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960 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 3, MARCH 2005 A Linear Inverse Space-Mapping (LISM) Algorithm to Design Linear and Nonlinear RF and Microwave Circuits José Ernesto Rayas-Sánchez, Senior Member, Fernando Lara-Rojo, Member, IEEE, and Esteban Martínez-Guerrero, Member, IEEE Abstract—A linear inverse space-mapping (LISM) optimization algorithm for designing linear and nonlinear RF and microwave circuits is described in this paper. LISM is directly applicable to microwave circuits in the frequency- or time-domain transient state. The inverse space mapping (SM) used follows a piecewise linear formulation, avoiding the use of neural networks. A rigorous comparison between Broyden-based “direct” SM, neural inverse space mapping (NISM) and LISM is realized. LISM optimization outperforms the other two methods, and represents a significant simplification over NISM optimization. LISM is applied to several linear frequency-domain classical microstrip problems. The physical design of a set of CMOS inverters driving an electrically long microstrip line on FR4 illustrates LISM for nonlinear design. Index Terms—Aggressive space mapping (ASM), Broyden, computer-aided design (CAD), high-speed digital design, interpolating neural networks, inverse space mapping (SM), neural models, nonlinear transient design, optimizing expensive functions, RF and microwave design, surrogate models. I. INTRODUCTION S PACE-MAPPING (SM) optimization techniques have been proposed in numerous innovative ways to efficiently design microwave circuits using very accurate, but computationally expensive models, typically full-wave electromagnetic (EM) simulators. A comprehensive review on SM for microwave modeling and design is the research by Bandler et al. [1]. All of the algorithmic SM approaches to microwave engineering design have been illustrated with linear frequency-domain design problems, although the original formulation of SM [2], as well as some other more advanced versions [3]–[6], consider a general formulation that, in principle, could also be applied for transient-domain design. Furthermore, in some of the SM algorithms, the frequency variable is intelligently manipulated to improve the parameter-extraction (PE) process, as in [7] and [8], making these particular SM techniques applicable only for frequency-domain linear problems. An interesting formulation to nonlinear EM optimization by SM has been recently developed [9], where the mapping inversion process is merged with the harmonic-balance analysis into the solution of a nonlinear system Manuscript received April 22, 2004; revised August 2, 2004. This work was supported in part by the Consejo Nacional de Ciencia y Tecnología, Mexican Government under Grant 010581, Grant I39341-A, and Grant PFPN-03-42-8. The authors are with the Department of Electronics, Systems and Informatics, Instituto Tecnológico y de Estudios Superiores de Occidente, Tlaquepaque, Jalisco 45090, México (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842482 of equations. In this paper, we describe an integrated transientand frequency-domain SM-based design algorithm. Artificial neural networks (ANNs) have also been extensively used for efficient electromagnetics-based design and optimization of microwave circuits [10]. All of these ANN techniques for microwave design have been developed either for the frequency or transient domains. An integrated transientand frequency-domain ANN-based algorithmic (online) design approach has not yet been reported. Powerful techniques for developing EM-based neural models in the frequency domain [11]–[13] and time domain (trained from frequency-domain EM data [14]–[16] or from time-domain measured samples [17]) have been proposed. Advanced ANN techniques for EM-based modeling of passive microwave components for frequency-domain and transient analysis have been recently developed [18]. Once these neural models are trained, they can be added to linear and nonlinear circuits to efficiently incorporate EM effects during optimization. Neural inverse space-mapping (NISM) optimization was the first SM algorithm that explicitly made use of the inverse of the mapping from the fine to the coarse model parameter spaces [6], [19]. A statistical procedure to PE is employed in NISM to avoid the need for multipoint matching and frequency mappings. An ANN whose generalization performance is controlled through