514-1 Trigonometric Fourier series Any periodic waveform f(t=f(t+n) can be expressed y a Fourier series provided that (1)If it is discontinuous, there are only a finite number of discontinuous in the period T; (2) It has a finite average value over the period t 3 )It has a finite number of positive and negative maximums in the period T
§14-1 Trigonometric Fourier series Any periodic waveform f(t)=f(t+T) can be expressed by a Fourier series provided that (2) It has a finite average value over the period T; (3) It has a finite number of positive and negative maximums in the period T. (1) If it is discontinuous, there are only a finite number of discontinuous in the period T;
Example: Find the Fourier series f∫(t) 10 10 f()=at(0<at<2) 2兀 10 2 4丌 5 22 2兀 2丌10 110at n Dat cos natd(at= sin not+-cos not 0 丌02丌 丌2丌n 0 2丌10 110 G at sin natd(at) cos not+-sinnot 丌02兀 x2兀 n 0 10 10 2丌cos2+0 2n兀 2 n兀 oo 10 10 10 10 sinnot ∴∫(D)=5-- sinat-sin2a sInsa 5 2兀 3兀 ∑ n=1
Example: Find the Fourier series (0 2 ) 2 10 ( ) f t = t t 5 2 10 2 0 = = a ) cos ( ) 2 10 ( 1 2 0 a t n t d t n = ) sin ( ) 2 10 ( 1 2 0 b t n td t n = n n 10 2 cos2 0 2 10 2 = − + = − f t t t t sin3 3 10 sin2 2 10 sin 10 ( ) = 5 − − − t 2 4 f (t) 100 cos 0 1 sin 2 1 10 2 0 2 = = + n t n n t n t 2 0 2 sin 1 cos 2 1 10 = − + n t n n t n t = = − 1 10 sin 5 n n nt