# Showing papers in "IEEE Transactions on Circuits and Systems in 1981"

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TL;DR: In this paper, a general condition for high-frequency synthesis applicability is given, together with a method which allows direct converter design from the desired input-output characterizations, and a general model is introduced for highfrequency synthesis converters.

Abstract: This paper deals, both from the Fourier analysis and the circuit design point of view, with a large family of electronic power converters which synthesize the assigned slow-varying waveforms via highfrequency switching, thereby needing very little reactive elements. A general condition for high-frequency synthesis applicability is given, together with a method which allows direct converter design from the desired input-output characterizations. Furthermore, a general model is introduced for high-frequency synthesis converters; as a consequence, they are characterized as two-port, multipole, time varying, linear circuit elements. Finally, as a major application example, a new AC-AC, three-to-three phase converter is introduced. The new converter displays several attractive features: sinusoidal waveforms, bidirectionality, separate control over amplitude, frequency, phase, and power factor. Moreover, depending on which side is taken as an input, it can either step up or step down the voltage. For these reasons the new converter can be regarded to as a generalized transformer.

404 citations

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TL;DR: In this paper, the authors present a framework for computing the minimal sampling period of a given digital filter structure when the speed of arithmetic operations is given but the number of processing units is unlimited.

Abstract: This paper presents a framework for Fiding efficient multiprocessor realizations of digital filters. Based on simple graph-theoretic concepts, a method is derived for determining the minimal sampling period of a given digital filter structure when the speed of arithmetic operations is given but the number of processing units Is unlimited. It Is shown how realistic hardware implementations can be found and evaluated by using the timing diagram of this maximal rate realization as a starting point. The minimal sampling periods of several common digital filter structures are given in terms of addition and multiplication times.

305 citations

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TL;DR: In this paper, a block adaptive filtering procedure is proposed in which the filter coefficients are adjusted once per each output block in accordance with a generalized least mean-square (LMS) algorithm.

Abstract: Block digital filtering involves the calculation of a block or finite set of filter outputs from a block of input values This paper presents a block adaptive filtering procedure in which the filter coefficients are adjusted once per each output block in accordance with a generalized least mean-square (LMS) algorithm Analyses of convergence properties and computational complexity show that the block adaptive filter permits fast implementations while maintaining performance equivalent to that of the widely used LMS adaptive filter

303 citations

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TL;DR: In this paper, it was shown that, by the possible use of circuit equivalences, circuits satisfying rather light assumptions possess the semistate description, and an appropriate reduction with attendant signal-flow graph for design is obtained in the linear time-invariant case.

Abstract: It is shown that, by the possible use of circuit equivalences, circuits satisfying rather light assumptions possess the semistate description {\cal Q} \dot{x} + \cal B(x, t)= \cal D u y ={\cal F} x where u = input, y = output, x = semistate, and {\cal Q}, {\cal D} ,{\cal F} are constant operators. The semistate can be chosen as tree branch voltages and link branch currents; a determination of consistent initial semistates is given which stems from a forward stepping solution equation. An appropriate reduction with attendant signal-flow graph for design is obtained in the linear time-invariant case.

293 citations

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TL;DR: The system of equations derived here has the advantage of depending on only one parameter \omega rather than on the two parameters t, and puts into evidence the fact that the spectra of the optimal reconstruction filters can be pieced together directly, without additional computation, from the elements of the system's inverse matrix.

Abstract: A deterministic signal x(t) band limited to |\omega| is passed through m linear time-invariant filters (channels) to obtain the m outputs z_1(t),\cdots,Z_m(t) . If the filters are independent in a sense to be defined, then It Is shown that the common input x(t) may be reconstructed from samples of the outputs (Z_k) , each output being sampled at m \Pi samples per second or (1/m) th the rate associated with the Input signal. A rigorous derivation of this result Is given which proceeds from a minimum error energy criterion and leads to a system of linear algebraic equations for the optimal reconstruction filters. The system of equations derived here, which differs from the system given recently by Papoulis [1], has the advantage of depending on only one parameter \omega rather than on the two parameters \omega and t ; it also puts into evidence the fact that the spectra of the optimal reconstruction filters can be pieced together directly, without additional computation, from the elements of the system's inverse matrix. Lastly, the solutions of the system obtained in the Papoulis formulation are shown to be time-varying linear combinations of the simpler one-parameter solutions.