a network growing strategy approximates the inverse mapping at each iteration. The ANN starts from a two-layer perceptron and automatically migrates to a three-layer perceptron when the amount of nonlinearity found in the inverse mapping becomes significant. The NISM step consists of evaluating the current neural network at the optimal coarse model solution. In this paper, we describe in detail the linear inverse spacemapping (LISM) algorithm to design by optimization proposed in [20], and compare its performance with other SM-based optimization algorithms. LISM follows a piecewise linear formulation to implement the inverse of the mapping, avoiding the use of neural networks. LISM approximates the inverse of the mapping function at each iteration by linearly interpolating the last pairs of coarse and fine model design parameters, where is the number of optimization variables. This change significantly simplifies the implementation of the algorithm with respect to the NISM version. It also allows us to generalize the algorithm to be directly applicable for frequency-domain problems, for time-domain steady-state problems, and for time-domain transient-state problems, the latter being particularly relevant in high-speed digital design. The same statistical procedure 0018-9480/$20.00 © 2005 IEEE RAYAS-SÁNCHEZ et al.: LISM ALGORITHM TO DESIGN LINEAR AND NONLINEAR RF AND MICROWAVE CIRCUITS to PE is used in LISM as in NISM. LISM also follows an aggressive formulation in the sense of not requiring up-front fine model evaluations. LISM can be applied to design linear circuits and nonlinear circuits. A rigorous comparison between Broyden-based direct SM, NISM, and LISM is realized using a synthetic example in the frequency domain. LISM optimization is contrasted with NISM, and further illustrated by a classical problem of highspeed digital signal propagation: the physical design of a set of CMOS inverters driving an electrically long microstrip line on FR4. II. SM FOR DESIGN BY SOLVING A SYSTEM OF EQUATIONS Here, we follow the SM notation [1]. Let the vectors and represent the design parameters of the coarse and fine models, respectively . The corresponding optiand . The fine model is mizable responses are in vectors assumed to be a high-fidelity (or high accuracy) representation whose evaluations are expensive, while the coarse model is a low-fidelity representation that can be intensively evaluated with no significant cost. set of characterizing responses available in the two models and , as described in [19]. An implicit assumption in (2) is that the PE process is unique (given a set of fine model characterizing responses, there is a unique vector of coarse model parameters whose coarse model responses match those of the fine model). If PE is nonunique, alternative formulations to (2) can be followed, as in [21]. C. Solving a System of Nonlinear Equations An SM-based optimization algorithm can be formulated to find the fine model parameters that make the fine model response sufficiently close to the optimal coarse model response. , i.e., by This is realized by iteratively solving finding an approximate root of the system of nonlinear equations (3) At the space-mapped solution, so that . Clearly, this SM formulation does not aim at finding the actual optimal fine model solution , which corresponds to the direct minimization of the original objective function using the fine model A. Optimizing the Coarse Model SM-based algorithms start by directly optimizing the coarse model using conventional optimization methods that typically require many function evaluations (1) where is the objective function (usually minimax) is the expressed in terms of the design specifications, and optimal coarse model design. Vector contains the operating conditions, which consists of any required combination of independent variables according to the nature of the simulation, such as the operating frequencies, time samples, bias levels, excitation levels, rise time, fall time, initial conditions, temperature, etc. B. PE In the SM context, the PE process consists of a local alignment of the two models, i.e., consists of finding the coarse model design , whose corresponding responses are as close as possible to the fine model responses at the current fine model design . The PE process can be formulated as a nonlinear multidimensional vector function , where is evaluated by solving (2) PE is a key sub-problem in any SM algorithm, and many