Solution 2
3.1 The modulated signal is u(t) = m(t)c(t) = Am(t)cos(2π4 × 103 t) ] [ 250 π 200 t) + 4sin(2π t + ) cos(2π4 × 103 t) = A 2cos(2π π π 3 200 200 = Acos(2π(4 × 103 + )t) + Acos(2π(4 × 103 − )t) π π π π 250 250 +2Asin(2π(4×103 + )t+ )−2Asin(2π(4×103 − )t− ) π 3 π 3 Taking the Fourier transform of the previous relation, we obtain [ ] 250 250 200 200 2 jπ 2 jπ 3 3 u(f ) = A δ(f − ) + δ(f + ) + e δ(f − ) − e δ(f + ) π π j π j π ] 1[ δ(f − 4 × 103 ) + δ(f + 4 × 103 ) ∗ 2 A 200 200 = [δ(f − 4 × 103 − ) + δ(f − 4 × 103 + ) 2 π π π π 250 250 + 2e−j 6 δ(f − 4 × 103 − ) + 2ej 6 δ(f − 4 × 103 + ) π π 200 200 ) + δ(f + 4 × 103 + ) δ(f + 4 × 103 − π π π π 250 250 + 2e−j 6 δ(f + 4 × 103 − ) + 2ej 6 δ(f + 4 × 103 + )] π π To find the power content of the modulated signal we write u2 (t) as 200 200 )t) + A2 cos2 (2π(4 × 103 − )t) π π π 250 π 250 )t + ) + 4A2 sin2 (2π(4 × 103 − )t − ) + 4A2 sin2 (2π(4 × 103 + π 3 π 3 + terms of cosine and sine f unctions in the f irst power
u2 (t) = A2 cos2 (2π(4 × 103 +
Hence, ∫
T 2
P = lim
T →∞
u2 (t)dt =
− T2
A2 A2 4A2 4A2 + + + = 5A2 2 2 2 2
3.7 1. The spectrim of u(t) is U (f ) =
20 [δ(f − fc ) + δ(f + fc )] 2 2 + [δ(f − fc − 1500) + δ(f − fc + 1500) + δ(f + fc − 1500) + δ(f + fc + 1500)] 4 10 + [δ(f − fc − 3000) + δ(f − fc + 3000) + δ(f + fc − 3000) + δ(f + fc + 3000)] 4 1