Fourier analysis Indian Institute f Technology Madras傅里叶分析印度马德拉斯技术学院.docVIP

UFourier analysis Fourier series: A review Let be a real valued periodic function with time period that satisfies following conditions [Dirchelet’s conditions]. Then, it can be expressed in either of the following equivalent forms. a) Where is called fundamental frequency and , harmonic where - is an integer. b) In (a), the coefficients and are real valued, while in (b), coefficient may be complex. Both forms (a) and (b) are equivalent. (b) can be obtained from (a) by following substitutions. Thus, , and are related as follows. In other words, , the usage of exponential form also introduces the concept of negative frequency. It can be interpreted in the following way. represents a phasor of unit magnitude rotating in anti-clockwise direction at speed of rad/s. represents a phasor of unit magnitude rotating in clockwise direction at speed of rad/s. The coefficients , and can be computed using the following expressions. = = Since, many of the voltage and current phasor estimation methods are based upon Least square estimation, it is a worthwhile intellectual exercise to show that Fourier series truncated at some ‘n’ can be interpreted through Least square estimation theory. We define the concept of mean square error (MSE). MSE is defined as The interval for periodic function should be time period , while should be sine/cosine function (or) exponential function of the Fourier series. Our job is to find the coefficient so as to minimize the mean square error. Thus, the optimization problem is given by Consider appr