A Design of an Active OTA-C Filter Based on DESA Algorithm

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1 POSTER 2018, PRAGUE MAY 10 1 A Design of an Active OTA-C Filter Based on DESA Algorithm Dalibor BARRI 1,2 1 Dept. of Microelectronics, Czech Technical University, Technická 2, Prague 6, Czech Republic 2 STMicroelectronics, Pobřežni 620/3, , Prague 8, Czech Republic barridal@fel.cvut.cz Abstract. With the increasing demands for design and optimization of analog integrated circuits (IC) in practical applications, a few of theoretical and practical algorithms for design and optimization of IC have been proposed. In this paper is presented a new DESA algorithm (Differential Evolution and Simplex Algorithm) that is based on the stochastic and heuristic algorithm. The combination of the two different algorithms brings a very powerful algorithm for design and optimization of any analog integrated circuits such is operation amplifier, current mirror or complex structure as is OTA-C filter. As the reference analog block was chosen active low-pass OTA-C filter. Keywords OTA-C filter, differential evolution algorithm, Nelder-Mead algorithm, DESA algorithm 1. Introduction In the present time, the computing technology and a new calculation approaches offer advanced optimization methods. These methods are currently acknowledged as a powerful tool to find out an optimal solution for complicated tasks. Typically, the character of these tasks has many variables that are independent itself. For this kind of operation, it is very complicated to find the optimal solution. Therefore, it is very important to us to have an advanced algorithm that could help us, like is proposed Modern optimization methods are categorized into the three main parts such is heuristic, deterministic and stochastic methods. The first part is a heuristic algorithm that is a technique designed for solving a problem more quickly when direct search methods [1] are too slow, or for finding an approximate solution when traditional methods fail to find an exact solution. A well-known example of a heuristic algorithm is the conventional Traveling Salesmen Problem (TSP). The problem is as follows: given a list of cities and the distances between each city, what is the shortest possible route that visitor visits each city exactly once? A heuristic algorithm used to solve this problem quickly is the nearest neighbour (NN) algorithm (also known as the Greedy Algorithm) [2] or Nelder-Mead algorithm (simplex algorithm) [3]. The second part is deterministic global optimization methods that take advantage of the analytical properties of the problem to generate a sequence of points that converge to an optimal global solution. Each step and the following step is clearly defined. For this kind of algorithm, we get the same result for the same input data. It is main difference from other optimization algorithms. In this category, we can sort algorithms as is, e.g., linear programming, and non-linear programming [4]. The last category are stochastic or random algorithms that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involve random objective functions or random constraints. Stochastic optimization methods also include methods with random iterates. That method very well finds the global maximum or minimum, but the drawback of that is unpredictable computing time. Stochastic optimization methods include the following kind of algorithm as is, e.g., simulated annealing [5], evolution algorithms, and quantum annealing algorithm. In this paper, we invented a new algorithm that combines stochastic and heuristic algorithm to have the best optimization performance for the dedicated task. The new invented algorithm is calling DESA algorithm is a combination of Differential Evolution algorithm (DE) [6] and Simplex Algorithm (SA) [3]. The combination of two and more different optimization methods is a very actual topic at the present time. The task for design and optimization are much more complex than it was in past years and required time for find out the correct solution is still shorter and shorter. Therefore, it is necessary to find the best algorithm for the solving of them. 2. DESA Algorithm The base of DESA algorithm is a stochastic method based on the evolution algorithm that is expanded by heuristic simplex optimization method. The combination of these methods ensures advance optimization performance as is short computing time, precision, and immunity stagnation in local values.

