear system. For example, if an input g1(t) is applied to a system whose impulse response is g2(t), then the output of the system is given by the convolution of g1(t) with g2(t). Letting t − τ = x, Equation 3.42 may be rewritten as () g1(t)*g2(t)∫−∞∞g1(t − x)g2(x)dx
Digital Signal Processing
Published in Richard C. Dorf, Circuits, Signals, and Speech and Image Processing, 2018
W. Kenneth Jenkins, Alexander D. Poularikas, Bruce W. Bomar, L. Montgomery Smith, James A. Cadzow, Dean J. Krusienski
Most of the properties listed in Table 14.5 for the DFT are similar to those of the Z-transform and the DTFT, although there are some important differences. For example, Property 5 (time-shifting property) holds for circular shifts of the finite length sequence s[n], which is consistent with the notion that the DFT treats s[n] as one period of a periodic sequence. Also, the multiplication of two DFTs results in the circular convolution of the corresponding DT sequences, as specified by Property 7. This latter property is quite different from the linear convolution property of the DTFT. Circular convolution is simply a linear convolution of the periodic extensions of the finite sequences being convolved, where each of the finite sequences of length N defines the structure of one period of the periodic extensions.
Identification of Impulse Responses in Heat Transfer: Parameterization, Doses, Partial Time Moments
Published in Heat Transfer Engineering, 2023
The purpose of this paper is to test a method for identifying the impulse response (output) of a dynamical thermal system using discrete noisy transient measurements of the excitation (input) and of the response (output). Here, the input is either a dissipated thermal power or a temperature change in some part of the domain, while the output is the variation of a local temperature at any point in the system. An impulse response explains the causal relationship between an input and an output and is linked to a convolutive model since the response is its convolution product with the input. The conditions of validity of this type of model are detailed in the first section of the paper. The commutative property of a convolution product implies that the identification problem, for known input and output is mathematically equivalent to a linear input estimation problem, for known output and impulse response.
A data-driven machine learning approach for the 3D printing process optimisation
Published in Virtual and Physical Prototyping, 2022
Phuong Dong Nguyen, Thanh Q. Nguyen, Q. B. Tao, Frank Vogel, H. Nguyen-Xuan
CNNs are MLP networks whose hidden layers are convolutional layers. Convolution is a linear operation between two matrices or functions, and the result is a new matrix or function. Convolution is often used in works related to image processing and image signals. In convolution, one matrix is called the input, and the other is called the kernel or filter. Normally, the filters do not change but have a fixed value. When performing convolution, we slide the kernel matrix over the input matrix. In the sliding process, we multiply the value of the kernel by the value of the input matrix at the corresponding position, and then add them all. After the sliding process, the new matrix contains the values we calculate during the sliding process. The model is shown in Figure 9.
Development of a digital astronomical intensity interferometer: laboratory results with thermal light
Published in Journal of Modern Optics, 2018
Nolan Matthews, David Kieda, Stephan LeBohec
We have successfully employed two different types of digital correlators, off-line and real-time. In the off-line correlator, the digitized data from each channel are scaled, truncated to 8-bits and merged into a single continuous data stream by a Virtex-5 FPGA (PXIe7965R). The data stream is then recorded to a high speed (700 MB/s) 12TB RAID disc. A software routine using LabVIEW can later be used to retrieve intensity correlations between channels as a function of the digital time lag, typically up to 1s in steps of 4 ns. Due to the large number of samples, the data are read in blocks of 512 samples. The convolution theorem gives the correlation between two signals as the inverse Fourier transform of the product of the Fourier transforms of the signals. This is implemented by use of the NI Multi-core Analysis and Sparse Matrix toolkit cross-correlation virtual instrument (VI), which optimizes the computation by utilizing separate computing cores.