Signals In Natural Domain MCQs for Electrical Engineering (EE) Exam

The period of a sinusoidal signal of the form x(t) = A*sin(ωt + φ) is given by T = 2π/ω, where ω is the angular frequency. In this case, ω = 18π, so the period is T = 2π/(18π) = 1/9 seconds. The phase angle (in degrees) does not affect the period. Therefore, the period of the given signal is 1/9 seconds.

The function obeys the scaling/homogeneity property, but doesn’t obey the additivity property, thus not being linear.

Sin x is a periodic function, but log x is not a periodic function. Thus y is log t, where t= sin x, thus y is not periodic.

Reduces to 1 – 2t, which is a strictly decreasing function.

Correct option is not C.
first t is missing
if t is there fundamental frequency would be 2pi*w

Stability of a System

Stability is a crucial property of a system that determines its behavior and performance. In the context of linear time-invariant (LTI) systems, stability refers to the boundedness of the system's output for bounded input signals. An LTI system is said to be stable if and only if every bounded input signal produces a bounded output signal.

There are three types of stability for LTI systems:
1. Stable: A system is stable if its output remains bounded for any bounded input.
2. Unstable: A system is unstable if its output becomes unbounded for at least one bounded input.
3. Marginally Stable: A system is marginally stable if its output remains bounded for some bounded input signals but not for all bounded input signals.

Analysis of the Given System

The given system is represented by the transfer function h(t) = exp([-1-2j]t), where exp represents the exponential function and [a+bi] represents a complex number with a real part a and an imaginary part b.

To determine the stability of this system, we need to analyze the behavior of its transfer function. In this case, the transfer function is a complex exponential function with a negative real part (-1) and a complex part (-2j).

Boundedness of the Exponential Function

The boundedness of an exponential function depends on its real part. A complex exponential function with a negative real part is always bounded. This is because the exponential term decreases exponentially as time increases.

In our case, the real part of the complex exponential function is -1, which is negative. Therefore, the exponential function exp([-1-2j]t) is bounded for any value of t.

Conclusion

Since the given system's transfer function is a bounded complex exponential function, it implies that the system's output will always remain bounded for any bounded input signal. Therefore, the system is marginally stable as it remains bounded for some bounded input signals but not for all bounded input signals.

Hence, the correct answer is option C) Marginally Stable.

Question Analysis:
The given question is asking whether the function y[2n] = x[2n] is linear in nature or not. To answer this question, we need to understand the concept of linearity and analyze the given function.

Concept of Linearity:
In mathematics, a function is said to be linear if it satisfies two properties: additivity and homogeneity.

1. Additivity: The property of additivity states that for any two inputs x and y, the function satisfies the following equation:
f(x + y) = f(x) + f(y)

2. Homogeneity: The property of homogeneity states that for any input x and scalar a, the function satisfies the following equation:
f(ax) = af(x)

If a function satisfies both additivity and homogeneity, it is considered to be linear.

Analysis of the Given Function:
The given function y[2n] = x[2n] can be rewritten as:
y[n] = x[n]

To determine whether this function is linear or not, we need to check if it satisfies the properties of additivity and homogeneity.

Additivity:
To check the additivity property, let's consider two inputs x1[n] and x2[n] and their corresponding outputs y1[n] and y2[n]:
y1[n] = x1[n]
y2[n] = x2[n]

Now, let's calculate the sum of the inputs and their corresponding outputs:
x1[n] + x2[n] = y1[n] + y2[n]

Since the sum of the inputs is equal to the sum of the outputs, the function satisfies the additivity property.

Homogeneity:
To check the homogeneity property, let's consider an input x[n] and a scalar a, and their corresponding outputs y[n] and z[n]:
y[n] = x[n]
z[n] = ax[n]

Now, let's calculate the output when the input is multiplied by the scalar:
az[n] = a(x[n])

Since the output when the input is multiplied by the scalar is equal to the scalar multiplied by the output, the function satisfies the homogeneity property.

Conclusion:
From the analysis above, we can conclude that the given function y[2n] = x[2n] is linear in nature. It satisfies both the additivity and homogeneity properties, which are the defining properties of linearity. Therefore, the correct answer is option 'A' - Yes.