2 2 D. BARRI, A DESIGN OF AN ACTIVE OTA-C FILTER BASED ON DESA ALGORITHM 2.1 The Evolution Part of DESA Algorithm The evolution part is working with the population of i individuals that are randomly dispersed within the design space in the initialization part. Each one individual is represented by a list of variables j, that are passed into the fitness (cost) function. That fitness function defined the quality of the individual solutions and therefore also indirectly control the searching area of possible solutions. The DESA algorithm is using sets of D-dimension vectors x i, G, where all parts of the vector are independent itself (for example zeros and poles transfer function of a filter) for each generation G. In each generation exists a number of populations NP that are under evolution processing and in each evolution period we have the same number of populations. The first step toward to start realization of the algorithm is the initialization of the 0 th population by randomly generated value for each variable. These values are generated in already predefined ranges. Now, on the initializated 0 th population the first step of the DE algorithm is applied. It generates a new noisy vector v i, G+1 by the following expression: 2.2 Integration of the Simplex Method into DESA Algorithm In the previous section was explained evolution part of DESA algorithm, which offers NP new possible solutions. The algorithm is working in two phases. The first evolution phase ensures diversity members of the population in the generation. The result of the first phase is the intermediate result. In the second phase of DESA algorithm is percent part of generation from the intermediate population (PPGIP) passed to simplex algorithm, that is searching minimum value in the nearest area. The final process integration of simplex algorithm into DESA algorithm is shown in the Fig. 2 and 3. v i,g +1 = x r 1,G+F ( x r 2,G x r3,g) (1) where r 1, r 2, r 3 are randomly generated values from the range r 1, r 2, r 3 {1, 2,, NP}, F is multiple weigh coefficient in interval F0,1, x i,g is an i current individual vector in the generation G. After the first operation (1), the crossing process is realized to the ensures the diversity of all individuals. The result of that crossing process is vector u i,g+1 shown in the following Fig. 1. In the figure is present crossing process for item j = 3, j = 4, and j = 5. The crossing process is also influenced by crossing factor (CR). If CR is equal to 1 the crossing process will be done, if CR is equal to 0 the crossing process will be not done. Fig. 2. The final process integration of simplex algorithm into the Fig. 1. The crossing process of DE algorithm. To decide if the tested vector u i,g +1 will be a member of the next population G+1 is that vector compare with the current vector x i, G. If the tested vector u i,g+1 has a better result of fitness function than x i, G current vector, is the current vector replaced by the tested vector. In opposite case, the current vector will keep all parameters into the next generation G+1. Fig. 3. A Simple example of searching for global minimal value by The first randomly generated individuals of population G are highlighted by yellow color (a); intermediate population, green individuals will be optimized by SA algorithm, red individuals are directly passed to the next generation G+1 (b); the final generation G+1, blue individuals are the result of SA algorithm, red individuals are same as in the intermediate population (c).

POSTER 2018, PRAGUE MAY 10 3 2.3 The Control Parameters of DESA Algorithm DESA algorithm is based on iteration of DE algorithm and direct searching in an area by SA.

3 POSTER 2018, PRAGUE MAY The Control Parameters of DESA Algorithm DESA algorithm is based on iteration of DE algorithm and direct searching in an area by SA. Its control parameters are F, CR, NP, and PPGIP. This section more describes each control parameter. F is multiple mutation factor that is applied during generation noisy vectors. We recommend setting that parameter to a range F = The smaller value leads to premature convergence and, on the contrary, to stagnation of the population. CR is crossing factor that ensures diversity of individuals in the population and thus partially prevents stagnation. We recommend setting that parameter in the range CR = The smaller values lead to a slowdown in finding optimal solution and higher values increase the risk of stagnation and rapidly reduce the number of possible solutions. NP is the number of individuals in the population of one generation. The value of this parameter is published in the publication [6, 7] as a multiple values of independently optimized variables D. In [7] the value of this parameter is recommended to set to NP equal to 20D. Otherwise, then it is in [6], where the value of this parameter is recommended to set to NP equal to 10D. We recommended setting NP parameter to 10D. The more extensive population reduces the risk of stagnation, but on the other hand, the algorithm becomes computational more complicated and finding a solution is so far away. We do not recommend the lower number of individuals in the population, because it reduces the ability to find optimal solution and also it can easier and early achieves convergence to one solution. However, there is also an application where it is possible. Choosing of population size is individual matters and depends on the nature of the case. PPGIP is the percentage part of the intermediate population of the previous generation. We recommended choosing a value in a range PPGIP = Greater or lesser values led to getting a slower calculation of optimization. 2.4 The Fitness Function of DESA Algorithm The fitness function is not embedded part of DESA algorithm. The algorithm only causes a request for calculation of it. It is an independent function which calculates fitness value for each individual in the population at each generation step. The fitness function is defined by a user of DESA algorithm who decide the complexity of the task and based on it he defines the fitness function. The algorithm uses a one fitness function for the definition of transmission characteristic that fulfils a frequency request of the lowpass active OTA-C filter. The frequency request is defined as a set of attenuation {δ 1, δ 2,, δ N,} for frequencies higher than the cutoff frequency. That fitness function (2) is defined as the sum of the absolute value of the deviations of the functional values of the characteristic at the δ i points from the setpoint the attenuation at these points. N K 1 (δ)= E (δ i ) e(δ i ) (2) i=1 where E(δ i ) is requested value of the attenuation in the δ i point, e(δ i ) is the real value of the attenuation in the δ i point, δ i are points of interest in our case these points represent the frequency of interest, N is the total number of the points of interest. 3. Experiment 3.1 OTA-C Filter Parameters To verify complexity and practicability of DESA algorithm, the active low-pass OTA-C filter was chosen as reference analog circuit. The OTA-C filter should follow frequency criteria defined be tolerant schema Fig. 4. Fig. 4. Tolerant schema of the transmission characteristic of the OTA-C filter. 3.2 The Transmission Characteristic of the OTA-C Filter The position of zeros and poles of the demand transmission function is graphically shown in the following Fig. 5. The red rectangle and blue lines defined the areas where the poles respectively zeros can be defined. Based on that constraints DESA algorithm found the optimal solution (Fig. 6). DESA algorithm finds the final solution for transmission function in 15 minutes. The algorithm was working with the following parameters: NP = 300, F = 0.8, CR = 0.8, and PPGIP = 0.4. For evaluating was used a laptop with the Intel Pentium M, 1.73 GHz processor

