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Fractal Based Ultra-Wideband Antenna Design
Published in Praveen Kumar Malik, Planar Antennas, 2021
C. Muthu Ramya, R. Boopathi Rani
The German mathematician David Hilbert introduced an algorithm for Hilbert curve in 1891 [40]. The curve has been utilized as a space-filling structure. In antenna design, the Hilbert curve configuration has been used for reducing the size and producing multiband frequency. The geometry of Hilbert curve is constructed with the procedure shown in Figure 8.6. Every step generated with the four replicas of the previous shape and additional line sections. This shape possesses four basic properties. They are self-avoidance, self-similarity, space-filling, and simplicity [41]. From Figure 8.6, it can be observed that the full length of the Hilbert curve greatly grows with respect to of iteration. However, the overall dimension of the structure is kept unchanged. This is achieved by means of the electrically reducing the size of the structure. The self-similarity dimension for Hilbert curve fractal structure Ds is 2.
BBO based location optimization of target nodes using single mobile anchor node in WSNs
Published in Rajesh Singh, Anita Gehlot, Intelligent Circuits and Systems, 2021
Parulpreet Singh, Gurleen Kaur Walia, Gagandeep Singh Walia
In 1891, David Hilbert proposed a space-filling curve which on his name is referred to as the Hilbert curve [6]. While the curve was being planned it was made sure that it must visit all the spots in the whole target area. The most important step is to make sure that the right order of the curve should be selected so that the full possibility of the network is ensured. The least order should always be selected to cover the entire network. Assume considering N sub areas we consider the whole network and all the sub areas are sub-squares of size 1. The sub-square size is given by Equation (3). l=L2k
Miniaturized and optimized half mode SIW bandpass filter design integrating Hilbert cells as DGS
Published in Electromagnetics, 2023
Mohammed El Amine Chaib, Nabil Cherif, Mehadji Abri, Hadjira Badaoui
DGS involves etching shapes into the ground plane, which cause a change in the current distribution, where several forms have been used including the fractal shapes as Sierpinski gasket, Koch curve, and Hilbert curve. Based on self-similarity and space filling properties, Hilbert curve has been proposed by the German mathematician David Hilbert in 1891 (Sagan 1994). This means that Hilbert curve is modified from stage to the next stage by iterating the original curve (Chen et al. 2007). Due to the specific transmission characteristics provided by this curve (YAN et al. 2010), it has been investigated to design many microwave and millimeter components such as filters (Fellah et al. 2021; Yan, Zeng, and Zhang 2010), antennas (Zhu, Hoorfar, and Engheta 2003). Not long after in 2003, SIW (Substrate Integrated Waveguide) technology also appeared as a solution by providing as essential advantage: Q factor, reduction of size, and the low-cost (Rayas-Sanchez and Gutierrez-Ayala 2008).
The process intensification of CO2 absorption in Hilbert fractal reactor fabricated by a 3D printer
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2019
Suqi Zhang, Yue Lu, Jian Sun, Yu Gu, Xiaodan Zhang, Zhiyong Tang
The design of Hilbert fractal reactor was inspired by the Hilbert curve. The Hilbert curve is a continuous space-filling curve that fills a square and is typically defined as the limit of a sequence of curves defined iteratively and that has only short vertical and horizontal jumps between the points in a square grid with a size of 2 × 2, 4 × 4, 8 × 8, 16 × 16, or any other power of 2. At all stages, each curve has neither self-intersections nor touching points (Kuan et al. 2009). From the perspective of fractal dimension, the fractal dimension of 1 represents a line, the fractal dimension of 2 represents an area, and the fractal dimension of Hilbert curve is 2, which means that this curve can be considered as an area. In our experiment, the fractal reactor is fabricated with the 3rd Hilbert fractal curve because of its large open ratio and long carved open perimeter length (Kuan et al. 2009). These photographs of two reactors are seen in Figure 2.
Improved security in the genetic algorithm-based image steganography scheme using Hilbert space-filling curve
Published in The Imaging Science Journal, 2019
Gyan Singh Yadav, Aparajita Ojha
A Hilbert space-filling curve or simply a Hilbert curve is a continuous fractal curve drawn by joining point locations in a particular space depending upon the value of an iteration number. These locations can be mapped onto the image pixel locations (see Figure 4). If the number of iteration is z then point locations will be generated in a row \ column. Hence, the total number of points will be or . Hilbert curve generation algorithm starts with initialization of complex numbers p, q and , and subsequent points will be generated as follows.