Description

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).
Programming Problems:
• Use Matlab to solve the programming problems.
• For your solutions, you need to submit a zipped file on Google classroom with the following:
– program files (.m) 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: “A1 GroupNo RollNo Name.zip”. • Codes/reports submitted without a zipped file or without following the naming convention will NOT be checked.

1) [CO1] (9 points) A discrete signal x[n] is given as shown in Fig. 1. Using x[n], two more signals y[n] and z[n] are generated, as per the following definitions:
• Even{y[n]} = x[n] for n ≥ 0 and Odd{y[n]} = x[n] for n < 0
• Even{z[n]} = x[n] for −∞ < n < ∞. Assume that z[n] = 0 for n < 0
i) (6 pts) Find and sketch y[n] and z[n].
ii) (3 pts) For the three signals i.e. x[n], y[n], and z[n], check and justify whether any of these are odd/even functions.

Figure 1: Signal x[n]
√ √
2) [CO1] (9 points) For the signal g(t) = ( 2 + 2j)ejπ/4e(−1+j2π)t, sketch the following:
1
i) (3 pts) Real{g(t)}
ii) (3 pts) Imag{g(t)}
iii) (3 pts) g(t + 2) + ¯g(t + 2), where ¯g(t) denotes the complex conjugate of g(t).
3) [CO1] (11 points) Two students of the Signal and Systems course are instructed to generate periodic signals of period T seconds using triangular pulses. Student-A generated a signal of the form s1(t) = at/T for 0 ≤ t < T as depicted in Fig. 2 (left), where a is a positive quantity that denotes the amplitude of the signal. In comparison, student-B generated a signal s2(t) as shown in Fig. 2 (right).
i) (2 pts) Write the mathematical expression of signal s2(t) for 0 ≤ t < T.
ii) For both the signals, compute the following signal parameters:
a) (1 pt) Peak or maximum value
b) (1 pt) Energy
c) (1 pt) Power
d) (2 pts) Root-mean-square (RMS) value
(1)
e) (1 pt) Mean or average value
!
Avg(2) f) (1 pts) Mean absolute value
!
MAV(3)
g) (2 pts) Sketch the derivate of the signal s1(t).

Figure 2: Signals s1[t] and s2[t]
4) [CO2] (6 points) A system S is described by the relation y(t) = x(at + b), where x(t) is the input signal and y(t) is the output signal.
i) (1 pt) Determine the values of b for which the system remains memoryless. Take a = 100.
ii) (1 pt) Will the system be memoryless if b = −t2 yielding the system of form y(t) = x(at − t2)? Take a = 97. iii) (2 pts) If the input x(t) = cos(t), will the system be causal? Justify. iv) (2 pts) Another system S2 is described by the relation y(t) = ex(at+b). Is it stable? Justify.
Note: Each part of this problem is to be solved individually.
Programming Problems:
5) [CO1] (6 points) Generate and plot each of the following sequences over the indicated intervals.
i) (3 pts) x[n] = n[u[n] − u[n − 10]] + 10e−0.3(n−10)[u[n − 10] − u[n − 20]], 0 ≤ n ≤ 20
ii) (3 pts) y[n] = cos[0.03πn] + u[n], 0 ≤ n ≤ 50
6) [CO1] (4 points) Let z[n] = u[n] − u[n − 10]. Decompose z[n] into its even and odd components and plot these in three individual subplots for the interval −20 ≤ n ≤ 20.