Linear Functions Based on Walsh Functions
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Abstract
In this paper, the method of constructing an orthonormal system of linear functions for approximating the function of changing the wind speed by linear approximations is considered in order to optimize the operation of the wind generator and to maximize energy recovery from renewable sources. In the construction of this system, the system of orthonormal Walsh functions was taken as the basis. In the application of the Gram-Schmidt process to the system of the basic system of functions an analytic expression for linear functions based on Walsh functions was obtained. The developed system of functions possesses all properties, such as the system of continuous Walsh functions. The discrete version of the developed functions has all the parameters of the base system of functions, and also has the same graphs as the system of discrete Walsh functions. Most common ways of organizing functions within the system were considered. It is established that the error of the approximation of the function of changing the speed of the wind does not depend on the way of the ordering of functions. An example of an approximation of the function of change in wind speed during the day is given with the help of linear functions based on Walsh functions with a dimension of 8, as well as errors of the approximation of the wind speed change function are calculated. The mean square error of approximation of linear processes does not exceed 88%, and the average relative error of approximation in knots is 40%. This system is subjected to the Gibbs effect. That is, the reduced accuracy of approximation at the ends of the approximation interval. Given the Gibbs effect, the developed system allows for the approximation of linear processes with a mean square error that does not exceed 33%, the relative error of approximation in units of approximation decreases to 23%. The choice of the dimension of the system affects the frequency spectrum of the approximating signal, which in turn affects the accuracy of the approximation. Namely, with the increase in the number of functions in the spectrum of the approximating function, the content of high-frequency harmonics increases. Therefore, when increasing the signal frequency, which is subject to approximation, it is expedient to increase the dimension of the system. In turn, for the low-frequency signal approximation, it is expedient to use as little as possible the dimensions of the system. Thus, by choosing the appropriate dimension of the system, we can minimize the error of approximation.
Ref. 11, fig. 2, add. 1.
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