different techniques have been proposed to realize it [1]. In this paper, we take an statistical approach when solving (2), aligning and , but the complete not only the optimizable responses 961 (4) For most practical problems, solving (4) by conventional optimization methods is prohibitive. In spite of the fact that solving the system of nonlinear equadoes not guarantee to find , it can still tions be used as an efficient practical design procedure to find the fine model parameters that yields a desired fine model response. Other SM algorithms have been formulated [22]–[24] that aim at finding at the expense of a larger number of fine model evaluations. III. BROYDEN-BASED “DIRECT” SM In a Broyden-based “direct” SM, the mapping equation is directly solved by using Broyden’s rank 1 updating formula [25]. This formulation corresponds to the so called aggressive space mapping (ASM) [26]. Here, the next iterate is predicted by (5) where the step solves the linear system (6) and matrix is an approximation of the Jacobian of with respect to at the current iterate. It is initialized by the identity matrix and updated by using (7) 962 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 3, MARCH 2005 The Broyden-based “direct” SM optimization algorithm can be summarized as follows. where (11) Algorithm: Broyden-based “Direct” SM A detailed description on how to choose the complexity of the ANN is found in [19]. begin find solving (1) , B. LISM using (2) Here, the interpolating function piecewise linear mapping defined by repeat until stopping criterion for solve is implemented by (12) using (2) and the columns of are stored in . where The linear inverse mapping is “trained” by interpolating the last pairs of designs by solving the optimization problem end (13) IV. NISM AND LISM TECHNIQUES where the th error vector is given by In the inverse SM techniques, an approximation of the inverse of the mapping function is developed at each iteration. This is over a realized by optimizing an interpolating function set of corresponding designs, where is a vector of weights. The next iterate is predicted by simply evaluating the current inverse mapping at the optimal coarse model solution (8) contains the optimal weights for the current inverse where mapping. The inverse SM algorithms can be summarized as follows. Algorithm: Generic Inverse SM begin find solving (1) , initialize repeat until stopping criterion (14) C. Common Aspects in NISM and LISM In both NISM and LISM algorithms, the vector of weights is initialized to implement a unit inverse mapping. Both algorithms also use the same optimization method (the scaled conjugate gradient available in the MATLAB Neural Network Toolbox1) for training the inverse mapping, i.e., for solving (10) and (13). Since NISM uses a two-layer perceptron during the first iterations [19], if LISM and NISM are implemented with the same PE method, then they predict the same iterates during the first iterations. Clearly, LISM represents a significant simplification over NISM, mainly because the problem of controlling the generalization performance of the ANN is avoided, which is done in NISM by controlling the complexity of the ANN (the number of hidden neurons) and the training error (the number of epochs and the magnitude of the minimum acceptable training error). using (2) train V. COMPARISON BETWEEN BROYDEN, NISM, AND LISM end A. NISM Here, the interpolating function ANN defined by is implemented by an (9a) (9b) It is seen that Broyden-based “direct” SM, NISM, and LISM algorithms require a fine model evaluation per iteration. To make a fair comparison between the three algorithms, they are implemented using exactly the same PE procedure following the statistical approach described in [19]. They also use the same termination criteria: 1) when the maximum absolute error in the solution of the system of nonlinear equations is small enough; 2) when the relative change in the fine model design parameters is small enough; or 3) when a maximum number of iterations is reached as follows: (15a) (9c) contains and the columns of and . The ANN is trained to interpolate all the accumulated pairs of designs by solving (15b) where (10) (15c) where and . 