Coupling Matrix Synthesis Using Groebner Basis

In this paper, novel approaches to synthesize admittance function polynomials and canonical <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> +2 coupling matrices for narrowband lossy filters are presented. The methods are simpler and more general than the ones found in the literature. The polynomial synthesis approach is fully analytical and also very useful for lossless polynomial synthesis with simpler derivations. The coupling matrix synthesis method is based on a lossy transversal network model, which can also accommodate direct source to load coupling. Unlike the lossless transversal coupling matrix, the lossy coupling matrix model requires the assumption of complex <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> -inverters and additional resistive elements in the network. The complex <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> -inverter circuit model is defined and explained in detail in this paper. The lossy transversal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> +2 matrix can be systematically rotated to obtain a number of practical realizations. Parallel-coupled pairs and folded lossy configurations are shown as examples. Moreover, the synthesis of novel networks with different return-loss levels at source and load is presented. A performance comparison with a predistorted filter is also included in this paper.

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10−8.

In this paper a new approach to the synthesis of coupling matrices for microwave filters is presented. The new approach represents an advance on existing direct and optimization methods for coupling matrix synthesis in that it will exhaustively discover all possible coupling matrix solutions for a network if more than one exists. This enables a selection to be made of the set of coupling values, resonator frequency offsets, parasitic coupling tolerance etc that will be best suited to the technology it is intended to realize the microwave filter with. To demonstrate the use of the method, the case of the recently – introduced ‘extended box’ (EB) coupling matrix configuration is taken. The EB represents a new class of filter configuration featuring a number of important advantages, one of which is the existence of multiple coupling matrix solutions for each prototype filtering function, eg 16 for 8th degree cases. This case is taken as an example to demonstrate the use of the synthesis method – yielding one solution suitable for dual-mode realization and one where some couplings are small enough to neglec

ABSTRACTA novel coupling matrix synthesis method for dual‐band bandpass filter is presented in this article. The output admittance poles and zeros of the single‐band filter prototype are firstly obtained and then transformed into dual‐band prototype. Then the coupling matrix with predefined topology is optimized according to these poles and zeroes directly. This method can be achieved with no need to construct the characteristic function of dual‐band filter and reconfigure the coupling matrix. Symmetric and asymmetric dual‐band bandpass filter coupling matrix synthesis has been carried out to illustrate and validate this method. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 58:603–606, 2016

The direct and inverse scattering problems in two-dimensional acoustics at a fixed frequency are considered. It is shown how the null-field approach can be modified so that the Q matrix (which in a straightforward manner gives the transition matrix) is obtained as a series instead of an integral. For an obstacle which is a perturbation of a circle this series form gives an approximate, very explicit, expression for the transition matrix. A few numerical examples are given to show the utility and limitations of this approximation. For the inverse problem, the series form of Q gives a system of nonlinear polynomial equations which are solved by the imbedding method. Some numerical examples show that quite accurate results are obtained by this method in cases where the system of equations can be kept small. The linear approximation of the system of polynomial equations yields a method that works surprisingly well and which is also promising for the more difficult three-dimensional and vector problems.

The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NP-hard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorithms have large exponential complexity and cannot solve in practice systems with n ≥ 15. Kipnis and Shamir [9] have recently introduced a new algorithm called “relinearization”. The exact complexity of this algorithm is not known, but for sufficiently overdefined systems it was expected to run in polynomial time.In this paper we analyze the theoretical and practical aspects of relinearization. We ran a large number of experiments for various values of n and m, and analysed which systems of equations were actually solvable. We show that many of the equations generated by relinearization are linearly dependent, and thus relinearization is less efficient that one could expect. We then develop an improved algorithm called XL which is both simpler and more powerful than relinearization. For all 0 < ε ≤ 1/2, and m ≥ εn 2, XL and relinearization are expected to run in polynomial time of approximately \( n^{\mathcal{O}(1/\sqrt \varepsilon )} \). Moreover, we provide strong evidence that relinearization and XL can solve randomly generated systems of polynomial equations in subexponential time when m exceeds n by a number that increases slowly with n.KeywordsQuadratic EquationExhaustive SearchAsymptotic ComplexityExact ComplexityMultivariate EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃ ℝ, which has been studied earlier by other communities. The class ∃ ℝ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP ⊆ ∃ ℝ. We prove that the art gallery problem is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃ ℝ. As an illustration of our techniques, we can show that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. In the ∃ ℝ-hardness proof for the art gallery problem we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it can be used to obtain ∃ ℝ-hardness proofs for other problems. In particular, ETR-INV has been very recently used to prove ∃ ℝ-hardness of other geometric problems.

