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Unformatted text preview: Note #5. Fourier Series ECSE-2410 Signals & Systems - Fall 2009 0. Introduction. Read Text Section 3.3.2. The exponential Fourier series is defined as , where = = k t jk k e a t x ) ( is the fundamental frequency. 1. Derivation of Basic Equation. Example 1. Find the exponential Fourier coefficient, , associated with the second harmonic of a general signal, . 2 a ) ( t x Solution. Multiply entire equation through by , i.e. , = = k t jk k e a t x ) ( t j e 2 (Equation A) ( ) = = k t k j k t j e a e t x 2 2 ) ( Now realize that the integral of ( ) ( ) t j t e t j sin cos + = over one period, T 2 = 2 = T is ( ) ( ) . sin cos = + = dt t j dt t dt e T T T t j This is clear from figure below for ( t cos ) where the positive area is equal to negative area; same for ( t sin ) , and consequently, the same for . t j e T ) cos( t T 2 1 t Now back to Equation A above. Integrating over one period, ( ) ( ) ( ) ( ) ( ) T a dt a dt e a dt e a dt e a dt e a dt e a dt e t x T T t j T t j T t j T t j k T t k j k t j T 2 2 2 2 2 2 1 1 2 2 1 1 2 2 1 ) ( = + + + = + + + = = = L L L L Therefore, Note #5. Fourier Series. p.1 T a dt e t x t j T 2 2 ) ( = and T dt e t x T a t j T 2 , ) ( 1 2 2 = = . If the periodic signal , and its period T are known, then we can calculate exponential Fourier coefficient, . End of Example. ) ( t x 2 a The same procedure can be used to find the exponential Fourier coefficient, , for any harmonic . Thus the general equation for finding any harmonic, , of any periodic signal, , is k a k k ) ( t x T dt e t x T a t jk T k 2 , ) ( 1 = = . Example 2. Find and sketch the exponential Fourier coefficients, , for the periodic signal, , shown. k a ) ( t x t k a t x ) ( ... ... 1 1 2 3 4-2 -1 First find the fundamental frequency, . In this case 4 = T and consequently, 2 2 = = T . Next, find the average value of over one period (the DC value) ) ( t x 2 1 1 4 1 ) ( 4 1 ) ( 1 ) ( 1 1 1 2 2 = = = = = = dt dt t x dt t x T dt e t x T a T k t jk T Note that the period T chosen in the integral is from -2 to +2 ( i.e. , from 2 T to 2 T + ). Although you can use any period, certain choices simplify the problem more than others. One rule of thumb: if is even , then choose the period from ) ( t x 2 T to 2 T + . (This issue is discussed in the next problem.) Now find all the other exponential Fourier coefficients, , k a k . ( ) ( ) ( ) , sin 2 1 1 4 1 4 1 1 4 1 ) ( 4 1 ) ( 1 2 2 ) 1 ( 2 1 2 2 1 1 2 2 1 1 2 2 2 = = = = = = k k k e e jk jk e dt e dt e t x dt e t x T a jk jk t jk t jk t jk t jk T K , (Equation B) Note #5. Note #5....
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This note was uploaded on 05/03/2009 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.
- Spring '07