1MATLAB Neural Network Toolbox, ver. 4.0 (R12), The MathWorks, Natick, MA, 2000. RAYAS-SÁNCHEZ et al.: LISM ALGORITHM TO DESIGN LINEAR AND NONLINEAR RF AND MICROWAVE CIRCUITS 963 Fig. 1. Synthetic example to compare the performance of Broyden-based “direct” SM optimization, NISM optimization, and LISM optimization. (a) Coarse model consisting of a canonical lumped RLC parallel resonator x = [R ( ) L (nH) C (pF)] . (b) Fine model consisting of the same resonator with some parasitic elements, with R = 0:5 ; L = 0:13 nH, x = [R ( ) L (nH) C (pF)] . Fig. 2. Starting point for the SM optimization of the lumped RLC parallel resonator: optimal coarse model response (-); R (x ) and fine model response ( ) at the optimal coarse model solution R (x ), where x = [50 0:2370 11:8792] . Consider the following synthetic example. Both the coarse and fine models are illustrated in Fig. 1. The coarse model consists of a canonical parallel lumped resonator [see Fig. 1(a)], nH pF . whose design parameters are The fine model consists of the same parallel lumped resonator with some parasitic elements: a parasitic series resistor assoand a parasitic series inductance ciated to the inductance associated to the capacitor [see Fig. 1(b)]. The fine model design parameters are nH pF . We nH. These parasitic values imtake pose a severe misalignment between the two models, and makes very nonlinear. the mapping function The design specifications are (assuming a reference impedance of 50 ) from 1 to 2.5 GHz and from 2.95 to 3.05 GHz. from 3.5 to 5 GHz, and Performing direct minimax optimization of the coarse model by conventional methods (we used a sequential quadratic programming (SQP) method available in the MATLAB Optimization Toolbox2), we find the optimal coarse model solution . The optimal coarse model reand the fine model response at the optimal sponse coarse solution are illustrated in Fig. 2. We first apply the Broyden-based direct SM algorithm described in Section III. The algorithm cannot solve this problem. 2MATLAB Optimization Toolbox, ver. 2.1 (R12), The MathWorks, Natick, MA, 2000. Fig. 3. Results after applying Broyden-based “direct” SM optimization to the lumped RLC parallel resonator, for three different attempts. (a) Coarse model response (-) at x and fine model responses at the three SM solutions found ( ; ; ). (b) Fine model minimax objective function values at each iteration, for the three attempts (- -; - -; - -). Broyden-based “direct” SM optimization fails at solving this problem, becoming unstable. 2 2 Fig. 3 shows the final responses and the fine model objective function values at each iteration for three different attempts. It is seen that the Broyden-based “direct” SM algorithm becomes calculated in (7) beunstable. This is because matrix calculated in comes ill conditioned, making the next step (6) unreliable. We then apply the NISM algorithm described in Section IV-A. Since the mapping function for this problem is very nonlinear, to illustrate the difficulty in controlling the generalization performance of the ANN at each iteration, we use 3000, 5500, and 10 000 epochs. Fig. 4 shows the final responses and the fine model objective function values at each iteration for each different number of epochs. In the three cases, NISM terminates due to the number of iterations that have reached the maximum. NISM finds an excellent fine model response when 3000 epochs are used at the space mapped-solution . From Fig. 4(b), it is confirmed that too large a number of epochs yields too small a training error, which makes the ANN “over fit” the training samples, and to deteriorate its generalization performance, predicting the following iterates with a large error. 964 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 3, MARCH 2005 Fig. 4. Results after applying NISM optimization to the lumped RLC parallel resonator. (a) Coarse model response (-) at x and fine model response at the NISM solution using 3000 epochs ( ) using 5500 epochs ( ) and using 10 000 epochs ( ). (b) Fine model minimax objective function values at each NISM iteration using 3000 epochs (- -) using 5500 epochs (- -) and using 10 000 epochs (- -). This problem illustrates the difficulty in controlling the generalization performance of the ANN when the mapping between the two models is severely nonlinear. 