---
title: "Even and odd parts of a signal"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:even-odd-parts"
source: https://www.bctengineer.com/q/semester_5/data_communication/even-odd-parts
site: BCT Engineering
---

# Even and odd parts of a signal

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2074 Chaitra, 4+4 marks
- Video: [Even & Odd Components of a Signal Example 1](https://www.youtube.com/watch?v=GrpDgOjR5ZA) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/even-odd-parts
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Derive an expression to find even and odd part of signal x(t). Find even and odd part of a signal x(t) = 0.5(t+1) for -1 ≤ t ≤ 1.

## Short answer

Even part: x_e(t)=½[x(t)+x(-t)], odd: x_o(t)=½[x(t)-x(-t)]. For x(t)=0.5(t+1) on [-1,1], x_e(t)=0.5 and x_o(t)=0.5t over that interval.

## Answer

#### Derivation
Any signal can be written as the sum of an even part and an odd part: x(t) = xₑ(t) + xₒ(t), where xₑ(-t) = xₑ(t) and xₒ(-t) = -xₒ(t).

Replace t by -t: x(-t) = xₑ(t) - xₒ(t).

Adding the two equations: x(t) + x(-t) = 2xₑ(t)
Subtracting: x(t) - x(-t) = 2xₒ(t)

**xₑ(t) = ½ [x(t) + x(-t)]**

**xₒ(t) = ½ [x(t) - x(-t)]**

#### Numerical
x(t) = 0.5(t + 1) = 0.5t + 0.5 for -1 ≤ t ≤ 1 (zero outside).

x(-t) = 0.5(-t + 1) = 0.5 - 0.5t, for -1 ≤ t ≤ 1.

xₑ(t) = ½ [(0.5t + 0.5) + (0.5 - 0.5t)] = **0.5**, for -1 ≤ t ≤ 1

xₒ(t) = ½ [(0.5t + 0.5) - (0.5 - 0.5t)] = **0.5t**, for -1 ≤ t ≤ 1

Both are zero outside this interval.

Check: xₑ + xₒ = 0.5 + 0.5t = x(t).

```text
x(t): ramp from 0 at t = -1 to 1 at t = 1
  1 |              /
0.5 |        /
  0 +--/-----+-----+--> t
      -1     0     1

xe(t): rectangle of height 0.5 on -1 <= t <= 1
0.5 |  +-----------+
  0 +--+-----+-----+--> t
      -1     0     1

xo(t): line from -0.5 at t = -1 to 0.5 at t = 1
 0.5 |              /
   0 +--------/--------> t
-0.5 |  /
       -1     0     1
```

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the even part of x(t)?
   A. (1/2)[x(t) - x(-t)]
   B. x(t) + x(-t)
   C. (1/2)[x(t) x(-t)]
   D. (1/2)[x(t) + x(-t)]
   Answer: D. Adding x(t) = xe + xo and x(-t) = xe - xo gives 2xe. (Key point: Even part formula)

2. What is the odd part of x(t)?
   A. (1/2)[x(t) - x(-t)]
   B. (1/2)[x(t) + x(-t)]
   C. x(-t) - x(t)
   D. (1/2)[x(t)]
   Answer: A. Subtracting the two equations gives 2xo. (Key point: Odd part formula)

3. For x(t) = 0.5(t+1) on -1 <= t <= 1, what is the even part?
   A. 0.5t
   B. 0.5
   C. 0.5t + 0.5
   D. 0
   Answer: B. x(t) + x(-t) = 1, so half of it is 0.5. (Key point: Numerical even part)

4. What is the odd part of x(t) = 0.5(t+1) on -1 <= t <= 1?
   A. 0.5
   B. 0.5(t+1)
   C. 0.5t
   D. -0.5t
   Answer: C. x(t) - x(-t) = t, so half is 0.5t. (Key point: Numerical odd part)

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