HW4_solution

HW4_solution - MAE107 Homework #4 Solution Prof. M'Closkey...

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MAE107 Homework #4 Solution Prof. M’Closkey Problem 1 Solution I’m providing the details of the analytical solution since the “ﬂip and drag” graphical approach is straightforward. Since the impulse response is causal and the input is zero prior to t =0the convolution reduces to (note the integral limits) y ( t )= Z t 0 h ( t τ ) u ( τ ) dτ. We consider three time intervals in order to build the analytical solution. Case 1. “ t ” is conFned to the interval I 1 =[ 0 , 2]. Note that u ( t ) = 1 on this interval and that the argument of h in the convolution also coincides with I 1 . Thus, we need to derive and expression for h on I 1 : h ( t )=1 t/ 2 ,t I 1 , so y ( t Z t 0 h ( t τ ) u ( τ ) = Z t 0 (1 ( t τ ) / 2) = ± τ 1 2 + 1 4 τ 2 ² t 0 = t 1 4 t 2 I 1 Case 2. “ t ” is conFned to the interval I 2 =[2 , 4]. Note that h ( t )=2 t/ 2for t I 2 . y ( t Z t 0 h ( t τ ) u ( τ ) = Z 2 0 h ( t τ ) u ( τ ) | {z } 1 + Z t 2 h ( t τ ) u ( τ ) | {z } 0 = Z t t 2 h ( σ ) (set σ = t τ ) = Z 2 t 2 h ( σ ) + Z t 2 h ( σ ) = Z 2 t 2 (1 σ/ 2) | {z } h on I 1 + Z t 2 (2 2) | {z } h on I 2 = ± σ 1 4 σ 2 ² 2 t 2 + ± 2 σ 1 4 σ 2 ² t 2 =1 t I 2 1

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Case 3. “ t ” is confned to the interval I 3 =[4 , 6]. y ( t )= Z t 0 h ( t τ ) = Z t t 2 h ( σ ) (same steps as Case 2.) = Z 4 t 2 h ( σ ) + Z t 4 h ( σ ) | {z } 0 = Z 4 t 2 (2 σ/ 2) =9 3 t + 1 4 t 2 t I 3 When t< 0, y ( t )=0andwhen t> 6, y ( t ) = 0. Summarizing the cases, y ( t 0 0 t 1 4 t 4 t [0 , 2] 1 t [2 , 4] 9 3 t + 1 4 t 2 t [4 , 6] 0 6 The plot oF y is shown below: 2 1 0 1 2 3 4 5 6 7 8 0.5 0 0.5 1 1.5 seconds y 2
Here is the Matlab code for generating this Fgure: T = 0.01; t1 = [-2:T:0]; y1 = zeros(1,length(t1)); t2 = [0:T:2]; y2 = t2- 0.25*t2.^2; t3 = [2:T:4]; y3 = ones(1,length(t3)); t4 = [4:T:6]; y4 = 9 - 3*t4 + 0.25*t4.^2; t5 = [6:T:8]; y5 = zeros(1,length(t5)); t = [t1 t2 t3 t4 t5]; y = [y1 y2 y3 y4 y5]; figure(1) plot(t,y,’b’,’LineWidth’,2); grid on axis([-2 8 -0.5 1.5]) xlabel(’seconds’) ylabel(’y’) Problem 2 Solution We need to compute χ ( ω )= Z −∞ h ( t ) e jωt dt, where h ( t 0 t< 0 . 1 5 t [ 0 . 1 , 0 . 1] 0 t> 0 . 1 . Thus, χ ( ω Z −∞ h ( t ) e dt = Z 0 . 1 0 . 1 5 e dt = 5 ( e j 0 . 1 ω e j 0 . 1 ω ) . The code to graph this frequency response function is listed below: clear close all om = logspace(-1,3,500); H = 5*(exp(-j*0.1*om) - exp(j*0.1*om))./(-j*om); 3

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figure(1); loglog(om/2/pi,abs(H),’LineWidth’,2) grid on axis([0.01 100 1e-3 1e1]) xlabel(’Hz’) ylabel(’magnitude’) figure(2) semilogx(om/2/pi,unwrap(angle(H))*180/pi,’LineWidth’,2) grid on axis([0.01 100 -20 200]) The Bode plots are shown in Fig. 1. Note the deep notches start at 5 Hz and are thereafter repeated every 5 Hz. This is due to the fact that sinusoids with frequencies 5 k Hz, k =1 , 2 , 3 ,... ,hav e a whole number of periods in the interval [ 0 . 1 , 0 . 1]. Since the impulse response is constant over
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This note was uploaded on 10/09/2009 for the course MAE 107 taught by Professor Tsao during the Winter '06 term at UCLA.

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HW4_solution - MAE107 Homework #4 Solution Prof. M'Closkey...

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