Minimax Optimization of Networks by Grazer Search

A new optimization method called grazer search has been developed. This method is suitable for nonlinear minimax optimization of network and system responses. A linear programming problem using gradient information of one or more highest ripples in the response error function to produce a downhill direction followed by a linear search to find a minimum in that direction is central to the algorithm. Unlike the razor search method due to Bandler and Macdonald, the present method overcomes the problem of discontinuous derivatives characteristic of minimax objectives without using random moves. It can fully exploit the advantages of the adjoint network method of evaluating partial derivatives of the response function with respect to the variable parameters. Sufficient details are given to enable the grazer search method to be readily programmed and used. Although the method is intended for the computer-aided solution of an extremely wide range of design problems, it is largely compared with other methods on microwave network design problems, for which the solutions are known. Its reliability and efficiency on more arbitrary problems, examples of which are also included, is thereby established.     

Optimization Methods for Computer-Aided Design

This paper surveys record automatic optimization methods which either have found or should find useful application in the optimal design of microwave networks by digital computer. Emphasis is given to formulations and methods which can be implemented in situations when the classical synthesis approach (analytic or numerical) is inappropriate. Objectives for network optimization are formulated including minimax and least pth. Detailed consideration is given to methods of dealing with parameter and response constraints by means of transformations or penalties. In particular, the formulation of problems in terms of inequality constraints and their solution by sequential unconstrained minimization is discussed. Several one-dimensional and multidimensional minimization strategies are summarized in a tutorial manner. Included are Fibonacci and Golden Section search, interpolation methods, pattern search, Rosenbrock's method, Powell's method, simplex methods, and the Newton-Raphson, Fletcher-Powell, and least squares methods. Relevant examples of interest to microwave circuit designers illustrating the application of computer-aided optimization techniques are presented. The paper also includes a classified list of references.     

The design of weighted minimax quadrature mirror filters

A technique for converting the constrained nonlinear optimization problem encountered in the design of weighted minimax quadrature mirror filters into an iterative unconstrained nonlinear optimization problem is presented. This renders the design of weighted minimax quadrature mirror filters possible. The technique is very efficient, typically taking about seven iterations to converge. A rapidly converging iterative procedure for solving the above nonlinear unconstrained optimization problem is also presented. This procedure typically requires less than five iterations to converge     

Cascaded Network Optimization Program

A user-oriented computer program package is presented that will analyze and optimize certain cascaded linear time-invariant electrical networks such as microwave filters and all-pass networks. The program is organized in such a way that future additions or deletions of performance specifications, constraints, optimization methods, and circuit elements are readily implemented. Presently, a variety of two-port lumped and distributed elements, all-pass C-type sections and all-pass D-type sections can be treated as fixed or variable between upper and lower bounds on the parameters. Adjoint network sensitivity formulas are incorporated. The Fletcher-Powell or Fletcher optimization methods can be called to optimize in the least pth sense of Bandler and Charalambous an objective function incorporating simultaneously, at the user's discretion, input reflection coefficient, insertion loss, group delay, and the parameter constraints (if any). The program is particularly flexible in the way in which response specifications are handled at any number of, in general, overlapping frequency bands. The package, which is written in Fortran IV, has been tested on a CDC 6400 digital computer.     

Design centring and tolerance assignment problems

A reduction in the cost of mass-produced electric circuits may be achieved by the optimal assignment of component nominal values and tolerances. A method is proposed for the transformation of this design problem into an unconstrained minimisation that can be carried out using minimax algorithms.     

Practical Least pth Optimization of Networks

A new and practical approach to computer-aided design optimization is presented. Central to the process is the application of least pth approximation using extremely large values of p, typically 1000 to 1 000 000. It is shown how suitable and reasonably well conditioned objective functions can be formulated, giving particular emphasis to more general approximation problems as, for example, in filter design. It is demonstrated how easily and efficiently extremely near minimax results can be achieved on a discrete set of sample points. Highly efficient gradient methods can be employed and, in network design problems, the use of the adjoint network approach for evaluating gradients results in greater savings in computer effort. A comparison between the Fletcher-Powell method and the more recent Fletcher method is made on the application of least pth approximation, using a range of values of p up to 1 000 000 000 000 on transmission-line transformer problems for which optimal minimax solutions are known. This is followed by filter design examples subject to certain constraints.     

Computer Optimization of Inhomogeneous Waveguide Transformers

The problem of designing broadband multisection stepped rectangular waveguide impedance transformers, when the input and output guides have different cutoff frequencies but propagate the same mode, is formulated in general terms for direct optimization by digital computer. The formulation is sufficiently flexible to allow nonideal junction discontinuity effects and mismatched terminations to be taken into account during optimization. Constraints placed on the width, height, or length of any section need be dictated only by considerations for dominant mode propagation and the requirement of small (but not necessarily negligible) junction discontinuities. The objective of the present formulation is a minimax equal-ripple response over a predetermined frequency band satisfying the constraints selected for the particular problem. The ripple search strategy to locate the maximum reflection coefficient within the band and the razor search strategy to minimize it, as described by Bandler and Macdonald in another paper, were employed. Constrained optimum equal-ripple solutions to examples previously published by Young, Matthaei et al. and Riblet are presented. They demonstrate the considerable improvements made possible by the present formulation with regard to performance, reduction in number of sections, and physical size. The approach used in this paper should also find application in the design of broadband microwave matching or equalizing networks consisting of noncommensurate components and for which exact synthesis techniques may be unavailable.     

