Os Sinais e Sistemas

UNIVERSIDADE DE BRASÍLIA, FACULDADE GAMA

Sinais e Sistemas - Lista 3 Gabarito

17 de novembro de 2015

1. Calcule a Transformada de Fourier dos seguintes sinais:

( )a) x[n] =                u[n − 1]

1 n−1( )b) x[n] =

21 |n−1|

2

(                )d) x[n] = sin                n + cos(n)

c) x[n] = u[n − 2] − u[n − 6]π

e) x[n] = sin ( 5π n) + cos ( 7π n)

2

3        3

Resposta:

( )a) x[n] =                u[n − 1]

1 n−12

 

Para x[n] = (1/2)n−1u[n − 1]

X (e jΩ) =

 ∞

 

n=−∞ x[n]e−jΩn

X (e jΩ) =

X (e jΩ) =

j Ω

 ∞ (1/2)n−1e− j Ωn n=1

 ∞

(1/2)ne− j Ω(n+1)n=0Ω        1        

X (e

 ) = e−j

 (1−(1/2)e− j Ω )

( )b) x[n] =

1 |n−1|2

 

Para x[n] = (1/2)|n−1|

X (e jΩ) =

 ∞

n=−∞

n=1

n=0−∞ x[n]e−jΩn

∞

X (e jΩ) =

 (1/2)−(n−1)e− j Ωn +        (1/2)n−1e− j Ωn

---

0

n=−∞

n=0

2

1−(1/2)ej Ω

2

1−(1/2)ej Ω

(1−(1/2)e− j Ω )

1.25−cosΩ

    (1/2)−(n−1)e− j Ωn =  ∞  (1/2)(n+1)e j Ωn = ( 1 )        1        

X (e j Ω) = ( 1 )        1        + e− j Ω        1        = 0.75e−jΩ

c) x[n] = u[n − 2] − u[n − 6]

 

x[n] = u[n − 2] − [n − 6] = δ[n − 2] + δ[n − 3] + δ[n − 4] + δ[n − 5]

X (e jΩ) =

j Ω

 ∞

n=−2 j∞Ω

 x[n]e−jΩn

d) x[n] = sin ( π n) + cos(n)         

−3 j Ω