---
title: "Sketch x[n] built from unit steps"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:draw-dt-signal-steps"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/draw-dt-signal-steps
site: BCT Engineering
---

# Sketch x[n] built from unit steps

Signals and Systems · Chapter 1: Signals

- Asked in: 2079 Asoj, 3 marks
- Video: [Sketching of signals || how to draw waveform of a signal || signals and systems](https://www.youtube.com/watch?v=LXhji5HKPes) — easy study plus
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/draw-dt-signal-steps
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Draw the following signal x[n] = u[n+2] − u[n−3] + nu[n−3] − nu[n−6].

## Short answer

x[n] = 1 for −2 ≤ n ≤ 2, equals n for 3 ≤ n ≤ 5, and 0 elsewhere. The discrete-time plot consists of samples: (−2,1), (−1,1), (0,1), (1,1), (2,1), (3,3), (4,4), (5,5).

## Answer

Evaluate each term range by range:

- $u[n+2] - u[n-3]$ = 1 for $-2 \le n \le 2$, else 0.
- $n u[n-3] - n u[n-6] = n\,(u[n-3]-u[n-6])$ = $n$ for $3 \le n \le 5$, else 0.

| $n$ | ≤ −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | ≥ 6 |
|---|---|---|---|---|---|---|---|---|---|---|
| $x[n]$ | 0 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 5 | 0 |

So $x[n] = \{1, 1, \underline{1}, 1, 1, 3, 4, 5\}$ for $n = -2$ to $5$ (underline marks $n = 0$).

```text
 x[n]
  5 |                          o
  4 |                       o  |
  3 |                    o  |  |
  2 |                    |  |  |
  1 |     o  o  o  o  o  |  |  |
    |     |  |  |  |  |  |  |  |
 ---+--+--+--+--+--+--+--+--+--+--+--> n
      -3 -2 -1  0  1  2  3  4  5  6
```

## Check yourself

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

1. u[n+2] - u[n-3] equals 1 for:
   A. -2 <= n <= 2
   B. -2 <= n <= 3
   C. 0 <= n <= 5
   D. 2 <= n <= 3
   Answer: A. u[n+2] turns on at n = -2 and u[n-3] turns on at n = 3, leaving 1 for n = -2..2. (Key point: Window u[n+2] - u[n-3])

2. n(u[n-3] - u[n-6]) equals n for:
   A. n = 3, 4, 5
   B. n = 3 to 6
   C. n = 0 to 5
   D. n = 2, 3, 4
   Answer: A. The window is 1 for 3 <= n <= 5. (Key point: Ramp segment)

3. x[4] is:
   A. 1
   B. 3
   C. 4
   D. 5
   Answer: C. At n = 4 only the ramp term contributes, giving 4. (Key point: Value at n = 4)

4. x[2] is:
   A. 2
   B. 1
   C. 0
   D. 3
   Answer: B. n = 2 lies in the first window (value 1), not the ramp window. (Key point: Value at n = 2)

## Related in Signals and Systems

- [DT unit step; periodicity of e^(jωn)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-unit-step-periodicity) — Signals and Systems, chapter 1, asked 5×
- [Fourier transform of unit step](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unit-step) — Signals and Systems, chapter 3, asked 3×
- [Unit step and delta: CT and DT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aunit-step-delta-relation) — Signals and Systems, chapter 1, asked 2×
- [CT unit step; periodicity of e^(jωt)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aunit-step-ct-periodicity) — Signals and Systems, chapter 1, asked 2×
- [CT unit impulse, step and ramp relationship](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-impulse-step-ramp) — Signals and Systems, chapter 1, asked 2×
- [DT unit impulse, step and ramp relationship](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-impulse-step-ramp) — Signals and Systems, chapter 1, asked 2×

## Related in other subjects

- [Control signal types and hardware control unit](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Acontrol-signals-hardware) — Computer Organization and Architecture, chapter 3
- [Steps of virtual (unit) load method](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aunit-load-method-steps) — Theory of Structures I, chapter 3, asked 4×
- [Steps to establish small scale unit](https://www.bctengineer.com/question.md?key=semester_7%2Forganization_and_management%3Aentrepreneurship-small-scale-steps) — Organization and Management, chapter 8
- [Steps in designing a microprogram control unit](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Amicroprogram-design) — Computer Organization and Architecture, chapter 3