# Article

 Article title ALGORITHM OF INTEGRATION OF THE GENERATING EQUATIONS FOR THE CALCULATION OF ELEMENTARY FUNCTIONS Authors M.I. Ledovskoy, E.S. Sinyutin Section SECTION II. METHODS, MODELS AND ALGORITHMS OF INFORMATION PROCESSING Month, Year 03, 2015 @en Index UDC 519.6:004.383 DOI Abstract In the present paper the task of creation of algorithm with low consumption of energy for a calculation of elementary functions in wireless touch systems where energy consumption is ensured at the expense of accumulation of energy from environment is considered. In an algorithm basis the tabular-algorithmic method of functions evaluation and a mode of definition of the correction is supposed by a path of integration of the generating equations describing behavior of function between the nearest tabular value and a preset value of argument. Original implementation of a mode of integration of the generating equations is offered. Integration process is unrolled on argument bits, and also the variable step equal to weight of current bit of argument is used. On an example of functions sin(x), cos(x) the possibility of use of suitable methods of integration is investigated: Euler"s methods of 1st and 2nd order, a Runge-Kutta method of 4th order from which Euler"s method of 2nd order is chosen. Association of a methodical error of algorithm from order of a method of integration and number of bit of argument is received. The step of the table of values of functions for the considered methods of integration is defined. Matching of offered algorithm with a method of linear interpolation and algorithm CORDIC is reduced. In comparison with algorithm CORDIC the number of executable cycles on the average decreases in 2 times. In comparison with a method of linear interpolation the amount of tabular values of functions several times decreases. For example, for function cos(x) the amount of tabular values decreases in 16 times. Thus the gang of operations of algorithm is limited to simple operations of addition (sub-traction) and shift. Outcomes of the experimental analysis of a methodical, tool and full error of algorithm in system MATLAB are reduced. In the conditions of the limited digit capacity of the microcontroller and calculations with the fixed point the algorithm error accepts admissible values, commensurable with a margin error roundoffs of data. Outcomes of the present paper can find application in development algorithmic and the software for wireless touch units on the basis of microcontrollers, and also other sorts of embedded systems with low consumption of energy. Download PDF Keywords Embedded systems; functions calculation; table-algorithmic method; algorithm CORDIC; algorithm of numerical integration. References 1. Новиков А. Как снизить энергопотребление встраиваемого приложения на базе микроконтроллера // Электронные компоненты. – 2015. – № 1. – С. 56-60. 2. Беляев А.О., Юдина Е.В., Синютин Е.С. Перспективные беспроводные датчики системы кардиомониторирования и эргометрии для комфортного съема биофизиологических показателей. // Труды Конгресса по интеллектуальным системам и информационным технологиям «IS&IT'14». В 4 т. Т. 2. – М.: Физматлит, 2014. – С. 117-122. 3. Bachmann C., Ashouei M., Pop V., Vidojkovic M., H. de Groot, and Gyselinckx B. Low-Power Wireless Sensor Nodes for Ubiquitous Long-Term Biomedical Signal Monitoring // IEEE Com. Mag. – Jan. 2012. – Vol. 50, No.1. – P. 36-43. 4. Гольцова М. Аккумулирование кинетической энергии из окружающей среды // Электроника НТБ. – 2011. – № 7. – С. 78–85. 5. Harb A. Energy Harvesting: State-of-the-Art // Renewable Energy 36 (Elsevier). – 2011. – Vol. 36, No. 10. – P. 2641-2654. 6. Попов Б.А., Теслер Г.С. Вычисление функций на ЭВМ: справочник. – Киев: Наукова думка, 1984. – 600 с. 7. Strollo A.G. M., De Caro D. and Petra N. Elementary functions hardware implementation using constrained piecewise polynomial approximations // IEEE Trans. Comput. – 2011. – Vol. 60. – P. 418-432. 8. Lars E. Bengtsson. Lookup Table Optimization for Sensor Linearization in Small Embedded Systems // Journal Sensor Technology. Doi:10. 4236/jst. 2012. 24025, Dec. 2012. 9. Байков В.Д., Смолов В.Б. Специализированные процессоры: Итерационные алгоритмы и структуры. – М.: Радио и связь, 1985. – 288 с. 10. Torres, Omar A. Design and implementation of a CORDIC rotator and software integration for low-power exponent computation," The University of Texas Digital Repository (UTDR), 2013–12. Available at: http://hdl.handle.net/2152/24052. 11. Moroz L., Nagayama S., Mykytiv T., Kirenko I., Boretskyi T. Simple Hybrid Scaling-Free CORDIC Solution for FPGAs," International Journal of Reconfigurable Computing Volume 2014 (2014), Article ID 615472, 4 pages, http://dx.doi.org/10.1155/2014/615472. 12. Теслер Г.С. Адаптивные экономичные асинхронные итерационные методы «цифра за цифрой» // Математические машины и системы. – 1999. – № 1. – С. 43-52. 13. Pongyupinpanich Surapong, FaizalAryaSamman and Manfred Glesne. Design and Analysis of Extension-Rotation CORDIC Algorithms based on Non-Redundant Method // International Journal of Signal Processing, Image Processing and Pattern Recognition. – March, 2012. – Vol. 5, No. 1. 14. Дайнеко Д. Реализация CORDIC-алгоритма на ПЛИС // Компоненты и технологии. – 2011. – № 12. – С. 36-46. 15. K. Sharat, B.V. Uma, D.M. Sagar. Calculation of Sine and Cosine of an Angle using the CORDIC Algorithm // International Journal of Innovative Technology and Research (IJITR). – February–March 2014. – Vol. No.2, Issue No. 2. – P. 891-895. 16. 16. Chadha A., Jyoti D., Bhatia M.G. Design and Simulation of an 8-bit Dedicated Processor for calculating the Sine and Cosine of an Angle using the CORDIC Algorithm // IEEE International Conference on Computational Intelligence and Computing Research (ICCIC), arXiv:1111.1086 (2011). 17. Бахвалов Н.С., Жидков Н.П., Кобельников Г.М. Численные методы. – М.: Бином. Лаборатория знаний, 2012. – 636 с. 18. Каляев А.В. Теория цифровых интегрирующих машин и структур. – М.: Советское радио, 1970. – 472 с. 19. Ледовской М.И. Алгоритм вычисления функций SIN(X) и COS(X) для 16-разрядных микроконтроллеров // Известия ЮФУ. Технические науки. – 2014. – № 4 (153). – С. 164-170. 20. Ледовской М.И. Моделирование алгоритма инерциальной навигации в MATLAB-SIMULINK // Ползуновский вестник. – 2011. – № 3/1. – С. 9-11.