ECE250 Assignment 1 Solved

0. 1) [CO1]  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]forn≥0andOdd{y[n]}=x[n]forn<0
   • Even{z[n]}=x[n]for−∞<n<∞. Assumethatz[n]=0forn<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).

1. i)  (2 pts) Write the mathematical expression of signal s2(t) for 0 ≤ t < T.
2. 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

e) (1 pt) Mean or average value

f) (1 pts) Mean absolute value

􏰀1􏰂T 􏰁1/2

s(t)2dt (1) 􏰀1􏰂T 􏰁

s(t)dt (2) 􏰀1􏰂T 􏰁

MAV{s(t)} = T g) (2 pts) Sketch the derivate of the signal s1(t).

|s(t)|dt (3)

RMS{s(t)} = T Avg{s(t)} = T

0

0

0

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.

1. i)  (1 pt) Determine the values of b for which the system remains memoryless. Take a = 100.
2. ii)  (1 pt) Will the system be memoryless if b = −t2 yielding the system of form y(t) = x(at − t2)? Take a = 97.
3. iii)  (2 pts) If the input x(t) = cos(t), will the system be causal? Justify.
4. 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) (3pts)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.