MIT 13 42 - Fourier Series (8 pages)

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Fourier Series



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Fourier Series

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Pages:
8
School:
Massachusetts Institute of Technology
Course:
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 Ao An cos n o t Bn sin n o t 1 n 1 where o 2 T is considered the fundamental frequency and the coefficients are written as 1 T f t dt T 0 2 An 2 T f t cos n o t dt T 0 3 Bn 2 T f sin t n o t dt T 0 4 Ao 2004 2005 A H Techet 1 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 2 The Fourier series can be written more compactly using complex notation ei t cos t i sin t f t Ce in o t 5 n n C n 1 T f t e in o t dt 0 T 6 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 f t dt 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 f t is absolutely integrable The Fourier Transform of f t is f such that f f t e i t dt 8 The inverse Fourier Transform of f is f t f t 2004 2005 A H Techet 1 2 2 f ei t d 9 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 2 Example 1 Let 1 x t 0 t T1 t T1 10 Take Fourier transform of x t x x t e i t dt x T1 1 e i t dt T1 2sin T1 2sin T1 11 12 Next we take the inverse Fourier transform of x x t x t 1 2 1 2 x ei t d 13 2sin T1 i t e d 14 So we arrive back at the original function x t 1 x t 0 t T1 t T1 15 Example 2 Given some function in frequency space x such that 1 x 0 W1 W1 16 We can take the inverse Fourier transform of this function x t 1 2 x e i t d x t 2004



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