4 4 D. BARRI, A DESIGN OF AN ACTIVE OTA-C FILTER BASED ON DESA ALGORITHM and the optimal solution was found in 57 generation. DESA algorithm is running in Maple software from the The Maplesoft TM company [8]. 3.3 Realization of the OTA-C Filter The final realization of the active OTA-C filter is shown in the Fig. 7, where the operation transconductance amplifiers g m02 and g m03 represent a voltage operational amplifier with the gain (3). A u = g m02 g m03 (3) where g m02, g m03 are transconductance of the OTA The final values of the used components are listed in the following Tab. 1, and the final achievement amplitude parameter values are listed in the Tab. 2. As we can see (Tab. 2) the result data are the same or better than it was required. Fig. 5. Position of zeros (green (circles) and poles (block cross) of the normalized transmission function. C I pf g m μa/v g m μa/v C I pf g m μa/v C x(3) pf C I pf g m μa/v g m μa/v C I pf g m μa/v g m μa/v Tab. 1. The final value of the used components evaluated by Amplitude parameters Frequency [khz] Requested attenuation [db] Real attenuation [db] < > Fig. 6. Designed normalized transmission function of ideal OTA-C filter. Tab. 2. Amplitude parameter result. Fig. 7. The final design of the active OTA-C filter.

5 POSTER 2018, PRAGUE MAY Summary and Conclusions In this paper, a new DESA algorithm has been presented. The invented algorithm combines two different optimization methods such as stochastic and heuristic methods. The combination of these two different methods brings a very powerful algorithm. It allows to design any analog integrated circuits such is operation amplifier, transconductance operation amplifier or complex structure as is OTA-C filter. As the reference analog block for verification of DESA algorithm, a low-pass active OTA-C filter was chosen. The final achievement circuit data are same or better than it was required. That design approach has been verified, and therefore it is recommended to extend that problem even to optimize any electrical circuit. Acknowledgements Research described in the paper was supervised by Doc. Ing. Jiří Jakovenko, Ph.D., FEE CTU in Prague and supported by the CTU SGS grant no. SGS17/188/OHK3/3T/13. References [1] HOOKE, R., and JEEVES, T. A. Direct search solution of numerical and statistical problems, In Journal of the Association for Computing Machinery, 1961, vol. 8, issue 2, p [2] SEVKLI Z., and SEVILGEN F. Variable neighborhood search for the orienteering problem, In Computer and Information Sciences ISCIS, 2006, p [3] NELDER, J. A., MEAD, R. A Simplex Method for Function Minimization. In The Computer Journal, 1965, vol. 7, no. 4, p [4] PRESS, W. H., TEUKOLSKY, S. A., VETTERLING, W. T., FLANNERY, B. P. Numerical Recipes in C: The Art of Scientific Computing Second Edition. Cambridge University Press, p , 1992, ISBN [5] KIRKPATRICK, S., GELATT, C. D., and VECCHI, M. P. Optimization by simulated annealing, In Science, 1983, vol. 220, no. 4598, p [6] STORN, R. On the Usage of Differential Evolution for Function Optimalization., Fuzzy Information Processing Society, NAFIPS, Berkeley, CA (USA), 1996, p , ISBN [7] PRICE, K. V. An Introduction to Differential Evolution. Advanced Topics In Computer Science Series, Maidenhead (UK), McGraw-Hill Ltd., 1999, p , ISBN [8] Waterloo Maple, Inc. [on-line], [cit.: ] About Authors... Dalibor BARRI was born in Prague, Czech Republic in He received his B.Sc. and M.Sc. degree in Electronics from the Czech Technical University (CTU), Prague, in 2005 and 2007, respectively. He worked at EMicroelectronics for five years as an analog IC front-end designer. At present time, he works at STMicroelectronics as an analog IC back-end designer. He is a Ph.D. student, and his topic of the thesis is to invent a novel tool for an automatic or semi-automatic layout of the analog integrated circuits.

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