Exercícios Resolvidos de Sinais e Sistemas - Análise de Fourier e Transformadas

T 12 w0 2πT 2π12 π6 rads 2ptos xt a0 Σ bk cosw0 k t ck senw0 k t k1 xt é um sinal par pois xt xt logo ck 0 a0 2T 012 xt dt 21206 xt dt 16 01 1 dt 15 2 dt 16 t01 2t15 a0 16 1 10 2 16 1 8 96 32 bk 4T 012 xt cosw0 k t dt 412 01 1 cosw0 k t dt 15 2 cosw0 k t dt bk 13 senw0 k tw0 k 01 2 senw0 k tw0 k 15 bk 13 1w0 k senw0 k sen0 2w0 k sen5 w0 k senw0 k bk 13 w0 k senw0 k 2 sen5 w0 k 2 senw0 k bk 13 π k 6 senπ6 k 2 sen5 π6 k bk 2π k 2 sen5 π6 k senπ6 k 2º xt 2 sen2π t 3 sen6π t 2ptos xt Σ ak ej w0 k t k 2 ej 2π t3 2j ej 2π t3 2j ej 6π t2j ej 6π t2j Σ ak ej w0 k t k 12j ej 6π t 1j e3j ej 2π t 1d e3j ej 2π t 12j ej 6π t Σ ak ej w0 k t k w0 k 2π T k 6π 2π T k 2π 2π T k O menor valor positivo de T que satisfaz as equações é T 1 visto que k Z Logo w0 2π T 2π 1 2π rads Assim temos a3 12j j2 a1 1j e3j j e3j a1 1j e3j j e3j a3 12j j2 ak 0 k 1 e 3 3º a ht h1t h2t e2t ut cos3t j e2t ut sen3t ht e2t ut cos3t j sen3t ht e2t ut ej 3t ht e2j3 t ut Hjw Fht from to e2j3 t ut ej w t dt Hjw from 0 to e2j3jw t dt e2j3jw t2j3jw 0 Hjw 12j3w e e0 12 j3w b gt h1t ej 3t h2t ej 3t gt ej 3t h1t h2t ej 3t ht Gjw Hjw w0 Hjw 3 logo Gjw 1 2 j3 w 3 1 2 jw 4º h1t e2t sen 5t ut e2t ej5t ej5t 2j μt h2t e5t μt a Hs ℒ h1t h2t H1s H2s com RDC ℝ H1s ℝ H2s h1t e2t ej5t ej5t 2j μt 1 2j e2 j5t e2 j5t μt H1s 1 2j e2 j5t μt 1 2j e2 j5t μt H11s H12s H11s to 1 2j e2 j5t μt est dt 1 2j 0 to e2 j5 λt dt H11s lim T 1 2j 0 to T e2 j5 λt dt lim T 1 2j e2 j5 λT 2 j5 λ 0 H11s lim T 1 2j e2 j5 λT 2 j5 λ 1 2 j5 λ Quando Re2 j5 λ Re2 j5 σ jω 2 σ 0 σ 2 e2 j5 σ jω e Diverge Quando Re2 j5 σ jω 2 σ 0 σ 2 e2 σ j5 jω e 0 converge log H11s 1 2j 2 j5 λ para σ 2 H12s to 1 2j e2 j5t μt eλt dt 1 2j 0 to e2 j5 λt dt H12s lim T 1 2j e2 j5 λT 2 j5 λ 1 2 j5 λ Quando Re2 j5 λ Re2 j5 σ jω 2 σ 0 σ 2 e2 j5 λ e Diverge Quando Re2 j5 σ jω 2 σ 0 σ 2 e2 σ j5 λ e 0 converge Logo H12s 1 2j 2 j5 λ para σ 2 Portanto H1s 1 2j 1 2 j5 λ 1 2 j5 λ 2 j5 λ 2 j5 λ 2j 2 j5 λ 2 j5 λ para σ 2 σ 2 σ 2 H2s ℒ h2t to e5t μt eλt dt 0 to e5 λt dt H2s lim T 0 to T e5 λt dt lim T e5 λT 5 λ 1 5 λ H2s lim T e5 λT 5 λ 1 5 λ Quando Re5 λ Re5 σ jω 5 σ 0 σ 5 e5 σ jω e 0 converge Quando Re5 σ jω 5 σ 0 σ 5 e5 σ jω5 σ jω e Diverge Logo H2s 1 5 λ para σ 5 Como Hs H1s H2s Hs 1 2j 1 2 j5 λ 1 2 j5 λ 1 5 λ ROC σ 2 σ 5 σ 2 b Hs 2 j5 λ 2 j5 λ 2j2 j5 λ2 j5 λ 1 5 λ Hs 10j 2j2 j5 λ2 j5 λ5 λ 5 2 j5 λ2 j5 λ5 λ Hs 5 2 j5 λ 2 j5 λ 5 λ σ 2 zeros 7 polos 2 j5 λ 0 λ 2 j5 2 j5 λ 0 λ 2 j5 5 λ 0 λ 5 c Não existe a transf de Fourier de ht Fim