Análise de sinais - Série de Fourier e Transformada de Laplace

2 EE SS 20221 1º T 8 w0 2πT 2π8 π4 rads xt Σ αkejw0kt k to cos2πt4 sen2πt2 Σ αkejw0kt k to ejπt4 2 ejπt4 2 ejπt2 2j ejπt2 2j Σ αkejw0kt k to 14 ejπt2 12 ejπt4 14 ejπt2 12 ejπt4 2 2j ejπt4 1 2j ejπt4 Σ 14 ejπt2 12 14 ejπt2 14 ejπt 12 14 ejπt Σ αkejw0kt k to 14 ejπt 14 ejπt2 14 14 ejπt2 14 ejπt Σ αk ejπ4 kt k to k4 a4 1 4 k2 a2 1 4 k0 a0 1 4 k2 a2 1 4 k4 a4 1 4 logo por comparação temos a4 a4 1 4 a2 a2 1 4 a0 1 ak 0 k 0 2 4 2 xt é uma função ímpar pois xt xt t logo a0 bk 0 ck 4T xt senw0kt dt 0 to T ck 46 xt senπ3 kt dt 0 to 3 ck 23 1 senπ3 kt dt 1 to 2 ck 9 3 cosπ3 kt 1 πk3 1 to 2 ck 23 1πk cos2πk3 cosπk3 ck 23πk cos2πk3 cosπk3 ck 2πk cos2πk3 cosπk3 3 h1t 2 e3t 2 e3t t0 2 t0 2 e3t t0 h2t 6 e9t ut Hjω ℱht ℱh1t h2t ℱh1t ℱh2t Hjω H1jω x H2jω H1jω h1t ejωt dt 2 e3t ejωt dt 2 e3t ejωt dt from to H1jω 2 e3jωt dt 2 e3 jωt dt from to H1jω 2 e3jωt 3 jω from to 0 2 e3 jωt 3 jω from 0 to H1jω 2 1 0 3jω 2 0 1 3jω H1jω 2 3jω 2 3jω H2jω 6e9t ut ejωt dt 6 e9jωt dt 0 to H2jω 6 e9jωt 9 jω 0 H2jω 6 9jω e e0 6 9jω logo Hjω H1jω H2jω Hjω 23jω 23jω 69jω 4 a Hs ℒh1t h2t ℒh1t ℒh2t Com a RDC RDC H1s RDC H2s H1s from to h1t est dt from 0 to 4e2t est dt H1s lim τ from 0 to τ 4e2st dt lim τ 4 2s e2sτ 1 α quando Re2s 2σ 0 σ 2 e2σjωτ e 0 CONVERGE α quando Re2s 2σ 0 σ 2 e2σjω e DIVERGE logo H1s 4 2s para σ 2 4