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( nw t )+ B sin( nt)} (1) wo n o n o n =1 where wo = 2p/T is considered the fundamental frequency and the coefficients are written as 1 T ()A =� ft dt (2) o0T 2 T t )A =� f () cos( nw t dt (3) n 0 oT 2 T ()sin ( nw t dt (4) B =� f t )n 0 oT ©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 itw( e = cos wt + i sinwt ). ¥ f t int () =� Ce wo (5) n n =-¥ C� n = 1 Tft e -int () wo dt (6) 0T 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: ¥ () | dt <¥ (7) | f t 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 ft is f () such that()�w +¥ f� ( ) =�-¥ ft e w dt (8) w () -it The inverse Fourier Transform of f ()is ft� w ()+¥ w itwf ( ) = 21 p �-¥ f� ()edw (9) t ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/2/200513.42 Design Principles for Ocean Vehicles Reading #2 Example 1: Let 1��; tT1£ (10) ��0;() =�xt tT1> Take Fourier transform of xt :()+¥ ( ) 2sin wT1 (11)-itwx�(w ) =�-¥ xt e -iw tdt =�-TT 11 (1)e dt = w x�() = 2sin wT1 , -¥ <w <+¥ (12)w w Next we take the inverse Fourier transform of x�()w : wx t 1 +¥ � w it( ) = 2p �-¥ x()edw (13) () = 1 +¥ 2sin wT1 itwx t edw (14)2p �-¥ w So we arrive back at the original function xt()1��; tT1£ (15) ��0;() =�xt tT1> Example 2: Given some function in frequency space, x�()w , such that 1� w ��; w £ W1 (16) ��0; x() =� w >W1 We can take the inverse Fourier transform of this function sin Wtw 1 (17)x t 1 +¥ � w iwt 2p �-W1 (1)edw = pt( ) = 2p �-¥ x( )e dw = 1 +W1 it sin Wt() = , -¥ < t <+¥ (18)x t 1 p t ©2004, 2005 A. H. Techet 3 Version 3.1, updated 2/2/200513.42 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. d.c. 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�()w ; (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, ht , and()system input, xt , such that the output of the system is()+¥( () ( )dy t) = xth t -tt , (19) the Fourier transform of the output is written as +¥ y�(w ) = FT { �-¥ x() ( ) dt h t -tt } (20) +¥ +¥ -itt ( )de w dt (21)[ x()h t -tt ]-¥ it as e -iw (t+t -t ) = e -iw ( t-t ) iwt without changinge -wWe can then rewrite the exponential e our equation. Then let t1=-t and dt1 = dt , such that equation(21 ) becomest +¥ +¥ +¥ +¥iw (t1 +t ) dtdt1 = it1 iwt td 1 wy( ) = x()( t h t1 eedt (22)t h t )e -¥ x()()1� w �-¥ �-¥ ��-¥ Reordering the terms within the integrals we see that we have two separable integrals such that � wy( ) =� h t1 t� w ���-¥ ( )e -it1 dt1 �����{�-¥ x()e -iwt dt}. (23)�������������� h� () x� ()ww 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 2/2/200513.42 Design Principles for Ocean Vehicles Reading #2 y�(w) = h�()(). (24) w � x� w Where h�()w is the Fourier transform of the impulse response and is referred to as the TRANSFER FUNCTION, commonly written as H()w. y( ) = H ()() (25) � ww� x� w 4. Recap of Fourier Transform • Convolution: y t ( ) * xt( ) = h t () • Multiplication: y�(w) = H ()()w� x� w ( ) fi �() and y t y w ( ) + by t w• Linearity: If x t x w ( ) fi �() then ax t ( ) fi ax �(w) + by�(). 5. LTI Systems and Fourier Transforms To evaluate a LTI system you can use the Fourier transform and convolution to find the output yt() given the input and the transfer function. ( ) fi �() take the FT of the input.1. x t x w 2. y( ) = H ()() convolve the FT of the input and the transform function.x�� w w w () take the inverse FT to find the system output.3. y�(w) fi yt For a given a harmonic input ut() and transfer function H ()wwe 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 ot +( ) = xo cos( w y ) (26) ( ) = y cos( w yy 1) (27) y t t ++o o where y = xH w {(| () | is the amplitude of the response and y =argHwo )} is theo o o 1 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 …