204 citations

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TL;DR: In this article, a pair of complementary strays-insensitive switched-capacitor (SC) integrator circuits are analyzed to determine the errors in their transfer functions due to the finite gain and finite bandwidth of the op amp.

Abstract: A pair of complementary strays-insensitive switched-capacitor (SC) integrator circuits are analyzed to determine the errors in their transfer functions due to the finite gain and finite bandwidth of the op amp. The results are used to predict the transfer function deviation of biquadratic filter sections and LC ladder simulations. It is shown that while the effect of finite op amp gain is similar to that encountered in active-RC filters, SC filters are much more tolerant of the finite op amp bandwidth. However, the relationship between transfer function error and finite op amp bandwidth is an exponential one as contrasted to the linear relationship of active-R C filters. Experimental results are presented.

192 citations

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TL;DR: In this article, the concept of physically measurable operating point is defined and is related to that of strict local dissipativeness (which generalizes that of strong local passivity), and the behavior near a fold and cusp is examined.

Abstract: Some circuits exhibit jump behavior. for example, this occurs when the velocity field specified by the \dot{i}_L and \dot{\upsilon}_c of the inductor and capacitor characteristics cannot be "lifted" on to the resistive constraint manifold. The (jump) behavior is viewed as the limit as \epsilon \rightarrow 0 of the solutions of a regularized system of equations obtained by introducing suitably located \epsilon -parasitic L 's and C 's: this leads to a consistent way of defining discontinuous solutions. In particular, the behavior near a fold and cusp is examined. The concept of physically measurable operating point is defined and is related to that of strict local dissipativeness (which generalizes that of strict local passivity). Two examples are included.

175 citations

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TL;DR: In this paper, the authors derived limit theorems for estimating the wiring area on large chips in terms of average wire length, average number of wires emanating from each logic block, and wire trajectory parameters.

Abstract: Two-dimenslonal stochastic models for Interconnections in master slice LSI are described Several limit theorems are derived for estimating the wiring area on large chips in terms of average wire length \bar{R} , average number of wires emanating from each logic block \lambda , and wire trajectory parameters The expected value of the maximum number of tracks per channel on an N \times N chip is shown to be less than O(\ln N) as long as \bar{R} does not grow faster than O(\ln N) If \bar{R} > O(\ln N) , then the expected maximum number of tracks is O(\bar{R}) Simple bounds on the expected wiring area are given and numerical results compared to the earlier work by Helier et al

156 citations

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TL;DR: In this paper, Chaotically transitional processes in the forced negative-resistance oscillator were investigated using analog and digital computers and the difference between the almost periodic oscillations and the Chaotic Transition Process (CTP) was clarified.

Abstract: This paper deals with chaotically transitional phenomena which occur In the forced negative-resistance oscillator. Experimental studies using analog and digital computers have been carried out. The difference between the almost periodic oscillations and the chaotically transitional processes is clarified. Various strange attractors representing chaotically transitional processes and their average power spectra are given. They are discussed in detail and compared with the results obtained in the forced oscillatory systems.

144 citations

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TL;DR: A Hilbert space approach to the derivations of magnitude normalized signal and gain recursions is presented and normalized forms are expected to have even better numerical properties than the unnormalized versions.

Abstract: Recursive least squares ladder estimation algorithms have attracted much attention recently because of their excellent convergence behavior and fast parameter tracking capability, compared to gradient based algorithms. We present some recently developed square root normalized exact least squares ladder form algorithms that have fewer storage requirements, and lower computational requirements than the unnormalized ones. A Hilbert space approach to the derivations of magnitude normalized signal and gain recursions is presented. The normalized forms are expected to have even better numerical properties than the unnormalized versions. Other normalized forms, such as joint process estimators (e.g., "adaptive line enhancer") and ARMA (pole-zero) models, will also be presented. Applications of these algorithms to fast (or "zero") startup equalizers, adaptive noise- and echo cancellers, non-Gaussian event detectors, and inverse models for control problems are also mentioned.

143 citations

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TL;DR: A number of switched-capacitor building blocks useful in adaptive systems are presented and it is shown, how a number of these signal processing blocks can be combined together to produce an adaptive channel equalizer.

