Description

Issued on:

Guidelines for submission
Theory Problems:
• Submit a hard copy of your solutions in the wooden box kept on the 3rd Floor of Old Academic Block
(right side of the lift).
• Do all questions in sequence.
• Use A4 sheets (Plain)
• Staple your sheets properly
Programming Problems:
• Use Matlab or python to solve the programming problems.
• For your solutions, you need to submit a zipped file on Google classroom with the following:
– program files (.m) or (.ipynb) with all dependencies.
– a report (.pdf) with your coding outputs and generated plots. The report should be self-complete with all your assumptions and inferences clearly specified.
• Before submission, please name your zipped file as: “A2 GroupNo RollNo Name.zip”. • Codes/reports submitted without a zipped file or without following the naming convention will NOT be checked.

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

Figure 1: Signal x(t)
(1)
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.
1
2) [CO2] (8 points) Consider a system whose input x(t) and output y(t) satisfy the first-order differential equation (2).
) (2)
The system satisfies the condition of initial rest.
i) (1 pt) Determine the system output y1(t) when the input is x1(t) = e3tu(t).
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.
iii) (1 pt) Determine the system output y3(t) when the input is x3(t) = Ke2tu(t).
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) [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) (2 pts) 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:
if 0 ≤ t < 1

s1(t) = e2−t, if 1 ≤ t < 2
0, otherwise (3)
( −t, if 0 ≤ t ≤ 4 e
s2(t) =
0, otherwise 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: (4)
) (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 ) and x(t) = e−tsin(t)u(t). Plot the signals x(t), h(t) and y(t) for t = [0,20].