Chapter1 Assignment 62u5p

1. Construct the one-sided and two-sided spectrum of v(t) = -3 – 4sin(30π t). Calculate its power. 2. Sketch the amplitude spectrum of a rectangular pulse train for each of the following cases: τ =T0/5, τ =T0/2, τ =T0.

3. Find the spectrum of 1 , t < τ / 2 a) Π ( t / τ ) =  0 , t > τ / 2  Ae − bt , t > 0 b) v ( t ) =  ,t < 0  0 −b t c) v ( t ) = Ae d) v(t) = Asinc(2Wt) e) v(t) = B/[1+(2π t)2] f)

v(t) g) v ( t ) = AΠ ( t / τ ) cos ( ωc t )   t   A 1 −  , t < τ h) w ( t ) =   τ   0 , t >τ 

  cos ( ωc t )  l) w ( t ) =  cos ( 2ω t ) c  m) Au ( t ) cos ( ω0t )

,t <

2π ωc

,t >

2π ωc

A πt   t  1 + cos  Π    2 τ   2τ  2 At  t  Π  o) v ( t ) = τ τ  4. Calculate v(t)*w(t): n) v ( t ) =

v(t) 5. Given v(t) = (t-3)2. Evaluate ∞

a)

∫ v ( t )δ ( t + 4 ) dt

−∞

b) v ( t ) ∗ δ ( t + 4 ) c) v ( t ) δ ( t + 4 )

d) v ( t ) ∗ δ ( −t / 4 )

6. Let x ( t ) = AΠ ( t / τ ) be applied to the zero-order hold. Use frequency-domain analysis to find y(t) when τ = T ,τ = T ,τ ? T . 7. Find the pdf of Z=cos X where X is uniformly distributed over [0, 2π ]. 1 −x 8. Given p X ( x ) = e . Calculate mx, E[x2]. 2 9. Let v(t) = X + 3t where the r.v. X has E[x]=0 and E[x 2]=5. Find E[v(t)], Rv(t1, t2), and E[v2(t)]. 10. The random digital signal is an ensemble of rectangular pulse trains. All pulses have fixed duration D, but the delay parameter Td is a continuous r.v. uniformly distributed over 0
A1