Abstract: A number of switched-capacitor (SC) building blocks useful in adaptive systems are presented. These include a phase-lock loop, a tracking filter, a second-order equalizer with programmable gain, zeros' frequency and zeros' Q factor, a quadrature sine-wave generator, and a synchronous demodulator. It is shown, how a number of these signal processing blocks can be combined together to produce an adaptive channel equalizer. The operation of most of the circuits presented has been verified using discrete prototypes.

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TL;DR: In this paper, a new heuristic procedure based on iterative improvement is proposed, which repeats random generation of an initial solution and its Improvement by a sequence of local transformations, the best among the local optimum solutions is taken as a final solution.

Abstract: This paper deals with the optimum placement of modules on a two-dimensional board, which minimizes the total muting length of signal sets. A new heuristic procedure, based on iterative improvement, is proposed. The procedure repeats random generation of an initial solution and its Improvement by a sequence of local transformations. The best among the local optimum solutions is taken as a final solution. The iterative improvement method proposed here is different from the previous one, in the sense that it considers interchanging more than two modules at the same time and examines only a small portion of feasible solutions which has high probability of being better. Experimental results show this procedure gives better solutions than the best one up to now. The computation time for each local optimum solution grows almost linearly with regard to the number of modules.

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TL;DR: In this article, two efficient algorithms are presented for obtaining steadystate solutions of nonlinear circuits and systems driven by two or more distinct frequency input signals, which are particularly useful in cases where the steady state response is either not periodic, or is periodic but its period is too large for existing methods.

Abstract: Two efficient algorithms are presented for obtaining steadystate solutions of nonlinear circuits and systems driven by two or more distinct frequency input signals. These algorithms are particularly useful in cases where the steady-state response is either not periodic, or is periodic but its period is too large for existing methods. The first algorithm is applicable to any circuit or system driven by any number p\geq 2 of input frequencies. The second algorithm is restricted only to 2 input frequencies and is therefore significantly more efficient than the first algorithm. Both algorithms are formulated for systems described by an implicit system of nonlinear algebraic-differential equations, thereby obviating the need to write state equations. Numerous examples have been solved successfully using these two algorithms. A selection of some of these examples is given for illustrative purposes.

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TL;DR: The most costly phase of statistical design,statistical simulation, may be carried out only once, and equivalent or superior designs for intermediate size networks are obtained with less computational effort than previously published methods.

Abstract: A new statistical circuit design centering and tolerancing methodology based on a synthesis of concepts from network analysis, recent optimization methods, sampling theory, and statistical estimation and hypothesis testing is presented. The method permits incorporation of such realistic manufacturing constraints as tuning, correlation, and end-of-life performance specifications. Changes in design specifications and component cost models can be handled with minimal additional computational requirements. A database containing the results of a few hundred network analyses is first constructed. As the nominal values and tolerances are changed by the optimizer, each new yield and its gradient are evaluated by a new method called Parametric sampling without resorting to additional network analyses. Thus the most costly phase of statistical design,-statistical simulation, may be carried out only once, which leads to considerable computational efficiency. Equivalent or superior designs for intermediate size networks are obtained with less computational effort than previously published methods. For example, a worst-case design for an eleventh-order Chebychev filter gives a filter cost of 44 units, a centered worst-case design reduces the cost to 18 units and statistical design using Parametric sampling further reduces the cost to 5 units (800 analyses, 75 CPU seconds on an IBM 370/158).

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TL;DR: In this paper, the authors use noninteger order filters to make oscillators of fractional order greater than 2, which are unusual not only in their order but also in their properties, due to an uncommon form of the oscillator frequency as regards the direct production of low frequency sine waves.

Abstract: After recalling the major disadvantages of integer order sinusoidal oscillators, we use noninteger order filters to make oscillators of fractional order greater than 2. These oscillators are unusual not only in their order but also in their properties, due to an uncommon form of the oscillator frequency as regards the direct production of low frequency sine waves. An algebraic performance criterion for the quality of systems just above the threshold of oscillation gives the optimum fractional order ensuring the sharpest tuning and the greatest frequency stability. The fact that the special form of the oscillation frequency in such systems satisfies the requirements of linearization of frequency modulation characteristics, is used in low frequency FM at large deviation and high linearity.

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TL;DR: A set of computer aids is described which supports a hierarchical design methodology for digital VLSI circuits and aims to provide a design environment which allows for a significant reduction in time between the initial concept of a complex digital system and the generation of masks.

