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Introduction to fractal antennas and their role in MIMO applications
Published in Yadwinder Kumar, Shrivishal Tripathi, Balwinder Raj, Multifunctional MIMO Antennas, 2022
Biswajit Dwivedy, Tanmaya Kumar Das
The Koch snowflake or Koch island is a variant of fractal geometry constructed by applying the Koch curve fractal to each line segment of an equilateral triangle [20]. The generation of a Koch snowflake with different IOs is shown in Figure 1.2. The scaling law can be applied to this fractal geometry having one independent unit length n, and Equation (1.3) can be expressed as [20, 21] lnKoch/ϵn=fa/ϵn=a/ϵnD
Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
The Koch snowflake fractal is very similar to the Koch curve (Figure 10.4), except that it starts with an equilateral triangle instead of a straight line. It can also be generated by placing three copies of the Koch curve on the three sides of an equilateral triangle. As in the Koch curve, it can be shown that the length of the snowflake tends to infinity as the number of iterations tends to infinity, while the area enclosed remains finite at 1.6=(85) times the area of the original triangle. The Hausdorff–Besicovitch dimension of the Koch snowflake is the same as that for the Koch curve at 1.26185.
Design and Developments of UWB Antennas
Published in Chinmoy Saha, Jawad Y. Siddiqui, Yahia M.M. Antar, Multifunctional Ultrawideband Antennas, 2019
Chinmoy Saha, Jawad Y. Siddiqui, Yahia M.M. Antar
Other useful fractals commonly used are the Koch snowflake and the Hilbert curve. The common fractal geometries have applications in antenna engineering. The Koch snowflakes are used to develop new designs for miniaturized loop as well as microstrip patch antennas [26, 27]. The Sierpinski gasket has been used to develop multiband and broadband antenna elements [28–31]. Fractal antennas are multi-resonant antennas. Variation of fractal parameters has an impact on the primary resonant frequency of the antenna. A fractal antenna is capable of operating with excellent performance at many different frequencies simultaneously. This makes the fractal antenna an excellent design for wideband and multiband applications [32].
The area, centroid and volume of revolution of the Koch curve
Published in International Journal of Mathematical Education in Science and Technology, 2021
Thus the volume of revolution about the vertical axis of symmetry is Classroom Exercises The Koch snowflake, or Koch island is formed by joining three Koch curves together around an equilateral triangle of unit side. Calculate the snowflake's area and show that its volume of revolution about the axis of symmetry of the unit equilateral triangle is given by . What is the volume of revolution about the snowflakes other axis of symmetry?A fractal S is formed from an initiating line of unit line. The generator is formed by dividing the line into five segments of equal length, removing the third segment and replacing it with the upper three edges of a square. To create the k = 2 prefractal each of the seven line segments is replaced by a version of the generator as shown in Figure 4. Repeating this process an infinite number of times produces the fractal. Show that the length of the fractal is divergent. What is the area bounded by S and its initiator? Calculate the centre of area and the volume of rotation of S about the initiator.
Koch–Sierpinski Fractal Microstrip antenna for C/X/Ku-band applications
Published in Australian Journal of Electrical and Electronics Engineering, 2019
Mohd Gulman Siddiqui, Abhishek Kumar Saroj, Devesh Tiwari, Saiyed Salim Sayeed
Similarly for combined Koch and Sierpinski technique of proposed antenna, the iterations are shown step by step in Figure 8.The Koch snowflake transformation is applied at each step to achieve higher levels of fractalisation. The iterated function is based on the application of affine transformations, w, given by: