University of California, Irvine

EECS 50

1. (10 pts) The frequency response of an LTI system is given by H(Ω) = (1 + 4j sin(2Ω)) cos  3Ω 2  e −jΩ/2 (a) (4 pts) Find h[n]. (b) (3 pts) Find the output y1[n] if the input to the system is x1[n] = 1 + sin πn 2
(c) (3 pts) Find the output y2[n] if the input to the system is x2[n] which has DTFT X2(Ω) = 1 − 2 cos(Ω) Solution: (a) H(Ω) = (1 + 4j sin(2Ω)) cos  3Ω 2  e −jΩ/2 = (1 + 4j e j2Ω − e −j2Ω 2j )(e j 3 2 Ω + e −j 3 2 Ω 2 )e −jΩ/2 = (1 + 2e j2Ω − 2e −j2Ω) 1 2 (e jΩ + e −j2Ω) = 1 2 e jΩ + 1 2 e −j2Ω + e j3Ω + 1 − e −jΩ − e −j4Ω ⇒ h[n] = 1 2 δ[n + 1] + 1 2 δ[n − 2] + δ[n + 3] + δ[n] − δ[n − 1] − δ[n − 4] (b) We will use the fact that complex exponentials are eigenfunctions of LTI systems. x1[n] = 1 + sin( πn 2 ) = e j0n + 1 2j (e j πn 2 − e −j πn 2 ), so that y1[n] = H(0)e j0n + 1 2j [H( π 2 )e j πn 2 − H( −π 2 )e −j πn 2 ] = 1 + 1 2j [cos(3π 4 )e −jπ/4 e j πn 2 − cos(3π 4 )e jπ/4 e −j πn 2 ]