Abstract: The current status of a research program at Carnegie-Mellon University aimed at the formulation of a hierarchical design methodology for digital VLSI circuits is described In addition, this paper describes a set of computer aids which supports this methodology One of the goals of this work is to provide a design environment which allows for a significant reduction in time between the initial concept of a complex digital system and the generation of masks Another goal is to allow the designer to efficiently explore a number of design alternatives

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TL;DR: In this article, a comprehensive analysis of the mean-squared error (MSE) of adaptation for LMS is presented, based on the method developed in the 1968 dissertation by K. D. Senne, and it represents the most complete treatment of the subject published to date.

Abstract: In narrow-band adaptive-array applications, the mean-square convergence of the discrete-time real least mean-square (LMS) algorithm is slowed by image-frequency noises generated in the LMS loops. The complex LMS algorithm proposed by Widrow et aL is shown to eliminate these noises, yielding convergence of the mean-squared error (MSE) at slightly over twice the rate. This paper includes a comprehensive analysis of the MSE of adaptation for LMS. The analysis is based upon the method developed in the 1968 dissertation by K. D. Senne, and it represents the most complete treatment of the subject published to date.

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Massachusetts Institute of Technology

^{1}, University of California, Berkeley^{2}, Bell Labs^{3}, TRW Inc.^{4}TL;DR: In this article, the authors examine the conflicting definitions of passivity and demonstrate the contradictions between them with several examples, and propose a particular definition of nonlinear passivity which avoids these contradictions by eliminating the dependency on a state of zero stored energy.

Abstract: This paper is the first in a two-part sequence which aims to state rigorously the energy-based concepts which are fundamental to nonlinear network theory, passivity and losslessness, and to clarify the way they enter the input-output and the state-space versions of the subject. In this part we examine the conflicting definitions of passivity which exist in the literature and demonstrate the contradictions between them with several examples. We propose a particular definition of passivity which avoids these contradictions by eliminating the dependency on a state of "zero stored energy," and we show that it has the appropriate properties of representation independence and closure. We apply it to several specific classes of n -ports and derive equivalent passivity criteria. The exact conditions are given under which this definition is equivalent to one based on an internal energy function, and we use the concept of an internal energy function to provide a canonical network realization for a class of passive systems.

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TL;DR: In this article, the authors present two new structures for adaptive filters based on the idea of frequency sampling filters and gradient based estimation algorithms, which operate in real time with no batch processing of signals as is the case when using the discrete Fourier transform.

Abstract: We present two new structures for adaptive filters based on the idea of frequency sampling filters and gradient based estimation algorithms. These filters have a finite impulse response (FIR) and can be thought of as attempting to approximate a desired frequency response at given points on the unit circle. The filters operate in real time with no batch processing of signals as is the case when using the discrete Fourier transform. They result in a marked reduction in dimension of the timedomain problem of fitting an Nth-order FIR transversal filter to a collection of length 2 transversal filters and further to a collection of N scalar filters. The advantages of this are then discussed.

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TL;DR: In this paper, a user-oriented weight selection heuristics are developed and applied to the MCDM design of an MOSFET hand gate and then extended to generate a family of weighted p-norm methods for solving the mCDM problem.

Abstract: Tbe basis of most engineering design Is making tradeoffs among competing factors. This Is especially true in the design of electronic circuits. In this paper we examine multiple criterion optimization (MCO), one aspect of the multiple criterion decision making (MCDM) problem. Because the Ideas and methods of MCO may not be familiar to circuit designers a tutorial review of the area is included. We then develop a weighted oo-norm method for generating noninferior solutions to the MCO problem. User oriented weight selection heuristics are developed and we apply this technique to the MCO design of an MOSFET hand gate. Next we extend this technique to generate a family of weighted p-norm methods for solving the MCO problem. We also develop a canonic weight associated with the \rho -norm method that contains the weight selection technique of the \alpha -norm. This family of MCO methods allows a new and useful interpretation to be given to many of the optimization methods that have been applied to the design of electronic circuits.

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TL;DR: Chebyshev gain functions have been widely employed for tly, matching a complex load to a resistive generator as discussed by the authors, and such transfer functions do result in optimum response when the terminations are purely resistive.

