A number of systems are specified below in terms of their

Question

Assignment: Signals and System Problems

1. A number of systems are specified below in terms of their input-output relationships. For each case, determine if the system is linear and/or time-invariant.

a. y(t)=|x(t) |+x(t)
b. y(t)=t x(t)
c. y(t)=e^(-t) x(t)
d. y(t)= ∫_(-∞)^tx(λ)dλ
e. y(t)= ∫_(t-1)^tx(λ)dλ
f. y(t)=(t+1)∫_(-∞)^tx(λ)dλ

2. Find a differential equation between the input voltage x(t) and the output voltage y(t) for the circuit shown below. At t = 0 the initial values are

i_L(0) = 1 A             V_C(0) = 2 V

Express the initial conditions for y(t) and dy(t)/dt

3. Find a differential equation between the input voltage x(t) and the output voltage y(t) for the circuit below. At t=0 the initial values are

V₁(0) = 2 V                    V₂(0) = -2 V

Express the initial conditions for y(t) and dy(t)/dt

4. Solve each of the first-order differential equations given below for the specified input signal and subject to the specified initial condition. Use the first-order solution technique

a. (dy(t)/dt)+4y(t)=x(t),               x(t)=u(t),                  y(0)= -1
b. (dy(t)/dt)+2y(t)=2x(t),             x(t)=u(t)-u(t-5),        y(0)= 2
c. (dy(t)/dt)+5y(t)=3x(t),             x(t)=δ(t),                  y(0)= 0.5
d. (dy(t)/dt)+5y(t)=3x(t),             x(t)=tu(t),                 y(0)= -4
e. (dy(t)/dt)+y(t)=2x(t),               x(t)=e^(-2t) u(t),      y(0)= -1

5. For each homogeneous differential equation given below, find the characteristic equation and show that at least some of its roots are complex. Find the homogeneous solution for t≥ 0 in each case subject to the initial conditions specified.

a. (d²y (t) / dt²) + 4(dy(t) / dt) + 13y (t) = 0,                                 y(0) = 5,       (dy(t) / dt)|_t=0 = 0
b. (d³y (t) / dt³) + 3(d²y (t) / dt²) + 4(dy(t) / dt) + 2y (t) = 0,        y(0) = 1,       (dy(t) / dt)|_t=0 = 0,         (d2y (t) / dt2)|_t=0 = -2