ECE 160Solution09. Modern Digital and Analog Communication Systems

2.1-1 Both ϕ(t) and w0(t) are periodic. The average power of ϕ(t) is Pg = T1 R0T ϕ2(t) dt = π1 R0π e-t/2
2 dt = 1-πe-π . The average power of w0(t) is Pg = T10 RoT0 wo2(t) dt = T10 R0T0 1 · dt = 1. 2.1-2 (a) Since x(t) is a real signal, Ex = R02 x2(t) dt. Solving for Fig. S2.1-2(a), we have Ex = R02(1)2 dt = 2, Ey = R01(1)2 dt + R12(-1)2 dt = 2 Ex+y = R01(2)2 dt = 4, Ex-y = R12(2)2 dt = 4 Therefore, Ex±y = Ex + Ey. Solving for Fig. S2.1-2(b), we have Ex = R0π(1)2 dt + Rπ2π(-1)2 dt = 2π, | Ey = R0π/2 | (1)2 dt + Rπ/ π 2(-1)2 dt + Rπ3π/2(1)2 dt + R32π/ π 2(-1)2 dt = 2π Ex+y = R0π/2(2)2 dt + Rπ/ 3π/ 2 2(0)2 dt + R32π/ π 2(-2)2 dt = 4π Ex-y = R0π/2(0)2 dt + Rπ/ π 2(2)2 dt + Rπ/ 3π/ 2 2(-2)2 dt + R32π/ π 2(0)2 dt = 4π Therefore, Ex±y = Ex + Ey. 0 2 2 2 0 (a) -2 (b) 2 0 -2 (c) 0 2 2 2 0 (a) -2 (b) 2 0 -2