Abstract: Chebyshev gain functions have been widely employed for tly, matching a complex load to a resistive generator. Such transfer functions do result in optimum response when the terminations are purely resistive. However, assuming the overall transducer gain characteristic has monotone decreasing stopband behavior, the equal ripple transfer function is shown to be not optimum for a complex load. Furthermore, equalizers simpler in structure and superior in frequency response to equal ripple designs can readily be synthesized. Indeed it appears that nonoptimality will generally result whenever analytic gain-bandwidth theory is used to determine the constants of a transfer function belonging to an a priori specified class. Examples are presented for the matching of LCR and CR loads.

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TL;DR: In this article, several new efficient residue-to-decimal conversion algorithms are presented, running from 12 to 36 bits with execution speed measured in nanoseconds, using a fixed-point dynamic range.

Abstract: One of the fundamental problems with residue arithmetic is the difficulty associated with residue-to-decimal conversion. In this paper, several new efficient conversion algorithms are presented. They span a practical fixed-point dynamic range.running from 12 to 36 bits with execution speed measured in nanoseconds. The mathematical foundations of the developed methods are discussed in detail and realizable architectures presented.

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TL;DR: In this article, a comparison between an optimal (branch-and-bound) algorithm and a suboptimal (loca search) algorithm for the design of finite wordlength finite-impulse-response (FIR) digital filters is presented.

Abstract: This paper presents a comparison between an optimal (branch-and-bound) algorithm and a suboptimal (loca search) algorithm for the design of finite wordlength finite-impulse-response (FIR) digital filters. Experimental results are described for 11 examples of length 15 to 35. We conclude that when computer resources are not available for the optimal method, it is still worth applying the local search method to the filter with rounded coefficients.

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TL;DR: The convergence properties of two different algorithms for the updating of the coefficients of an adaptive FIR digital filter are investigated and compared with one another in this paper, and it is shown that the convergence of the sign algorithm can always be assured but is much slower than that of the stochastic iteration algorithm if the same variance of the residual echo is to be obtained.

Abstract: The convergence properties of two different algorithms for the updating of the coefficients of an adaptive FIR digital filter are investigated and compared with one another. These algorithms are the stochastic iteration algorithm and the sign algorithm. In this latter algorithm a one-bit gradient estimation is used which makes its implementation very simple. The convergence is characterized by the residual echo variance after convergence, and a parameter that indicates the speed of the convergence. It is shown that the convergence of the sign algorithm can always be assured but is much slower than that of the stochastic iteration algorithm if the same variance of the residual echo is to be obtained.

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TL;DR: A new high-speed large moduli multiplier has been developed, which is the result of combining the quartersquared algorithm with recent breakthroughs in device technology.

Abstract: The residue number system has been recently shown to be a viable signal processing media. However, it does possess limitations. One of the most serious is overflow prevention through magnitude scaling. One method of overcoming this defect is to increase the dynamic range of the numbering system. To this end a new high-speed large moduli multiplier has been developed. The multiplier is the result of combining the quartersquared algorithm with recent breakthroughs in device technology. As a result, equivalent 18-bit full precision products can be obtained at a pipelined rate of 28.5 \times 103 multiplies per second.

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TL;DR: The method proposed is based on measurements of voltage using current excitations and has been developed for the location of a number of faults and utilizes certain algebraic invariants of faulty elements.

Abstract: This paper deals with multiple-fault detection for linear analog circuits. The method proposed is based on measurements of voltage using current excitations and has been developed for the location of a number of faults. It utilizes certain algebraic invariants of faulty elements. Computationally, it depends on checking the consistency or inconsistency of suitable sets of linear equations. The equations themselves are formulated via adjoint circuit simulations.

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TL;DR: The contention is that the matching problem is a very natural nonlinear analog of the classical prediction error problem, which might broaden the appeal of impedance matching theory since it is easy to remember and is intriguing to the many systems theorists who are schooled in linear prediction theory.

