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MIT2_017JF09_p07

# MIT2_017JF09_p07 - 1 A n = 2 π Z π 1 f t)cos nt dt = π...

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7 FOURIER SERIES CALCULATIONS 11 7 Fourier Series Calculations Compute the Fourier series coeﬃcients A 0 , A n , and B n for the following signals on the interval T = [0 , 2 π ]: 1. f ( t ) = 2 sin( t + π/ 4) + cos(5 t + π/ 3) Solution: use trigonometric identities to rewrite this as f ( t ) = 2 sin( t )cos( π/ 4) + 2 cos( A 0 = 0 , A 1 = 2 sin( π/ 4) = t )sin( π/ 4) + cos(5 t ) cos( π/ 3) + sin(5 t )sin( π/ 3) . Thus, 2 , B 1 = 2cos( π/ 4) = 2 , A 5 = cos( π/ 3) = 1 / 2 , B 5 = sin( π/ 3) = 3 / 2 , and all the other coeﬃcients are zero. 2. 1 ,t < T/ 2 f ( t ) = ( (biased square wave) 0 ,t T/ 2 Solution: A 0 is the mean value of the signal, or A 0 = 1 / 2 . Applying the formulas for the coeﬃcients, we get

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Unformatted text preview: 1 A n = 2 π Z π 1 f ( t )cos( nt ) dt = π π Z 1 cos( nt ) dt = sin( nt ) nπ | π = 0 1 B n = 2 π Z π 1 f ( t )sin( nt ) dt = π π Z 1 sin( nt ) dt =-cos( nt ) nπ | π = z ( n ) , where z is zero if n is even, and z is 2 /nπ if n is odd. Try this out in MATLAB! MIT OpenCourseWare http://ocw.mit.edu 2.017J Design of Electromechanical Robotic Systems Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT2_017JF09_p07 - 1 A n = 2 π Z π 1 f t)cos nt dt = π...

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