[SOLVED] ECE250 Signals and Systems Assignment 1

Description

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] 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

1. i)) Find and sketch y[n] and z[n].
2. ii)) 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]For the signal g(t) = (√ 2 + √2j)e jπ/4e(−1+j2π)t , sketch the following:

1i) () Real{g(t)}

1. ii) ) Imag{g(t)}

iii) g(t + 2) + ¯

g(t + 2), where ¯

g(t) denotes the complex conjugate of g(t).

3) [CO1] (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:
3. a) ( Peak or maximum value
4. b) ( Energy
5. c) (Power
6. d)  Root-mean-square (RMS) value

RMS{s(t)} = T1 Z 0T

s(t)2 dt! 1/2

(1)

1. e) ) Mean or average value

Avg{s(t)} = T1 Z 0T

s(t)dt!

(2)

1. f)  Mean absolute value

MAV{s(t)} = T1 Z 0T

|s(t)|dt!

(3)

1. g)  Sketch the derivate of the signal s1(t).

Figure 2: Signals s1[t] and s2[t]

4) [CO2] 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.

100. i) Determine the values of b for which the system remains memoryless. Take a = 100.
101. ii) 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.

1. iv) 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] ( Generate and plot each of the following sequences over the indicated intervals.

1. 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
2. ii) (3 pts) y[n] = cos[0.03πn] + u[n], 0 ≤ n ≤ 50

6) [CO1]  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.