Abstract: This paper begins by presenting a powerful method which is easy to apply to many broad-band circuit design problems. Anyone with a broad-band design problem, which in the narrow-band case amounts to finding a point inside of a certain circle (say on the Smith chart), might find the method here very useful (Section I). Gain equalization problems fall into this category and the main subject of this paper is a conceptually appealing, highly practical, and very flexible theory of gain equalization. The clever matching theory developed by Fano and Voula in principle handles passive one-ports well except for some difficulty in computing gain-bandwidth limitations. It converts the main problem into computing solutions to a system of nonlinear equations which are in practice so formidable that typical text book treatments [8], [9] never address the issue of solving them systematically. Also classical theory requires the load and gains to be specified as rational functions. Our theory does a good job on gain-bandwidth limitations, reduces all problems to ones of finding eigenvalues and eigenvectors of a given matrix, and only requires the load and gains be specified as data on a frequency band. Our theory is highly effective for multiports and so settles the old impedance matching problem for passive multiport circuits. The concrete results which we present here are: (1) Two numerically efficient ways to determine theoretical gain bandwidth limitations for one-ports and n-ports; (2) For one-ports a quick way to compute the frequency response function for the optimal coupling circuit directly from the answer obtained in (1). The recent advance of broad-band microwave technology has produced a need for more general and more flexible theories of gain equalization. The type of theory called for is based on measured data and avoids rational functions and spectral factorizations until late in the design process. One typically specifies a desired gain profile G(j\omega) and then wants to find the largest multiple \kappa G of it which is realizable. The procedure described herein is well suited to these needs since is requires only measured data and since determination of \kappa is automatic. A very different method for broadbanding which fills these needs was developed by Carlin [10]. It is a clever approach with quite a few compromises. One possible use of the lengthier rigorous procedure here would be to check the accuracy of Carlin's method. In addition to quantitative results we present some (much more easily learned) qualitative properties which every circuit (passive or active) designed to optimize gain possesses. They might be of considerable practical use in that any designer can learn them instantly and thereby obtain a certain (small) amount of general orientation very cheaply. The section on qualitative results, Section Ill, can be read independently of the rest of the paper (except for Fig. 1.1 and environs) and that might be best for some practical designers who have little taste for theory. Also in this paper we describe a certain viewpoint to the matching problem itself. From this perspective the matching problem is an elegant mathematical problem which fits solidly into a long line of classical mathematics. The classical mathematics underlies the computation of "prediction error" in Wiener's prediction theory. Our contention is that the matching problem is a very natural nonlinear analog of the classical prediction error problem. This formulation might broaden the appeal of impedance matching theory since it is easy to remember and is intriguing to the many systems theorists who are schooled in linear prediction theory (Section IV).

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TL;DR: This paper presents a stochastic optimization process, suitable for optimizing functions of a certain measure over generalized regions in R^n, and examples of a wide range of different optimization problems are given.

Abstract: The nonlinear optimization problem and statistical design problem can both be formulated as a region search problem. In this paper, we present a stochastic optimization process, suitable for optimizing functions of a certain measure over generalized regions in R^n . Conditions for an optimal process are discussed, and examples of a wide range of different optimization problems are given. These include the optimization of constrained, discontinuous and random functions in both discrete and continuous variable space. Design centering and tolerancing of large size systems subject to environmental disturbances are also treated.

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TL;DR: Algorithmic passivity shows the overall stability--or lack thereof--inherent in forward Euler, backward Euler and trapezoidal one-step integration approximations as well as Gear's two-step method.

Abstract: In this paper we introduce the concept of algorithmic passiv- in in ity and indicate its role in circuit and timing simulation programs Algorithmic passivity shows the overall stability--or lack thereof--inherent in forward Euler, backward Euler, and trapezoidal one-step integration approximations as well as Gear's two-step method Passivity can be applied to - Vi_ _ Vn-i+ i integration algorithms to provide sufficient conditions for the overall stability of simulation programs that may employ them on an intermixed basis

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TL;DR: In this paper, the authors discuss the nature of the recursive error surface and give examples of conditions under which local minima may exist, and conclude with a discussion of the effects of the non-quadratic error surface on gradient-search algorithms for recursive adaptive filters.

Abstract: For an adaptive filter with N adjustable coefficients or weights, the "error surface" is a plot, in N+ I dimensions, of the mean-squared error versus the N coefficient values. If the adaptive filter is nonrecursive, the error surface is a quadratic function of the coefficients. With recursive adaptive filters, the error surface is not quadratic and may even have local minima. In this correspondence we discuss the nature of the recursive error surface and give examples of conditions under which local minima may exist. We conclude with a discussion of the effects of the nonquadratic error surface on gradient-search algorithms for recursive adaptive filters.