Exercícios Resolvidos Sinais e Sistemas - Fourier Series e Transformadas

86 Sinais e sistemas 224 Considere a interconexão em cascata dos três sistemas LIT ilustrada na Figura P224a A resposta ao impulsso h2n é h2n un un 2 e a resposta ao impulso global é mostrada na Figura P224b a xn h1n h2n h2n yn b 1 5 10 11 8 4 1 1 0 1 2 3 4 5 6 7 n Figura P224 a Encontre a resposta ao impulso h1n b Encontre a resposta do sistema global para a entrada xn δn δn 1 247 Suppose an LTIC system has impulse response ht and input xt both shown in Fig P247 Use the graphical convolution procedure to determine yzsrt xt ht Accurately sketch yzsrt When solving for yzsrt flip and shift xt and explicitly show all integration steps 30 Find the signal energy of each of these signals a xt 2 rectt b xt Aut ut 10 c xt ut u10 t d xt rectt cos2πt e xt rectt cos4πt f xt rectt sin2πt g xt t 1 t 1 0 otherwise Answers 12 12 12 10A2 4 23 63 Determine the Fourier coefficients for the periodic sequence xn shown in Fig 67 From Fig 67 we see that xn is the periodic extension of 0 1 2 3 with fundamental period N0 4 Thus Ω0 2π 4 and ejΩ0 ej2π4 ejπ2 j By Eq 68 the discretetime Fourier coefficients ck are c0 14 Σ from n0 to 3 xn 140 1 2 3 32 c1 14 Σ from n0 to 3 xnjn 140 j1 2 j3 12 j12 c2 14 Σ from n0 to 3 xnj2n 140 1 2 3 12 c3 14 Σ from n0 to 3 xnj3n 140 j1 2 j3 12 j12 Note that c3 c41 c1 Eq 617 Fig 67 shows the sequence xn 59 Consider the triangular wave xt shown in Fig 513a Using the differentiation technique find a the complex exponential Fourier series of xt and b the trigonometric Fourier series of xt The derivative xt of the triangular wave xt is a square wave as shown in Fig 513b a Let xt Σ from k to of ck ejkω0t ω0 2π T0 5118 Differentiating Eq 5118 we obtain xt Σ from k to of jkω0ck ejkω0t 5119 Fig 513 shows a the triangular wave and b its derivative square wave 557 Another definition of bandwidth for a signal xt is the 90 percent energy containment bandwidth W90 defined by 12π from W90 to W90 Xω² dω 1π from 0 to W90 Xω² dω 09 Ex 5180 where Ex is the normalized energy content of signal xt Find the W90 for the following signals a xt eatut a 0 b xt sin at πt a From Eq 5155 xt eatut Xω 1 a jω From Eq 114 Ex from to xt² dt from 0 to e2at dt 1 2a Now by Eq 5180 1π from 0 to W90 Xω² dω 1π from 0 to W90 dω a² ω² 1aπ tan¹W90 a 09 12a From which we get tan¹W90 a 045π Thus W90 a tan045π 631a rads b From Eq 5137 xt sin at πt Xω paω 1 for ω a 0 for ω a Periodic Signals 27 Find the fundamental period and fundamental frequency of each of these functions a gt 10 cos50πt b gt 10 cos50πt π4 c gt cos50πt sin15πt d gt cos2πt sin3πt cos5πt 3π4 e gt 3 sin20t 8 cos4t f gt 10 sin20t 7 cos10πt g gt 3 cos2000πt 8 sin2500πt h gt g₁t g₂t g₁t is periodic with fundamental period T₀₁ 15μs g₂t is periodic with fundamental period T₀₂ 40μs Answers 120 μs and 8333 ⅓ Hz 125 s and 25 Hz π2 s and 2π Hz 2 s and 12 Hz 125 s and 25 Hz Not Periodic 04 s and 25 Hz 4 ms and 250 Hz