---
title: "Plot x(t) = t(u(t+3) − u(t−3))"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:plot-ramp-window"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/plot-ramp-window
site: BCT Engineering
---

# Plot x(t) = t(u(t+3) − u(t−3))

Signals and Systems · Chapter 1: Signals

- Asked in: 2071 Magh, 2 marks
- Video: [Plotting signals involving elementary signals: step, ramp, impulse](https://www.youtube.com/watch?v=YgB0aMVscdg) — Dr Waleed Al-Nuaimy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/plot-ramp-window
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Plot the signal x(t) = t(u(t+3) − u(t−3)).

## Short answer

The factor u(t+3)-u(t-3) yields a rectangular window from -3 to 3. Multiplying by t gives x(t)=t for -3≤t<3 and zero elsewhere. Key points: x(-3)=-3, x(3-)=3, with a line through the origin.

## Answer

The term $u(t+3) - u(t-3)$ is a rectangular window equal to 1 for $-3 \le t < 3$ and 0 elsewhere. Multiplying by $t$ keeps the ramp $t$ only inside the window:

$$
x(t) = \begin{cases} t, & -3 \le t < 3 \\ 0, & \text{otherwise} \end{cases}
$$

Key points: $x(-3) = -3$, $x(0) = 0$, $x(3^-) = 3$; the signal drops to 0 for $t \ge 3$ and is 0 for $t < -3$. It is a straight line of slope 1 through the origin, cut off at $t = \pm 3$ (an odd signal).

```text
          x(t)
            |      /| 3
            |    /  |
            |  /    |
  --+-------+/------+------> t
   -3      /0       3
    |    /  |
    |  /    |
 -3 |/      |
```

(The vertical jumps are at $t = -3$, from 0 down to $-3$, and at $t = 3$, from 3 down to 0.)

## Check yourself

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

1. What is u(t+3) - u(t-3)?
   A. A rectangular window equal to 1 for -3 ≤ t < 3 and 0 elsewhere
   B. A ramp from -3 to 3
   C. A window equal to 1 for 0 ≤ t < 6
   D. An impulse at t = 0
   Answer: A. u(t+3) turns on at t = -3 and u(t-3) cancels it at t = 3. (Key point: Window meaning)

2. What does x(t) equal for -3 ≤ t < 3?
   A. 1
   B. t
   C. 3
   D. t + 3
   Answer: B. Multiplying by the window keeps the ramp t inside it. (Key point: Value inside window)

3. What are x(-3) and x(3^-)?
   A. 0 and 0
   B. 3 and -3
   C. -3 and 3
   D. 1 and 1
   Answer: C. The line of slope 1 reaches -3 at the left edge and 3 just before the right edge. (Key point: Key values)

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