r2_fourier

# r2_fourier - 13.42 Design Principles for Ocean Vehicles...

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13.42 Design Principles for Ocean Vehicles Reading #2 13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. Fourier Series Figure 1. Periodic Signal Fourier series are very useful in analyzing complex systems with periodic inputs as they can be used to represent a periodic signal as a summation of scaled sines and cosines : ¥ f ( t ) = A + { A cos ( n w t ) + B sin nt ) } (1) w o n o n o n = 1 where w o = 2 p / T is considered the fundamental frequency and the coefficients are written as 1 T () A = ft dt (2) o 0 T 2 T t ) A = f () cos n w t dt (3) n 0 o T 2 T ()sin n w (4) B = f t ) n 0 o T ©2004, 2005 A. H. Techet 1 Version 3.1, updated 2/2/2005

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13.42 Design Principles for Ocean Vehicles Reading #2 The Fourier series can be written more compactly using complex notation it w ( e = co s w t + i sin w t ). ¥ f t i nt () = Ce w o (5) n n =-¥ C n = 1 T ft e - i w o dt (6) 0 T We can use Fourier series to represent a periodic, absolutely integrable function, f ( t ). N.B. An absolutely integrable function is one whose integral converges when between minus and plus infinity or which has a finite number of discontinuities that can be integrated around: ¥ | (7) | 2. Fourier Transform The Fourier transform (FT) converts a function of time into a function of frequency. The inverse Fourier transform (IFT) reverts the function in the frequency domain back to the time domain. We will assume that ft () is absolutely integrable. The Fourier Transform of is f () such that () w f ( ) = w (8) w - it The inverse Fourier Transform of f () is w w w f = 2 1 p f ed w (9) t ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/2/2005
13.42 Design Principles for Ocean Vehicles Reading #2 Example 1: Let 1 ; tT 1 £ (10) 0 ; () = xt 1 > Take Fourier transform of : () ( ) 2sin w T 1 (11) - it w x ( w ) =

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r2_fourier - 13.42 Design Principles for Ocean Vehicles...

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