---
title: "Even and odd part from stem plot"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:even-odd-part-stem-plot"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/even-odd-part-stem-plot
site: BCT Engineering
---

# Even and odd part from stem plot

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2071 Shrawan, 4 marks
- Video: [Extracting the even and odd parts of a signal](https://www.youtube.com/watch?v=zLyTwKLjgQk) — Anish Turlapaty
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/even-odd-part-stem-plot
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Find the odd and even part of the following signal. [Figure: stem plot x[n] for n = −4 to 4; x[n] = 1 for n = −4, −3, −2, −1, 0 and x[n] = 2 for n = 1, 2, 3, 4.]

## Short answer

From x[n]=1 (n=−4...0) and 2 (n=1...4): even part xe[n]=1.5 for 1≤|n|≤4, xe[0]=1. Odd part xo[n]=−0.5 for −4≤n≤−1, 0 at n=0, +0.5 for 1≤n≤4, using xe=(x[n]+x[−n])/2, xo=(x[n]−x[−n])/2.

## Answer

The stem plot gives x[n] = 1 for n = −4 … 0 and x[n] = 2 for n = 1 … 4.

Use xe[n] = ½{x[n] + x[−n]} and xo[n] = ½{x[n] − x[−n]}.

| n | x[n] | x[−n] | xe[n] | xo[n] |
|---|---|---|---|---|
| −4 … −1 | 1 | 2 | 1.5 | −0.5 |
| 0 | 1 | 1 | 1 | 0 |
| 1 … 4 | 2 | 1 | 1.5 | +0.5 |

#### Even part xe[n]
```text
 xe[n]
 1.5 |  o  o  o  o     o  o  o  o
 1.0 |              o
     +--+--+--+--+--+--+--+--+--+-- n
       -4 -3 -2 -1  0  1  2  3  4
```
xe[n] = 1.5 for 1 ≤ |n| ≤ 4, and xe[0] = 1. It is symmetric about n = 0.

#### Odd part xo[n]
```text
 xo[n]
 +0.5 |                 o  o  o  o
    0 +--+--+--+--+--o--+--+--+--+-- n
 -0.5 |  o  o  o  o
        -4 -3 -2 -1  0  1  2  3  4
```
xo[n] = −0.5 for n = −4 … −1, xo[0] = 0, and xo[n] = +0.5 for n = 1 … 4. It is antisymmetric.

**Check:** xe[n] + xo[n] = 1 for n ≤ 0 and 2 for n ≥ 1, which equals x[n].

## Check yourself

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

1. The even and odd parts are given by
   A. xe = 1/2{x[n]-x[-n]}, xo = 1/2{x[n]+x[-n]}
   B. xe = x[n]+x[-n], xo = x[n]-x[-n]
   C. xe = 1/2{x[n]+x[-n]}, xo = 1/2{x[n]-x[-n]}
   D. xe = x[n]x[-n], xo = x[n]/x[-n]
   Answer: C. Even part averages x[n] with its fold; odd part is half the difference. (Key point: Decomposition formulas)

2. What is xe[0] for this stem plot?
   A. 1.5
   B. 1
   C. 0
   D. 2
   Answer: B. x[0] = x[-0] = 1, so xe[0] = 1 and xo[0] = 0. (Key point: Even part at origin)

3. Which statement about the result is correct?
   A. xe[n] = 1.5 for 1 <= |n| <= 4 is symmetric; xo is antisymmetric
   B. xe is antisymmetric, xo is symmetric
   C. Both xe and xo are symmetric
   D. xo[n] = 1.5 for n > 0
   Answer: A. Even part is symmetric about n = 0; odd part flips sign (-0.5 for n<0, +0.5 for n>0). (Key point: Symmetry property)

4. How do you check the answer?
   A. xe[n] - xo[n] must equal x[n]
   B. xe[n] xo[n] must equal x[n]
   C. xe[n] must equal xo[n]
   D. xe[n] + xo[n] must equal x[n]
   Answer: D. The two parts must add back to the original: 1 for n <= 0 and 2 for n >= 1. (Key point: Verification)

## Related in Digital Signal Analysis and Processing

- [Even and odd part of a signal](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aeven-odd-part-piecewise) — Digital Signal Analysis and Processing, chapter 1, asked 2×

## Related in other subjects

- [Even and odd parts of a signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aeven-odd-parts) — Data Communication, chapter 2
- [Even/odd decomposition of a signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-decomposition) — Signals and Systems, chapter 1, asked 2×
- [Even/odd parts; time scaling and folding](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-parts-scaling-folding) — Signals and Systems, chapter 1
- [Output for a decaying stem-plot h[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-stem-plot-response) — Signals and Systems, chapter 6