2 2 Finally, we apply the LISM algorithm described in Section IV-B. Fig. 5 shows the final responses and the fine model objective function values at each iteration. LISM finds an excellent fine model response at the SM solution and terminates at the tenth iteration due to a sufficiently small error in the solution of the system of nonlinear equations. Since the fine model used is computationally very efficient, we can apply direct minimax optimization (using a conventional SQP optimization method). After 121 fine model evaluations starting from , the optimal fine model solution found is and the corresponding objec. The fine model tive function value is minimax objective function values at each iteration are shown in Fig. 6. Fig. 7 compares the optimal fine model response , , and the fine model the optimal coarse model response response at the space-mapped solution after applying LISM op. A comparison between and timization the fine model minimax objective function values at each itera- Fig. 5. Results after applying LISM optimization to the lumped RLC parallel resonator. (a) Coarse model response (-) at x and fine model response ( ) at the LISM solution. (b) Fine model minimax objective function values at each LISM iteration (- -). An excellent solution is found after ten iterations. Fig. 6. Fine model minimax objective function values at each iteration (-) of the conventional optimization method (SQP) directly applied to the fine model version of the lumped RLC parallel resonator. tion after applying Broyden-based direct SM, NISM, and LISM optimization is shown in Fig. 8. It is seen that LISM outperforms NISM and Broyden-based “direct” SM. Table I summarizes the numerical results obtained for this problem after applying the four optimization methods described. RAYAS-SÁNCHEZ et al.: LISM ALGORITHM TO DESIGN LINEAR AND NONLINEAR RF AND MICROWAVE CIRCUITS 965 Fig. 9. Block diagram of a set of CMOS inverters driving a capacitive load through an electrically long microstrip line. The segments of microstrip lines have a length L . The buffers are biased from a single dc voltage source. The effects of the dc power bus on the integrity of the propagated signal are usually neglected; they should be considered in a high-fidelity model (fine model). Fig. 7. Comparing results after direct conventional optimization of the lumped , fine model response () at the LISM solution, and coarse model response (-) at x . RLC parallel resonator: fine model response (- -) at x specifications, termination criteria, coarse and fine models, and optimal coarse model solutions were used as in [19]. The coarse and fine models for the first problem were implemented in MATLAB. For the second and third problems, the coarse models were implemented in APLAC,3 while the fine models were implemented in Sonnet.4 The results found after applying LISM optimization to these three problems were the same as those found in [19], as expected, since the number of NISM iterations needed for solving these three problems were no . greater than VII. CMOS DRIVERS FOR A LONG MICROSTRIP LINE Fig. 8. Comparing the optimal fine model minimax objective function value U (R (x )) with the fine model minimax objective function values at each iteration after applying Broyden-based direct SM, NISM, and LISM optimization to the lumped RLC parallel resonator. TABLE I RESULTS OF THE OPTIMIZATION OF THE LUMPED RLC PARALLEL RESONATOR VI. FREQUENCY-DOMAIN LINEAR EXAMPLES We applied LISM optimization to the following design problems: a two-section impedance transformer, a bandstop microstrip filter with open stubs, and a high-temperature superconducting (HTS) microstrip bandpass filter. The design Consider the problem of designing a set of CMOS inverters to drive an electrically long microstrip line. The most common practice to reduce the delay and other signal integrity problems when driving a long line is to place buffer stages at different locations along the line [27]. As an illustrative example, two intermediate buffer stages are inserted between the initial and final drivers on a long microstrip line, as shown in Fig. 9. is a trapezoidal pulse with a 3-V amThe input voltage plitude, 2.5-ns duration, and 100-ps rise time and fall time. The microstrip lines are on an FR4 half-epoxy half-glass substrate mil, loss tangent of 0.025, and dielecwith thickness using a width mil (50- lines). tric constant A typical 0.5- m CMOS process technology is assumed for all the inverters. The dc power line biasing each inverter is typically neglected (see Fig. 9), although it can have a significant impact on the integrity of the propagated signal. Both the coarse and fine models are implemented in APLAC. The high-fidelity model is shown in Fig. 10. It uses the Berkeley short-channel IGFET model (BSIM) (level 4) for each MOSFET, and the built-in component Mlin available in APLAC for all the microstrip line segments. The dc power line biasing each inverter is also modeled as shown in Fig. 10, assuming that it follows a completely different path than that one of the signal path. The low-fidelity model is shown in Fig. 11. It uses the level-1 (Shichman and Hodges) model for all the transistors. The main driven microstrip line (signal path) is modeled by segments of ideal lossless transmission lines using the built-in component Tlin available in APLAC. The effects of the dc power line bus are completely neglected. Due to the well-known inaccuracy of 3APLAC ver. 7.70b, APLAC Solutions Corporation, Helsinki, Finland, 2002. 