---
title: "Sketch {x(t) + x(−t)}u(t) from figure"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:sketch-even-part-figure"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/sketch-even-part-figure
site: BCT Engineering
---

# Sketch {x(t) + x(−t)}u(t) from figure

Signals and Systems · Chapter 1: Signals

- Asked in: 2069 Bhadra, 3+5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/sketch-even-part-figure
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define energy and power type signal with suitable examples. Sketch and label the signal y(t) = {x(t) + x(−t)}u(t) for given signal x(t) depicted below. [Figure: x(t) = 0 for t < −2; at t = −2 it jumps to −1 and rises linearly to 0 at t = −1; x(t) = 1 for −1 < t < 0; x(t) = 2 for 0 < t < 1; at t = 1 it drops to 1 and falls linearly to 0 at t = 2; x(t) = 0 for t > 2]

## Short answer

An energy signal has finite total energy; a power signal has finite average power. For t>0, using x(t) and its time reversal, y(t)=3 for 0<t<1, then 3-2t for 1<t<2, and 0 elsewhere.

## Answer

#### Energy and power signals
- **Energy signal:** total energy $E = \int_{-\infty}^{\infty}|x(t)|^2dt$ is finite and non-zero, and the average power is zero. Example: $x(t) = e^{-t}u(t)$ ($E = 0.5$ J); a single rectangular pulse of height $A$ and width $\tau$ ($E = A^2\tau$).
- **Power signal:** average power $P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2dt$ is finite and non-zero, so $E = \infty$. Example: $x(t) = A\sin\omega_0 t$ ($P = A^2/2$); the unit step $u(t)$ ($P = 1/2$).
- A signal cannot be both; some (e.g. $t\,u(t)$) are neither.

#### Sketch of $y(t) = \{x(t) + x(-t)\}u(t)$
From the figure, $x(t)$ is:

$$
x(t) = \begin{cases} t + 1, & -2 < t < -1 \\ 1, & -1 < t < 0 \\ 2, & 0 < t < 1 \\ 2 - t, & 1 < t < 2 \\ 0, & |t| > 2 \end{cases}
$$

Because of $u(t)$, $y(t) = 0$ for $t < 0$; only $t > 0$ is needed. For $t > 0$, $x(-t)$ takes the values of $x$ on the negative side:
- $0 < t < 1$: $-t \in (-1, 0)$, so $x(-t) = 1$.
- $1 < t < 2$: $-t \in (-2, -1)$, so $x(-t) = -t + 1 = 1 - t$.

Adding piece by piece:

| Interval | $x(t)$ | $x(-t)$ | $y(t)$ |
|---|---|---|---|
| $t < 0$ | - | - | 0 |
| $0 < t < 1$ | 2 | 1 | 3 |
| $1 < t < 2$ | $2 - t$ | $1 - t$ | $3 - 2t$ |
| $t > 2$ | 0 | 0 | 0 |

So $y(t)$ is 0 for $t < 0$, jumps to 3 at $t = 0$, stays at 3 up to $t = 1$, drops to 1 at $t = 1$, then falls linearly with slope $-2$ from $1$ at $t = 1$, crossing zero at $t = 1.5$, to $-1$ at $t = 2^-$, and jumps back to 0 at $t = 2$.

```text
 y(t)
  3 +-------+
    |       |
  2 |       |
    |       |
  1 |       +\
    |         \
  0 +-------+---\---+------> t
    |             \ |
 -1 |              \|
    0       1  1.5  2
```

(Note: $x(t) + x(-t) = 2x_e(t)$, so $y(t)$ is twice the even part of $x(t)$ kept for $t > 0$.)

## Check yourself

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

1. What defines an energy signal?
   A. Infinite energy and finite power
   B. Periodic with period T
   C. Finite non-zero energy and zero average power
   D. Zero energy
   Answer: C. Examples are a single pulse and e^(-t)u(t). (Key point: Energy signal)

2. Which is a power signal?
   A. A single rectangular pulse
   B. e^(-t)u(t)
   C. t u(t)
   D. A sin ω0 t with P = A^2/2
   Answer: D. Periodic signals have finite power and infinite energy; t u(t) is neither. (Key point: Power signal)

3. What does multiplying by u(t) do to y(t) = {x(t) + x(-t)}u(t)?
   A. Makes y(t) zero for t < 0
   B. Makes y(t) zero for t > 0
   C. Doubles y(t)
   D. Reflects y(t)
   Answer: A. u(t) is 0 for negative t, so only t > 0 is needed. (Key point: Effect of u(t))

4. What is y(t) for 0 < t < 1?
   A. 2
   B. 3
   C. 1
   D. 4
   Answer: B. x(t) = 2 and x(-t) = 1 there, so the sum is 3. (Key point: Value on 0<t<1)

## Related in Signals and Systems

- [Power of energy signal is zero](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-signal-example) — Signals and Systems, chapter 1
- [Energy and power of (3+4j)e^(2t)u(−t)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-complex-exponential) — Signals and Systems, chapter 1
- [Energy or power: x(t) = e^(−at)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-one-sided-exponential) — Signals and Systems, chapter 1, asked 6×
- [Energy or power: x(t) = e^(−3t)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-decaying-exponential) — Signals and Systems, chapter 1, asked 5×
- [Energy or power: e^(−5t)u(t+2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-shifted-exponential) — Signals and Systems, chapter 1, asked 5×
- [Energy or power: x(t) = e^(−a|t|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-two-sided-exponential) — Signals and Systems, chapter 1, asked 2×

## Related in other subjects

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- [ESD of an energy signal (ideal BPF example)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-energy-signal-ideal-bpf) — Communication Systems, chapter 1
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