ECE250 Assignment 2 Solved

0. Theory Problems:

   1) [CO2] (8 points) Consider an LTI system with input and output related through the equation (1)

   Figure 1: Signal x(t)

   􏰂t −∞

   i) (4 pts) What is the impulse response h(t) for this system.
   ii) (4 pts) Determine the response of the system when the input x(t) is as shown in Figure 1.

y(t) =

e−(t−τ)x(τ − 2)dτ (1)

1

2. 2)  [CO2] (8 points) Consider a system whose input x(t) and output y(t) satisfy the first-order differential equation (2).

   dy(t) + 2y(t) = x(t) (2) dt

   The system satisfies the condition of initial rest.

   1. i)  (1 pt) Determine the system output y1(t) when the input is x1(t) = e3tu(t).
   2. ii)  (3 pts) Determine the system output y2(t) when the input is x2(t) = αe3tu(t) + βe2tu(t), where α and β are real numbers.
   3. iii)  (1 pt) Determine the system output y3(t) when the input is x3(t) = Ke2tu(t).
   4. iv)  (3 pts) Determine the system output y4(t) when the input is x4(t) = Ke2(t−T)u(t − T). Show that

      y4(t) = y3(t − T).

3. 3)  [CO2] (10 points) We are given a certain linear time-invariant system with impulse response h0(t) . We are told that when the input is x0(t) the output is y0(t), which is sketched in Figure 2. We are given the following set of inputs to linear time invariant systems with the indicated impulse responses:

   i) (2 pts) x(t) = 2×0(t); h(t) = h0(t)
   ii) (2 pts) x(t) = x0(t) − x0(t − 2); h(t) = h0(t)

   iii) (2pts)x(t)=x0(t−2);h(t)=h0(t+1)
   iv) (2 pts) x(t) = x0(−t) − x0(t − 2); h(t) = h0(t)

   v) (2 pts) x(t) = x0(−t) − x0(t − 2); h(t) = h0(−t)

   Figure 2: Signal y0(t)

   In each of these cases, determine whether or not we have enough information to determine the output y(t) when the input x(t) and the system has impulse response h(t). If it is possible to determine y(t), provide an accurate sketch of it with numerical values clearly indicated on the graph.

4) [CO2] (9 points) Evaluate y[n] = x[n] ⊛ h[n], where x[n] and h[n] are shown in Figure 3. i) (5 pts) by an analytical technique.

ii) (4 pts) by a graphical method.

Figure 3: Signals x[n] and y[n] 5) [CO2] (5 points) Two signals s1(t) and s2(t) are defined as below:

et, if 0 ≤ t < 1
s1(t)=e2−t, if1≤t<2 (3)

0, otherwise 􏰃e−t, if0≤t≤4

s2(t) = 0, otherwise (4) Evaluate g(t) = s1(t) ⊛ s2(t), where ⊛ denotes the convolution operator.

Programming Problems:

1. [CO2] (10 points) A system S is represented by its impulse response: 1􏰀 􏰁

h(t) = 4 e−2t − e−t u(t) (5)

a) (5 pts) Find the response of the system, if the input is x(t) = cos(t)u(t). Plot the signals x(t), h(t) and y(t) for t = [0, 20].
b) (5 pts) Find the response of the system if h(t) = 21 (e−t − e−4t)u(t) and x(t) = e−tsin(t)u(t). Plot the signals x(t), h(t) and y(t) for t = [0, 20].