4em Suite, ver. 8.52, Sonnet Software Inc., 1020 North Syracuse, NY, 2002. 966 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 3, MARCH 2005 Fig. 10. High-fidelity model for the CMOS buffers driving an electrically long microstrip line. All the MOSFETs are using the BSIM (level 4) model. The built-in component Mlin available in APLAC is used for all the microstrip line segments on FR4 half-epoxy half-glass substrate with thickness H = 10 mil, width W = 19 mil, loss tangent of 0.025, and dielectric constant " = 4:5 (50- lines). The dc power line biasing each inverter is also modeled (assuming it follows a completely different path). Fig. 12. Starting point for LISM optimization of the CMOS inverters driving a long microstrip line: input trapezoidal voltage (v ), optimal coarse model output voltage (v ), and fine model output voltage (v ) at the optimal coarse model solution x = [11 11:5 11 10:5] (in micrometers). Fig. 13. Final results after applying LISM optimization to the CMOS inverters driving a long microstrip line: input trapezoidal voltage (v ), optimal coarse model output voltage (v ), and fine model output voltage (v ) at the LISM solution x = [23 18 21 19:5] (m). Fig. 11. Low-fidelity model for the CMOS buffers driving an electrically long microstrip line. All the MOSFETs are using the Shichman and Hodges (level 1) model, the microstrip lines are modeled by segments of ideal lossless transmission lines (" = 4:5; Z = 50 ), and the effects of the dc power line are neglected. the level-1 MOSFET model for submicrometer devices [28], the channel length in the coarse model was taken as a preassigned parameter [5] and adjusted to realize an initial compensation of the coarse model (notice that the coarse model is using channel lengths of 0.8 m instead of 0.5 m). V from 0 to The design specifications are 1 ns, V from 3 to 4.5 ns, and V from 6.5 to 8.5 ns. The design parameters are the channel widths for all the MOSFETs, assuming symmetric inverters . After applying conventional optimization (using an SQP method ), the following optimal coarse (in model solution is found: micrometers). The excitation signal, as well as the coarse and fine model responses at the optimal coarse solution , are shown in Fig. 12. After six LISM iterations, a solution is found for this problem. The fine model response at the LISM solution (in micrometers) is compared with Fig. 14. Fine model minimax objective function values at each LISM iteration during the optimization of the CMOS inverters driving a long microstrip line (--). the optimal coarse model response in Fig. 13. The fine model minimax objective function values at each LISM iteration is shown in Fig. 14. LISM requires seven fine model simulations to solve this problem. The algorithm took 1 h 53 min 11 s to find the solution using a 1.6-GHz Pentium 4 computer with 512 MB of RAM. RAYAS-SÁNCHEZ et al.: LISM ALGORITHM TO DESIGN LINEAR AND NONLINEAR RF AND MICROWAVE CIRCUITS VIII. CONCLUSION We have described an LISM optimization algorithm for designing microwave circuits in the frequency- or time-domain transient state. The inverse SM follows a piecewise linear formulation, avoiding the use of neural networks. 