Optimization of Microwave Networks (Short Papers)

The application of a new algorithm for minimax optimization is investigated. Unlike most of the previously published algorithms the new algorithm uses to its advantage certain obvious properties of the minimax function, namely, that the discontinuities in the first derivatives can be characterized by projections. An N-section transmission-line transformer is used as a test problem.     

Integrated model parameter extraction using large-scaleoptimization concepts

A robust approach to modelling parameter extraction in microwave circuit design is presented. The approach not only attempts to match DC and AC measurements under different bias conditions simultaneously, but also employs the DC characteristics of the device as constraints on Bias-dependent parameters, this improving the uniqueness and reliability of the solution. The approach is an expansion of the hierarchical modeling techniques recently proposed J.W. Bandler and S.H. Chen (1988). Based on J.W. Bandler and Q.J. Zhang's (1987) automatic decomposition concepts for large-scale optimization, a sequential model building method is proposed which can be combined with powerful l₁ optimization techniques to establish a model with simple topology and sufficient accuracy. Practical FET models are used to illustrate the formulation. A detailed numerical example is presented to show the effectiveness of the approach     

On Chebyshev design of linear-phase FIR filters with frequency inequality constraints

The alternation theorem is the basis of the Remez algorithm for unconstrained Chebyshev design of finite-impulse response (FIR) filters. In this paper, we extend the alternation theorem to the inequality-constrained case and present an improved Remez algorithm for the design of minimax FIR filters with inequality constraints in frequency domain. Compared with existing algorithms, the presented algorithm has faster convergence rate and guaranteed optimal solutions.     

Optimal apportionment of reliability and redundancy in seriessystems under multiple objectives

A multiobjective reliability apportionment problem for a series system with time-dependent reliability is presented. The resulting mathematical programming formulation determines the optimal level of component reliability and the number of redundant components at each stage. The problem is a multiobjective, nonlinear, mixed-integer mathematical programming problem, subject to several design constraints. Sequential unconstrained minimization techniques in conjunction with heuristic algorithms are used to find an optimum solution. A generalization of the problem in view of inherent vagueness in the objective and the constraint functions results in an ill-structured reliability apportionment problem. This multiobjective fuzzy optimization problem is solved using nonlinear programming. The computational procedure is illustrated through a numerical example. The fuzzy optimization techniques can be useful during initial stages of the conceptual design of engineering systems where the design goals and design constraints have not been clearly identified or stated, and for decision making problems in ill-structured situations     

Bandwidth efficient transmultiplexers. II. Subband complements andperformance aspects

For pt.I see ibid., vol.40, no.1, p.70-84 (1992). The performance issues related to the quadrature amplitude modulation (QAM) and vestigial sideband (VSB) transmultiplexers synthesized previously are examined. An analysis of the limitations of the configured systems regarding intersymbol interference and crosstalk suppression arising from the use of practical filters is made. A new design technique for an finite impulse response (FIR) low-pass prototype that takes the practical degradations into account is formulated. The procedure involves the unconstrained optimization of an error function. A performance evaluation reveals that for four of the five systems, the new method is superior to a minimax approach in that lower intersymbol interference and crosstalk distortions are achieved with a fewer number of filter taps. For the other transmultiplexer, the advantage of the optimized design over the minimax design is in the added flexibility of taking crosstalk into account, thereby diminishing the crosstalk distortion. The five transmultiplexers can be converted into new subband systems. The authors show how the optimized design approach formulated for the transmultiplexers over to the new subband systems     

Efficient Minimax Design of Networks without Using Derivatives

A new minimax network optimization algorithm not requiring derivatives has been developed. It is based on successive linear approximations to the nonlinear functions defining the problem. Adequate modeling of distributed parameter circuits for optimization purposes often involves parasitic, etc., which makes the gradient computation by the adjoint network method or related methods rather complicated, and often numerical errors are introduced in the gradients. Consequently, the algorithm is found to be of particular relevance in optimum design of practical microwave networks. The relative advantages of the proposed algorithm are established by comparison with known gradient and nongradient algorithms based on optimization of cascaded transmission-line transformers. The relevance to microwave filter design is demonstrated by an example which represents an improvement of analytical filter design results. Finally, optimum broad-band design of a practical coaxial transferred-electron reflection-type amplilier is carried out by means of the proposed method. The results are supported by experimental verification.     

