---
title: "Even/odd parts; time scaling and folding"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:even-odd-parts-scaling-folding"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/even-odd-parts-scaling-folding
site: BCT Engineering
---

# Even/odd parts; time scaling and folding

Signals and Systems · Chapter 1: Signals

- Asked in: 2073 Bhadra, 2+3 marks
- Video: [Operations on Time of Signals](https://www.youtube.com/watch?v=613MVjAoAHg) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/even-odd-parts-scaling-folding
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Derive the expression for finding even and odd part of signal x(t). Explain time scaling and time folding.

## Short answer

Even part: x_e(t) = (x(t)+x(-t))/2; odd part: x_o(t) = (x(t)-x(-t))/2. Time scaling y(t)=x(at) compresses for a>1 and expands for 0<a<1. Time folding y(t)=x(-t) reflects the signal about t=0.

## Answer

#### Even and odd parts of $x(t)$
Let any signal be written as the sum of an even part and an odd part:

$$
x(t) = x_e(t) + x_o(t) \qquad (1)
$$

where $x_e(-t) = x_e(t)$ and $x_o(-t) = -x_o(t)$. Replace $t$ by $-t$:

$$
x(-t) = x_e(-t) + x_o(-t) = x_e(t) - x_o(t) \qquad (2)
$$

Adding (1) and (2), and subtracting (2) from (1):

$$
\begin{aligned}
x_e(t) &= \frac{1}{2}\left[x(t) + x(-t)\right] \\
x_o(t) &= \frac{1}{2}\left[x(t) - x(-t)\right]
\end{aligned}
$$

These parts always exist for any $x(t)$, so **every signal can be decomposed into even and odd components**. The same holds for DT signals: $x_e[n] = \frac{1}{2}(x[n]+x[-n])$, $x_o[n] = \frac{1}{2}(x[n]-x[-n])$.

#### Time scaling
$y(t) = x(at)$, $a > 0$:
- $a > 1$: **compression** – the signal happens faster. If $x(t)$ is a pulse on $0 \le t \le 2$, $x(2t)$ is on $0 \le t \le 1$.
- $0 < a < 1$: **expansion** – $x(t/2)$ is on $0 \le t \le 4$.

For DT signals, $x[2n]$ keeps only every second sample (decimation), and $x[n/2]$ inserts samples (interpolation).

#### Time folding (reflection)
$y(t) = x(-t)$: the signal is reflected about the vertical axis $t = 0$. A value at $t = t_1$ moves to $t = -t_1$. Example: a ramp $t$ on $0 \le t \le 1$ becomes a ramp on $-1 \le t \le 0$. Folding is a key step in convolution.

```text
 x(t)          x(2t)        x(-t)
      /|           /|        |\
    /  |          / |        |  \
  /    |         /  |        |    \
-+-----+--> t  -+---+--> t  -+-----+--> t
 0     2        0   1       -2     0
```

## 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. xe(t) = [x(t) - x(-t)]/2
   B. xe(t) = [x(t) + x(-t)]/2
   C. xe(t) = x(t) + x(-t)
   D. xe(t) = x(t)x(-t)
   Answer: B. Adding x = xe + xo and x(-t) = xe - xo cancels the odd part. (Key point: Even part formula)

2. What is the odd part of x(t)?
   A. xo(t) = [x(t) + x(-t)]/2
   B. xo(t) = x(-t)
   C. xo(t) = [x(t) - x(-t)]/2
   D. xo(t) = x(t)/2
   Answer: C. Subtracting x(-t) from x(t) cancels the even part and leaves 2xo(t). (Key point: Odd part formula)

3. If x(t) is a pulse on 0 ≤ t ≤ 2, where is x(2t) non-zero?
   A. 0 ≤ t ≤ 4
   B. 0 ≤ t ≤ 2
   C. -2 ≤ t ≤ 0
   D. 0 ≤ t ≤ 1
   Answer: D. a = 2 > 1 compresses the signal by a factor of 2. (Key point: Time scaling)

4. What does y(t) = x(-t) do to a signal?
   A. Reflects it about t = 0
   B. Delays it
   C. Compresses it
   D. Negates its amplitude
   Answer: A. A value at t1 moves to -t1; this is a key step in convolution. (Key point: Time folding)

## Related in Signals and Systems

- [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 DT signals and periodicity](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-dt-periodicity) — Signals and Systems, chapter 1
- [Even and odd signals in CT and DT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-ct-dt-examples) — Signals and Systems, chapter 1
- [Even/odd signals; energy or power check](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-energy-power-check) — Signals and Systems, chapter 1
- [Even/odd decomposition is unique: {2,3,4,5,6}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-uniqueness-sequence) — Signals and Systems, chapter 1
- [Time shifting, scaling and inversion](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Atime-shift-scale-inversion) — Signals and Systems, 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 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×
- [Even and odd part from stem plot](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aeven-odd-part-stem-plot) — Digital Signal Analysis and Processing, chapter 1
- [RE: even a's followed by odd b's](https://www.bctengineer.com/question.md?key=semester_3%2Ftheory_of_computation%3Are-even-a-odd-b) — Theory of Computation, chapter 2