MIT 13 42 - Fourier Series (8 pages)

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Fourier Series



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Fourier Series

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8
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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 2005 A H Techet 1 2 W1 W1 1 e i t d sin W1t t t 3 sin W1t t 17 18 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 2 Notice the similarity between the two functions in examples 1 and 2 specifically equations 10 and 16 and also equations 12 and 18 Parseval s theorem explains that there exists a dual pair of functions with time and frequency interchanged i e a symmetric pair of functions a b c d Figure 2 Symmetric Functions a Function of time x t b Fourier transform of x t in the frequency domain c Function of frequency x d Inverse Fourier transform back to the time domain 2004 2005 A H Techet 4 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 2 3 Convolution and the Fourier Transform For LTI systems the Fourier transform turns the convolution integral into simple multiplication Given a continuous time LTI system with impulse response h t and system input x t such that the output of the system is y t x t h t d 19 the Fourier transform of the output is written as y FT x h t d x h t d e i t 20 dt 21 We can then rewrite the exponential e i t as e i t e i t e i without changing our equation Then let t1 t and dt1 dt such that equation 21 becomes y x h t1 ei t1 d t1 x h t1 ei t1 e i d t1 22 Reordering the terms within the integrals we see that we have two separable integrals such that y h t1 e i t1 dt1 x e i d 14424431442443 h 23 x Note the form of the two separate integrals on the RHS of equation 42 they are the Fourier transform of the input and the impulse response Now the Fourier transform out the system output is simply the multiplication of the Fourier transform of the input and impulse response 2004 2005 A H Techet 5 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 2 y h x 24 Where h is the Fourier transform of the impulse response and is referred to as the TRANSFER FUNCTION commonly written as H y H x 25 4 Recap of Fourier Transform Convolution y t h t x t Multiplication y H x Linearity If x t x and y t y then ax t by t ax by 5 LTI Systems and Fourier Transforms To evaluate a LTI system you can use the Fourier transform and convolution to find the output y t given the input and the transfer function 1 x t x take the FT of the input 2 y H x convolve the FT of the input and the transform function 3 y y t take the inverse FT to find the system output For a given a harmonic input u t and transfer function H we can easily write the output of our LTI system in terms of the amplitude and freque ncy of the input and the amplitude and argument phase of the transfer function x t xo cos ot 26 y t yo cos ot 1 27 where yo xo H o is the amplitude of the response and 1 arg H wo is the phase shift of the response from the input 2004 2005 A H Techet 6 Version 3 1 updated 2 2 2005 13 42 Design Principles for Ocean Vehicles Reading 2 From complex math we can write a function H in terms of its amplitude H and its argument H as follows H H ei H 28 Now lets look at the real part of a complex function u t Re x o ei ot Re x oei ot 29 where x o xo ei Taking the convolution y t h t x t we have y t h Re x oei o t d Re h e i o d x oei o t Re H o x oe i ot where H o H o ei H o Since we are only interested in the real part of y t the input was cosine we have y t xo H o cos o t H o …


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