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

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

Sketch Signal x(t) = u(t+2) – u(t−2) + u(t+1) – u(t−1)

📑 Table of Contents

• 🔹 Introduction
• 🔹 Step-by-Step Analysis
• 🔹 Graph of x(t)
• 🔹 Transformation: x(2t)
• 🔹 Transformation: x(1 − t)
• 🔹 Transformation: x(t) · u(t)
• 🔹 Handwritten Solution Images
• 🔹 Key Takeaways
• 🔹 Related Posts
• 🔹 Final Words

📌 Introduction

The signal x(t) = u(t+2) − u(t−2) + u(t+1) − u(t−1) is a combination of four shifted unit step functions. GTU’s Signals & Systems subject (EC branch) often asks you to test your concepts in time-domain signal analysis. This example helps visualize signal construction, overlapping effects, and common transformations like scaling, reversal, and gating.

🧠 Step-by-Step Analysis

The unit step function u(t) is defined as:

• u(t) = 0 when t < 0
• u(t) = 1 when t ≥ 0

Let’s examine each component in x(t):

• u(t + 2): This becomes active (value = 1) at t = −2
• u(t − 2): This becomes active at t = 2
• u(t + 1): This becomes active at t = −1
• u(t − 1): This becomes active at t = 1

Now calculate the effect of each difference:

• u(t + 2) − u(t − 2) is 1 in the range [−2, 2)
• u(t + 1) − u(t − 1) is 1 in the range [−1, 1)

So x(t) is the sum of two rectangular pulses:

• From −2 to −1: Only the first pulse is active ⇒ Value = 1
• From −1 to 1: Both pulses are active ⇒ Value = 2
• From 1 to 2: Only the first pulse is active ⇒ Value = 1
• Outside [−2, 2]: All values are 0

📈 Graph of x(t)

Original Signal x(t)

🔄 Transformation: x(2t)

In time compression by 2, every event happens twice as fast. This means the signal becomes narrower.

• The range [−2, 2] becomes [−1, 1]
• The center pulse [−1, 1] becomes [−0.5, 0.5]

Time-compressed Signal x(2t)

🔄 Transformation: x(1 − t)

This transformation performs:

• Time reversal: Replaces t with −t → reflects across the y-axis
• Time shift: Replaces t with t − 1 → shifts the signal to the right by 1 unit

Time-reversed and shifted Signal x(1 − t)

🔄 Transformation: x(t) · u(t)

This operation gates the signal to the right-hand side (t ≥ 0):

• Signal values for t < 0 become 0
• Signal remains unchanged for t ≥ 0

Right-sided Signal x(t) · u(t)

Handwritten Solution Images

🧠 Key Takeaways

• Unit step functions are essential for building piecewise signals.
• Overlapping step functions result in combined amplitudes.
• Transformations like compression, reversal, and gating let us analyze signals dynamically.

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

Find the natural response of the system described by difference equation

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

Determine the convolution sum of two sequences using graphical method

📝 Final Words

Sketching signals like x(t) = u(t+2) – u(t−2) + u(t+1) – u(t−1) builds your confidence in analyzing piecewise signals and applying transformations. Mastering these techniques is crucial for GTU EC exam preparation and a strong foundation in signal processing.