13.42 Design Principles for Ocean Vehicles Homework #2 Out: Thursday, 12 February 2004 Due: Thursday, 19 February 2004 1. Determine whether the following systems are linear and/or time invariant. a. 0 () ( ) t yt usds + α= ∫ b. 2() [ ( )] t t yt us ds α α + − = ∫ c. () ()() du t du t yt dt dtα= d. () () () ()yt yt yt ut α β γ+ + = 2. Determine whether the following systems are LTI systems. a. 1 cosa tω → 2 cos( )a tω φ→ + b. sin 5 t → 2 cos(10 )t π→ + 3. Fourier Transform a. Find the Fourier Transform of 0() ( )ft u t τ= − . b. Given that () 0fx → as x →∞ , and the Fourier Transform of f()x is what is the Fourier Transform of df dx? (Hint: Use partial integration.) c. Given that 0df dx→ as x →∞ , what is the Fourier Transform of 2 2 df dx ? ( )f α ,4. Transfer Function: a. Given the following linear system: tmx + cx + kx = f () it } it }where input f (t) = Re{ Fe ω and response x(t ) = Re{Xe ω , and X and F are both complex quantities, find the transfer function H(ω ). b. Using the same system, for which you have just found the transfer function, if the input is α f1() +β f () determine the system output, x(t).tt 2 5. Convolution Perform the following convolutions (from page 2.11 in Triantafyllou and Chryssostomidis, Environmental Description, Force Prediction and Statistics for Design Applications in Ocean Engineering): 6. A linear time-invariant system has a transfer function H(ω ), show that when the input is sinusoidal with frequency ω o, i.e. ft() = f cos(ω t +ψ )o o the output is also a sinusoidal function with the same
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