Analysis Of Linear Systems Dk Cheng Solution
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A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above.
In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions.
By analyzing the state of the art on shaped-beam in linear array pattern synthesis works as  can be referred where arbitrary shaped-beam patterns are generated by means of sub-array clustering solutions. Here, an optimal trade-off between the complexity of the array structure and the matching of a reference pattern is exploited. Another interesting alternative  proposes a superposition of scanned sum patterns for the shaped-beam pattern generation. With regard to the introduction of multiple sets of solutions within the array pattern synthesis problem, alternative methodologies from recent literature can be highlighted [5,6,7]. These works are mainly focused on solving mask-constrained problems through arbitrary array layouts and thus they can effectively deal with an arbitrary spacing between the elements. Otherwise, although some interesting attempts with the unit circle representation from aperiodic arrays have been made in , the methods inspired by Schelkunoff  (in the basis of the present work) certainly do not allow a direct management of non-uniformly spaced linear arrays.
An interesting feature of exploiting this new multiplicity of solutions is the facilitation of the physical realisation of the array. In this manner, the chance to select an adequate solution with regard to certain feeding network scenario (end-fed, centre-fed, as well as corporate fed arrays) is highly interesting. As a drawback, a slightly enlargement of the number of elements of the array has to be expected in comparison to the standard complex pattern case, if we want to keep the same number of ripples in the shaped region. Attending the quality of the results offered by this new procedure, both bandwidth and tolerance analyses of the entire multiplicity of solutions (the classical multiplicity and the one devised in the present work) are here proposed. In such a way, the potentials of the new set of solutions can be understood while their quality is evaluated within the framework of the application in linear array synthesis. Additionally, an extension to both circular and elliptical footprint patterns by means of a generalized procedure of the well-known Tseng and Cheng methodology  are applied. In such a way, the performance of the different solutions to produce extended 2-D far field patterns will add impact to the present study, motivating the provided development for space vehicle applications in order to facilitate both practical realisation and power consumption.
With regard to tolerance analysis, examples also devoted to report the performance of a classical multiplicity of solutions by generating the same shaped-beam pattern, have been developed in . In this work, statistical studies about the sensitivity of the performance to errors in element positions, excitation phases or amplitudes have been conducted. As limitation to highlight, the work was performed without introducing a model for the coupling effects neither taking care about the radiation field of each one of the dipoles which define the array (i.e., their element factors). So, for understanding the potentials of the multiplicity of solutions created by means of a distribution generating a pure real shaped-beam pattern, a tolerance analysis is here proposed. As it was already mentioned, the lack of modelling mutual coupling effects, as well as element factors, in previous strategies  adds motivation to conduct this specific study.
In a deep and general analysis about the nature of each one of the solutions for the RPC (in a similar way as the depicted in  for the CPC), if we have an equispaced linear array of 2N elements and it is desired to produce a symmetric shaped beam with 2M filled nulls (so 4M roots displace radially off the unit circle), then
To synthesize an equivalent pattern by means of the same number of elements, a distribution with just one couple of roots out of the Schelkunoff unit circle (at the same angular position) is necessary. So, in the case of the RPC for the 10-element linear array, 32/2=3 solutions will be generated. More precisely, there will be 1 RS solution (Solution 1), and 2 RA solutions (Solutions 2 and 3).
Sketch with regards to the applicability possibilities for the solutions with different nature: (A) end-fed, (B) center-fed, and (C) corporate-fed linear arrays.
Magnitudes (left) and phases (right) of the current excitations with maximum and minimum variability linked to the solutions of the synthesis in the RPC of 12-element linear array (see the pattern in Figure 2). The numbering of the solution agrees with Figure 1. Solutions 4 and 6 are mirror images of each other and because they are symmetric, they refer the same values in amplitude.
Detailed view of the active impedances which present the maximum and the minimum variability from the set of solutions for the RPC, in the case of the half-wavelength elements linear array of 12 elements: in absence of ground plane (left) and in presence of ground plane (right). The numbering of the solution agrees with Figure 1.
Detailed view of the active impedances which present the maximum and the minimum variability from the set of solutions for the RPC, in the case of the half-wavelength elements linear array of 10 elements: in absence of ground plane (left) and in presence of ground plane (right).
More concretely, by means of a detailed analysis of Table 1 and Table 4, it can be concluded that the CS solutions need the minimum rate of changes in length in order to obtain a resonant structure and also present the lowest dynamic range ratio and local smoothness of the set of solutions in the 12-element RPC, while the RS one has a similar behavior in the 10-element RPC (by means the comparison of Table 2 and Table 5). Otherwise, the set of solutions involving the greatest rate of change in lengths have been the RS solutions. These solutions coincide with the ones with maximum level of DRR and LS. Even, we can give a more concrete picture of the problem by ordering the solutions from minimum to maximum DRR, LS, and maximum rate of change in length of their elements. In this way, the list of the more advantageous to the more disadvantageous solutions for the 12-element RPC is: CS, CA, RS, and finally RA; while for the 10-element RPC is: RS and RA.
Detailed view of the active impedances which present the maximum and the minimum variability from the set of solutions for the RPC, in the case of the resonant linear array of 12 elements: in absence of ground plane (left) and in presence of ground plane (right). The numbering of the solution agrees with Figure 1.
Detailed view of the active impedances which present the maximum and the minimum variability from the set of solutions for the RPC, in the case of the resonant linear array of 10 elements: in absence of ground plane (left) and in presence of ground plane (right).
Nature of the solutions related to the maxima and minima bandwidths in terms of quality parameters of the pattern and selected terms for linear antenna array half-wavelength elements. Linear arrays in absence and presence of ground plane.
Nature of the solutions related to the maxima and minima bandwidths in terms of quality parameters of the pattern and selected terms for the resonant linear antenna array in absence and presence of ground plane.
Then, by performing the tolerance analysis for the resonant linear array structure, the results obtained for this case are shown in Figure 13 and Table 9 (again, both in absence and presence of ground plane.
From Figure 13, CPC in absence of ground plane presents enhanced tolerance to errors in voltage magnitudes in comparison with the RPC12 alternatives. Otherwise, RPC12 maxima improve the results of the CPC in the case of phase errors. It is particularly important to highlight the point that the results obtained by the RPC10 are particularly interesting in the case of the resonant structure for the linear array. More precisely, in the scenario of this array with improved efficiency in absence of ground plane, the solutions obtained by means of this procedure are always the best option among the others (RPC12 and CPC).
By analyzing the DRR results reflected in Table 10, we can observe how the CS solution of the linear array with 12 elements represents the more advantageous solution, while the RS solutions (in both cases the linear array has 10 and 12 elements) present the higher level of variability. So, in order to illustrate the performance of this case, a region of the layout of the planar array (the one with not dismissed elements) is shown in Figure 14, where both normalized amplitudes and phases are plotted. 153554b96e