GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 2(a) Solution

GTU Sem 4 – Signals & Systems – Summer 2021 – Question 2(a) Solution

Determine the energy and power of the signal x(t) = e-3t u(t)

Introduction

In signal analysis, determining whether a signal is an energy signal or a power signal is fundamental.

An energy signal has finite energy and zero average power, while a power signal has finite average power and infinite energy.

In this question from the GTU Summer 2021 Signals & Systems paper, we're given a continuous-time exponential signal multiplied by the unit step function:

x(t) = e-3t u(t)

Our task is to analyze this signal to determine whether it qualifies as an energy or power signal and calculate the respective values.

Table of Contents

• Introduction
• Question Statement
• Theory Part Related to the Question
• Solution in Detail
• Handwritten Solution Images
• Key Takeaways
• Related Posts

Question Statement

Q.2 (a) [Summer 2021 – 03 Marks]
Determine the energy and power of the signal:

x(t) = e-3t u(t)

This question tests your understanding of fundamental signal classifications. The given signal is a causal exponential signal that starts at t = 0 and decays exponentially for the positive time due to the presence of the u(t) (unit step function). You need to calculate the total energy and average power of this signal and conclude whether it is an energy signal or a power signal.

Theory Part Related to the Question

In signal analysis, continuous-time signals are often classified based on their energy and power characteristics. To determine whether a given signal is an energy signal or a power signal, we rely on two key definitions:

1. Total Energy of a signal x(t):

The energy of a signal is calculated using the formula:

E = ∫-∞∞ |x(t)|² dt

If the value of E is finite (i.e., E < ∞), then the signal is known as an energy signal.

2. Average Power of a signal x(t):

The power of a signal is defined as:

P = limT→∞ (1/2T) ∫-TT |x(t)|² dt

If the value of P is finite (i.e., P < ∞) and the total energy is infinite, then the signal is known as a power signal.

Now, consider the given signal:

x(t) = e-3t u(t)

Here, u(t) is the unit step function. It ensures that the signal is zero for t < 0 and equal to e-3t for t ≥ 0. Thus, we can simplify the domain of integration when calculating energy and power to the interval from 0 to ∞.

To proceed, we will calculate the energy and power over this range and determine the nature of the signal based on the results.

Solution in Detail

We are given the signal:

x(t) = e-3t u(t)

Since u(t) is the unit step function, the signal exists only for t ≥ 0. So, we can rewrite:

x(t) = e^-3t for t ≥ 0

x(t) = 0 for t < 0

Step 1: Calculate Energy

Using the energy formula:

E = ∫-∞∞ |x(t)|² dt = ∫0∞ |e-3t|² dt

= ∫0∞ e-6t dt

Now integrate:

E = [-1/6 e-6t]0∞

= (0 - (-1/6)) = 1/6

So, the energy of the signal is 1/6, which is a finite value.

Step 2: Calculate Power

Use the power formula:

P = limT→∞ (1/2T) ∫-TT |x(t)|² dt

Since the signal is zero for t < 0, we simplify the integration range:

P = limT→∞ (1/2T) ∫0T e-6t dt

= limT→∞ (1/2T) [-1/6 e-6t]0T

= limT→∞ (1/2T) × (1/6)(1 - e-6T)

As T → ∞, e-6T → 0, so:

P = limT→∞ (1/2T) × (1/6) = 0

So, the average power of the signal is 0.

Conclusion

Since the signal has finite energy and zero power, it is classified as an energy signal.

Handwritten Solution Images

Below is the handwritten solution showing the detailed steps for finding the energy and power of the signal x(t) = e-3t u(t).

Key Takeaways

• The given signal x(t) = e-3t u(t) is defined only for t ≥ 0 due to the unit step function u(t).
• The total energy of the signal is finite and equals 1/6.
• The average power of the signal is zero.
• Based on the signal classification, it is, therefore, an energy signal.
• This type of exponential decay signal is commonly used in system analysis and control theory.

Final Words

Understanding whether a signal is an energy or power signal is a fundamental concept in Signals and Systems. In this example, we clearly saw how an exponential signal multiplied by a unit step function behaves as an energy signal due to its finite energy and zero power.

Such questions are not only important for GTU exams but also form the foundation for deeper topics in communication systems and signal processing. Make sure to practice similar problems to strengthen your conceptual clarity.

For more step-by-step solutions like this, explore our other PYQ-based posts and PDF compilations.

GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 1(c) Solution

GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 2(b) Solution