Direct synthesis technique for coupling matrix of mixed topology filters that contain cross-coupled structures and a series of extracted-pole sections is proposed and demonstrated. Instead of evaluating elements one by one in conventional synthesis, the canonical coupling matrix, including all the cross-coupled and extracted-pole couplings, can be solved altogether one time by using the proposed technique. The direct synthesis is based on a process for the admittance function construction of a filter, yielding a mixed topology prototype where one or more extracted-pole sections are generated at the source port naturally. A sixth-order filter is carried out as an example, which verifies the validity of the proposed technique.

This paper presents a comparison between two techniques of S band interdigital diplexer design with i.e. with generalized multiplexer theory and coupling matrix synthesis. The design based on GMT, doubly terminated Chebyshev low pass filter response is used. The first two resonators of each channel are modified using formulas in GMT. The design is verified though MathCAD plots. In coupling matrix synthesis design, the general coupling matrix of each filter and coupling band width for two adjacent resonators are calculated. A novel technique is used for diplexer input design. The group delays are measured at the centre frequencies of channel filters. The physical diplexer design round rod type inter digital filter is selected. Both designs are simulated in HFSS with 2100 and 2250MHz with 100MHz channel band width.

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations. Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size. A corollary of our first result is that the proof systems Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate each other in the presence of Boolean axioms.

Coupling matrix synthesis has been recognized as a widely employed and powerful tool for microwave filter modeling and design. Generally, to obtain a coupling matrix whose topology maps to the actual physical structure, the rotation matrix is used for the matrix reduction and elimination. However, it is difficult to obtain a precise simplified matrix through the rotation matrix method for the corresponding planar resonant circuit filter structure. To overcome the shortcomings of conventional rotation matrices and similar equivalent simplification methods, based on the idea of optimization, we propose a simplification of the coupling matrix using a hybrid optimization algorithm. By the proposed algorithm, a simplified matrix that accurately consistent with the physical structure can be obtained. Firstly, we use the matrix M R that after the conventional equivalent transformation of the coupling matrix as the initial value of the hybrid optimization. Secondly, in order to further obtain the mapped coupling matrix, the scattering parameters corresponding to the coupling matrix is used to calculate the optimization fitness index, and the filter topology is used as the optimization constraint. Then, a global optimization algorithm is used to obtain a matrix M G which is similar to the initial coupling matrix engineering and the matrix structure is similar to the topology of the microwave filter. Finally, use a local optimization algorithm to further iteratively optimize the M G until it converges to meet the engineering accuracy requirements. Then, the simplified matrix M D that is approximately equivalent to the initial coupling matrix and corresponds to the filter topology can be obtained.

In geodesy and geoinformatics, most problems are nonlinear in nature and often require the solution of systems of polynomial equations. Before 2002, solutions of such systems of polynomial equations, especially of higher degree remained a bottleneck, with iterative solutions being the preferred approach. With the entry of Groebner basis as algebraic solution to nonlinear systems of equations in geodesy and geoinformatics in the pioneering work “Grobner bases, multipolynomial resultants and the Gauss Jacobi combinatorial algorithms : adjustment of nonlinear GPS/LPS observations”, the playing field changed. Most of the hitherto unsolved nonlinear problems, e.g., coordinate transformation, global navigation satellite systems (GNSS)’s pseudoranges, resection-intersection in photogrammetry, and most recently, plane fitting in point clouds in laser scanning have been solved. A comprehensive overview of such applications are captured in the first and second editions of our book Algebraic Geodesy and Geoinformatics published by Springer. In the coming third edition, an updated summary of the newest techniques and methods of combination of Groenbner basis with symbolic as well as numeric methods will be treated. To quench the appetite of the reader, this presentation considers an illustrative example of a two-dimension coordinate transformation problem solved through the combination of symbolic regression and Groebner basis.

In this paper, the multiport coupling matrix theory is adopted to represent the performance of OMT, such as return loss, isolation, cross-polarization. The initial values of the corresponding coupling matrix are synthesized according to the specified return loss and bandwidth by ignoring the weak cross couplings, and then the whole coupling matrix is optimized to achieve all the specifications including the cross-polarization and isolation. Two examples of single-band and dual-band OMTs are presented to validate our approach.

Systems of Polynomials for Detecting Orientable Matroids

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n 4 log n) + log w , the condition number k(M) satisfies k(M) ≤ w n 5/2 with probability at least 1-2w -1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length.