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

🙏Introduction:-

In this post, we will learn how to find the even and odd components of both continuous-time and discrete-time signals. This is a key concept in Signals and Systems and is frequently asked in GTU PYQ (Previous Year Questions) and exams for Electronics and Communication Engineering students.

📘 Table of Contents

• Introduction
• Question Statement
• What are Even and Odd Signals?
• Decomposition of Any Signal
  • Sub-Question 1: Continuous-Time Signal
    • Method 1: Step-by-Step Formula Method
    • Method 2: Shortcut Using Signal Symmetry
  • Sub-Question 2: Discrete-Time Signal
• Handwritten Solution
• Key Takeaways
• Final Words

📙Question Statement:-

Find the even and odd components of the following signals

(i) x(t) = cos t + sin t + cos t sin t 

(ii) x(n) =

(ii) x(n) = {1 1  1}
               ↓
               n = 0

What are Even and Odd Signals?

    Even Signal

• A signal x(t) is called even if: 

x(−t) = x(t)

• Even signals are symmetric about the vertical axis (t = 0).

    Odd Signal

• A signal x(t) is called odd if: 

x(−t) = −x(t)

• Odd signals have origin symmetry. They appear the same when rotated 180° about the origin.

Decomposition of Any Signal

Any signal x(t) can be written as a sum of its even and odd components: 

x(t) = xe(t) + xo(t)

Even part: 

xe(t) = ½ [x(t) + x(−t)]

Odd part: 

xo(t) = ½ [x(t) − x(−t)]

✏️ Sub-Question 1: Continuous-Time Signal

Given: 

x(t) = cos(t) + sin(t) + cos(t)·sin(t)

🔬 Method 1: Step-by-Step Formula Method

Step 1: Find x(−t)

cos(−t) = cos(t) (Even)
sin(−t) = −sin(t) (Odd)
cos(−t)·sin(−t) = −cos(t)·sin(t)(Odd)

x(−t) = cos(t) − sin(t) − cos(t)·sin(t)

Step 2: Even Component 

xe(t) = ½ [x(t) + x(−t)] 

xe(t) = ½ [(cos t + sin t + cos t·sin t) + (cos t − sin t − cos t·sin t)] 

xe(t) = ½ [2 cos t] 

xe(t) = cos t

Step 3: Odd Component 

xo(t) = ½ [x(t) − x(−t)]
 xo(t)= ½ [(cos t + sin t + cos t·sin t) − (cos t − sin t − cos t·sin t)]
 xo(t)= ½ [2 sin t + 2 cos t·sin t]
 xo(t)= sin t + cos t·sin t

✅ Final Answer:

Even component: cos(t)

Odd component: sin(t) + cos(t)·sin(t)

🔁 Method 2: Shortcut Using Signal Symmetry

We know from standard signal properties:

• cos(t) is an even signal ⇒ cos(−t) = cos(t)
• sin(t) is an odd signal ⇒ sin(−t) = −sin(t)
• cos(t)·sin(t) is a product of an even and odd function ⇒ Odd

Rule: Even × Odd = Odd

✅ Final Result Using Symmetry:

Even component: cos(t)
Odd component: sin(t) + cos(t)·sin(t)

✏️ Sub-Question 2: Discrete-Time Signal

Given:

{1 1  1}
   ↑
   n = 0

This means: 

x(−1) = 1 

x(0) = 1 

x(1) = 1

Step 1: Check Symmetry

Calculate x(−n): 

x(−n) = {1 1 1}
    ↑     n = 0 

x(−n) =  x(n)

So, the signal is even.

Step 2: Use the Formula 

Even component:

xe(n) = ½ [x(n) + x(−n)] 

xe(n) = ½ [x(n) + x(n)] 

xe(n) = {1 1 1}

    ↑     n = 0

Odd component:

xo(n) = ½ [x(n) − x(−n)] 

xo(n) = ½ [x(n) − x(n)] 

xo(n) = {0 0 0}

    ↑     n = 0

✅ Final Answer:

Even component:

{1 1  1}
   ↓
   n = 0

Odd component:

{0 0  0}
   ↓
   n = 0

 

📷 Handwritten Solution

Here is the scanned handwritten solution for your reference:

✍️Sub-Question 1:-

Method 2: Shortcut Using Signal Symmetry

⬇️ Download Image

✏️ Sub-Question 2: Discrete-Time Signal

⬇️ Download Image

🧠 Key Takeaways

• Even signals are symmetric around the y-axis.
• Odd signals are symmetric about the origin.
• Any signal can be split into even and odd parts using simple formulas.
• Using standard signal properties can help solve problems faster.

🔗 Related Topics

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

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

📌 Final Words

We hope this post helped you understand how to find the even and odd components of signals. This is an important topic in GTU exams and is core to mastering signals and systems. Bookmark this post and share it with friends preparing for ECE exams.

Stay tuned for more GTU PYQ Solutions for ECE Students.