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Unformatted text preview: 11.02.1 Chapter 11.02 Continuous Fourier Series For a function with period T , a continuous Fourier series can be expressed as [15] ∑ ∞ = + + = 1 ) sin( ) cos( ) ( k k k t kw b t kw a a t f (1) The unknown Fourier coefficients , a k a and k b can be computed as dt t f T a T ∫ = ) ( 1 (2) Thus, a can be interpreted as the “average” function value between the period interval ] , [ T . ∫ = T k dt t kw t f T a ) cos( ) ( 2 (3) k a − ≡ (hence k a is an “even” function) ∫ = T k dt t kw t f T b ) sin( ) ( 2 (4) k b − − ≡ (hence k b is an “odd” function) Derivation of formulas for , a k a and k b Integrating both sides of Equation 1 with respect to time, one gets ∫ ∫ ∫ ∑ ∫ ∑ ∞ = ∞ = + + = T T T k T k k k dt t kw b dt t kw a dt a dt t f 1 1 ) sin( ) cos( ) ( (5) The second and third terms on the right hand side of the above equations are both zeros, due to the result stated in Equation (1) of Chapter 11.01. Thus, [ ] ∫ = T T t a dt t f ) ( (6) T a = Hence, dt t f T a T ∫ = ) ( 1 (7) 11.02.2 Chapter 11.02 Now, if both sides of Equation (1) are multiplied by ) sin( t mw and then integrated with respect to time, one obtains ∫ ∑ ∫ ∫ ∫ ∑ ∞ = ∞ = + + = × T k k T T T k k dt t mw t kw b dt t mw t kw a dt t mw a dt t mw t f 1 1 ) sin( ) sin( ) sin( ) cos( ) sin( ) sin( ) ( (8) Due to Equations (1) and (3) of Chapter 11.01, the first and second terms on the right hand side (RHS) of Equation (8) are zero. Due to Equation (4) of Chapter 11.01, the third RHS term of Equation (8) is also zero, with the exception when m k = , which will become (by referring to Equation (2) of Chapter 11.01) ∫ ∫ + + = T T k dt t kw b dt t kw t f 2 ) ( sin ) sin( ) ( (9) 2 T b k × = Thus, ∫ = T k dt t kw t f T b ) sin( ) ( 2 Similar derivation can be used to obtain k a , as shown in Equation (3) A FORTRAN Program for finding Fourier Coefficients a , k a , and k b Based upon the derived formulas for a , k a and k b (shown in Equations 24), a FORTRAN/MATLAB computer program has been developed. (The program is available at http://numericalmethods.eng.usf.edu/simulations/mtl/11fft/f_coeff_final.m ) Example 1 Using the continuous Fourier series to approximate the following periodic function ( π 2 = T seconds) shown in Figure 1. Continuous Fourier Series 11.02.3 Figure 1 A Periodic Function (Between 0 and π 2 )....
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida  Tampa.
 Spring '08
 Kaw,A

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