---
title: "CT unit impulse, step and ramp relationship"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:ct-impulse-step-ramp"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-impulse-step-ramp
site: BCT Engineering
---

# CT unit impulse, step and ramp relationship

Signals and Systems · Chapter 1: Signals

- Asked in: 2080 Chaitra, 4 marks
- Video: [Relation between Unit Impulse and Unit Step Signals](https://www.youtube.com/watch?v=SWt2PYiGgKQ) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-impulse-step-ramp
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Briefly explain CT unit impulse, unit step and unit ramp signal. What is the relationship between unit impulse, unit step and unit ramp signal?

## Short answer

Unit impulse δ(t) is zero for t≠0 with unit area. Unit step u(t) is 0 for t<0, 1 for t≥0. Unit ramp r(t)=t u(t). Relations: step integrates impulse, ramp integrates step; derivatives reverse: δ=u', u=r'.

## Answer

#### Unit impulse $\delta(t)$
Zero for all $t \neq 0$, with unit area: $\int_{-\infty}^{\infty}\delta(t)\,dt = 1$. It is the limit of a pulse of width $\Delta$ and height $1/\Delta$ as $\Delta \to 0$. Sifting property: $\int x(t)\delta(t-t_0)dt = x(t_0)$. Used to model sudden shocks and to define the impulse response.

#### Unit step $u(t)$
$$
u(t) = \begin{cases} 1, & t \ge 0 \\ 0, & t < 0 \end{cases}
$$
Represents switching on a constant (e.g. closing a DC switch at $t=0$).

#### Unit ramp $r(t)$
$$
r(t) = t\,u(t) = \begin{cases} t, & t \ge 0 \\ 0, & t < 0 \end{cases}
$$
Increases linearly with slope 1 from $t = 0$.

```text
 delta(t)        u(t)            r(t)
   ^ area 1     1 +-------          /
   |              |               /
   |              |             /  slope 1
 --+------ t    --+------- t  --+------- t
   0              0             0
```

#### Relationship
Each is the integral of the previous one:

$$
u(t) = \int_{-\infty}^{t}\delta(\tau)\,d\tau, \qquad r(t) = \int_{-\infty}^{t}u(\tau)\,d\tau
$$

and each is the derivative of the next one:

$$
\delta(t) = \frac{du(t)}{dt}, \qquad u(t) = \frac{dr(t)}{dt}, \qquad \delta(t) = \frac{d^2 r(t)}{dt^2}
$$

Impulse → (integrate) → step → (integrate) → ramp; differentiation goes the other way.

## Check yourself

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

1. The unit ramp is:
   A. r(t) = t u(t)
   B. r(t) = u(t)
   C. r(t) = delta(t) t
   D. r(t) = t^2 u(t)
   Answer: A. r(t) = t for t >= 0 and 0 otherwise, a line of slope 1. (Key point: Ramp definition)

2. The ramp is obtained from the step by:
   A. Differentiation
   B. Integration
   C. Squaring
   D. Time reversal
   Answer: B. r(t) is the integral of u from -infinity to t. (Key point: Integral chain)

3. delta(t) equals which derivative of r(t)?
   A. First
   B. Second
   C. Third
   D. None
   Answer: B. dr/dt = u(t) and du/dt = delta(t), so delta = d^2 r/dt^2. (Key point: Derivative chain)

4. The impulse is modelled as the limit of a pulse of:
   A. Width Delta and height 1/Delta as Delta -> 0
   B. Width 1 and height Delta
   C. Width Delta and height Delta
   D. Width 1/Delta and height 1
   Answer: A. Keeping unit area while narrowing the pulse gives the impulse. (Key point: Impulse definition)

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