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Broyden, “A class of methods for solving nonlinear simultaneous equations,” Math. Comput., vol. 19, pp. 577–593, 1965. [26] J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting aggressive space mapping,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 12, pp. 2874–2882, Dec. 1995. [27] R. J. Baker, H. W. Li, and D. E. Boyce, CMOS Circuit Design, Layout, and Simulation. Piscataway, NJ: IEEE Press, 1998. [28] B. Razavi, Design of Analog CMOS Integrated Circuits. New York: McGraw-Hill, 2001. José Ernesto Rayas-Sánchez (S’88–M’89–SM’95) was born in Guadalajara, Jalisco, México, on December 27, 1961. He received the B.Sc. degree in electronics engineering from the Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO), Guadalajara, México, in 1984, the Masters degree in electrical engineering from the Instituto Tecnológico y de Estudios Superiores de Monterrey (ITESM), Monterrey, México, in 1989, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 2001. Since 1989, he has been Professor with the Department of Electronics, Systems, and Informatics, ITESO. In 1997, he spent his sabbatical with the Simulation Optimization Systems Research Laboratory, McMaster University. He returned to ITESO in 2001. His research focuses on the development of novel methods and techniques for computer-aided modeling, design and optimization of analog wireless and high-speed electronic circuits and devices exploiting SM and ANNs. Dr. Rayas-Sánchez is a member of the Mexican National System of Researchers, Level I. He is currently the IEEE Mexican Council Chair, as well as the IEEE Region 9 Treasurer. He was the recipient of a 1997–2000 Consejo Nacional de Ciencia y Tecnología (CONACYT) Scholarship presented by the Mexican Government, as well as a 2000–2001 Ontario Graduate Scholarship (OGS) presented by the Ministry of Training for Colleges and Universities in Ontario, Canada. He was the recipient of a 2001–2003 CONACYT Repatriation and Installation Grants presented by the Mexican Government. He was also the recipient of a 2004–2006 SEP-CONACYT Fundamental Scientific Research Grant presented by the Mexican Government. 968 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 3, MARCH 2005 Fernando Lara-Rojo (M’98) was born in Guadalajara, México, on June 5, 1935. He received the B.Sc. degree in electronics engineering from the Instituto Politécnico Nacional (IPN), México City, México, in 1958, and the Ph.D. degree in physics from the Institute of Theoretical Physics, University of Kiel, Kiel, Germany, in 1973. From 1958 to 1968, he was an Engineer in telecommunication systems with Philips, Siemens, and the Secretaría de Comunicaciones y Transportes, Mexican Government (SCT). From 1973 to 1975, he was a Professor with Centro de Investigación y de Estudios Avanzados (CINVESTAV), México City, México, and from 1975 to 1980, with the Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Tonantzintla, México. Since 1984, he has been a Professor with the Electronics, Systems and Informatics Department, Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO), Guadalajara, México. He is mainly interested in machine intelligence and automated reasoning. Over the last seven years, his research has focused on applications of fuzzy logic and neurofuzzy systems. Dr. Lara-Rojo was the recipient of a 1968-1972 Instituto Nacional de la Investigación Científica [now the Consejo Nacional de Ciencia y Tecnología (CONACYT)] Scholarship presented by the Mexican Government. He has also been the recipient of three scholarships for scientific collaboration sabbaticals at German universities by the German Academic Exchange Service (DAAD). Esteban Martínez-Guerrero (M’03) was born in Puebla, México, on November 28, 1965. He received the B.Sc. degree in electronics engineering from the Universidad Nacional Autonoma de México (UNAM), México City, México, in 1990, the Masters degree in electrical engineering from the Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV—IPN), México City, México, in 1996, and the Ph.D. degree in integrated electronic devices from the Institut National des Sciences Appliquées de Lyon (INSA-Lyon), Lyon, France, in 2002. From 1998 to 2002, he collaborated with the Nanophysics and Semiconductor Research Group, Commissariat à l Energie Atomique (CEA)-Grenoble, Grenoble, France. Since 2002, he has been Professor with the Electronics, Systems, and Informatics Department, Instituto Tecnológico y de Estudios Superiores de Occidente (ITESO), Guadalajara, México. His research focuses on physical modeling, simulation and design of semiconductor devices, and growth and characterization of thin-film and semiconductor nanostructures, namely, quantum dots and quantum wells for optical telecommunication systems. Dr. Martínez-Guerrero is a member of the Mexican National System of Researchers, Level I. He is a member of the IEEE Electron Devices Society. He was the recipient of 1992–1995 and 1998–2002 Consejo Nacional de Ciencia y Tecnología (CONACYT) Scholarships presented by the Mexican Government.