New Algorithms for Network Optimization

Two new alogrithms suitable for computer-aided optimization of networks are presented. They are both based on the nonlinear least /spl rho/th approximation approach, which has been successfully applied by the authors to microwave network design problems requiring minimax or near-minimax solutions. A basic difference here is that, instead of requiring very large values of /spl rho/, any finite value of /spl rho/, greater than 1 can be used to produce extremely accurate minimax solutions. This paper discusses a six-variable transformer example where values of /spl rho/ equal to 2, 4, 6, 10, 100, 1000, and 10 000 have all been used separately to obtain substantially the same solution. Both the adjoint network method for gradient evaluation and the Fletcher method are employed for greater efficiency. Comparisons with the razor search and grazer search methods are made. Some far-reaching observations concerning minimax design are also made.     

Novel Algorithms for Quadratic Programming by Using Hypergraph Representations

In this paper novel algorithms are introduced for solving NP hard discrete quadratic optimization problems commonly referred to as unconstrained binary quadratic programming. The proposed methods are based on hypergraph representation and recursive reduction of the dimension of the search space. In this way, efficient and fast search can be carried out and high quality suboptimal solutions can be obtained in real-time. The new algorithms can directly be applied to the quadratic problems of present day communication technologies, such as multiuser detection and scheduling providing fast optimization and increasing the performance. In the case of multiuser detection, the achieved bit error rate can approximate the Bayesian optimum and in the case of scheduling better Weighted Tardiness can be achieved by running the proposed algorithms. The methods are also tested on large scale quadratic problems selected from ORLIB and the solutions are compared to the ones obtained by traditional algorithms, such as Devour digest tidy-up, Hopfield neural network, local search, Taboo search and semi definite relaxing. As the corresponding performance analysis reveals the proposed methods can perform better than the traditional ones with similar complexity.     

Synthesis of antenna arrays with spatial and excitation constraints

Synthesis of antenna arrays subject to spatial and excitation constraints to yield arbitrarily prescribed patterns in both the mean-squared and minimax sense are discussed. The spatial constraints may require that the interelement spacings be greater than a prescribed value or that the element locations lie within a specified region. The excitation constraints are of the form where the current-taper ratio is constrained to be less than or equal to a prescribed value. The technique employed consists of reducing the constrained optimization problem into an unconstrained one by the use of simple transformations of the independent variables. In such cases where explicit transformations are not available, a created response surface technique (CRST) has been used to convert the constrained optimization problem into a series of unconstrained optimizations. The optimization has been carried out using a nonlinear simplex algorithm. Numerical examples are given wherein both the linear and circular arrays are synthesized subject to constraints.     

Two iterative algorithms for finding minimax solutions

Two iterative minimax algorithms are presented with associated convergence theorems. Both algorithms consist of iterative procedures based on a sequence of finite parameter sets. In general, these finite parameter sets are subsets of an infinite parameter space. To show their applicabilities, several commonly used examples are presented. It is also shown that minimax problems with or without finite parameter sets can be solved by these two algorithms numerically to any assigned degree of accuracy     

一种设计近似完全重构非均匀余弦调制滤波器组的新算法

该文提出了一种设计近似完全重构非均匀余弦调制滤波器组的新算法。针对现有合并算法中非均匀滤波器组性能无法直接控制优化的缺点,新算法把非均匀滤波器组的设计问题归纳为一个关于原型滤波器的无约束优化问题,其中目标函数是非均匀滤波器组传递失真与原型滤波器阻带能量的加权和,最后利用线性迭代算法求解该优化问题。理论分析和数值实验表明,新算法获得的非均匀余弦调制滤波器组比现有算法设计的滤波器组整体性能更佳。     

Synthesis of nonuniformly spaced arrays using a general nonlinear minimax optimisation method

Antenna patterns can be synthesized using a new nonlinear minimax optimization method with sure convergence properties. Not requiring derivatives, the proposed method is general and easy to use so that it might be applied to a wide variety of nonlinear synthesis problems for which analytical solutions are not known. To test the algorithm a group of test problems for which exact analytical solutions are known has been considered, namely, optimization of Dolph-Chebyshev arrays by spacing variation. The method is further applied to find the element positions in nonuniformly spaced linear arrays with uniform excitation that produce minimized (equal) sidelobe levels, and comparisons are made with conventional Dolph-Chebyshev arrays.     

New peak-to-average power-ratio reduction algorithms for multicarrier communications

New peak-to-average power-ratio (PAPR) reduction algorithms for multicarrier systems are developed by modifying the modulation constellation in active subcarriers and the modulation symbols in unused subcarriers. The proposed algorithms yield optimal PAPR-reduction solutions. For real-baseband multicarrier systems, the proposed PAPR-reduction algorithm is developed using a fast linear programming approach and considerable performance improvement can be achieved relative to that achieved with several existing algorithms. For passband multicarrier systems, a new PAPR-reduction algorithm is constructed whereby the associated minimax optimization problem is solved using an accelerated least-p th algorithm. Simulation results are presented which demonstrate that the proposed algorithm outperforms an algorithm due to Jones and that improved PAPR reduction can be achieved when the proposed algorithm is combined with another algorithm known as selective mapping scheme.