---
title: "Unit step in terms of signum; energy or power"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:unit-step-signum-energy-power"
source: https://www.bctengineer.com/q/semester_6/communication_systems/unit-step-signum-energy-power
site: BCT Engineering
---

# Unit step in terms of signum; energy or power

Communication Systems · Chapter 1: Introduction

- Asked in: 2079 Chaitra, 4+4 marks
- Video: [Signum Function](https://www.youtube.com/watch?v=T_pGvvyjIkI) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/unit-step-signum-energy-power
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Represent Unit step signal in terms of Signum function. Also, determine whether a Unit step signal is energy or power type or neither of the two.

## Short answer

u(t) = 0.5[1 + sgn(t)]; its energy is infinite because ∫u² dt diverges. Average power P = lim(1/T) ∫0^(T/2) 1 dt = 0.5 W (normalised), making it a power signal.

## Answer

#### Unit step in terms of signum
The signum function is

$$
\text{sgn}(t) = \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}
$$

and the unit step is $u(t) = 1$ for $t > 0$, $0$ for $t < 0$.

Adding 1 to sgn$(t)$ gives $2$ for $t>0$ and $0$ for $t<0$; halving it gives the unit step:

$$
u(t) = \frac{1}{2}\left[1 + \text{sgn}(t)\right]
$$

Check: $t>0$: $\frac12(1+1) = 1$; $t<0$: $\frac12(1-1) = 0$; $t=0$: $\frac12$ (the mid-value used at the jump).

```text
 sgn(t)              u(t)
  1 |------          1 |------
    |                  |
 ---+------> t      ---+------> t
-1--|                  |
```

This relation is used to find the Fourier transform of $u(t)$: since $\mathcal{F}\{\text{sgn}(t)\} = \frac{1}{j\pi f}$ and $\mathcal{F}\{1\} = \delta(f)$,

$$
U(f) = \frac12\delta(f) + \frac{1}{j2\pi f}
$$

#### Energy or power?
**Energy:**

$$
E = \int_{-\infty}^{\infty} u^2(t)\,dt = \int_{0}^{\infty} 1\,dt = \infty
$$

So it is not an energy signal.

**Power:**

$$
\begin{aligned}
P &= \lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2} u^2(t)\,dt \\
&= \lim_{T\to\infty}\frac{1}{T}\int_{0}^{T/2} 1\,dt = \lim_{T\to\infty}\frac{1}{T}\cdot\frac{T}{2} = \frac12
\end{aligned}
$$

**Answer:** $u(t) = \frac12[1 + \text{sgn}(t)]$; the unit step has infinite energy and finite average power $P = 0.5$ W (normalised), so it is a **power signal**.

## Check yourself

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

1. Which expression gives the unit step in terms of the signum function?
   A. u(t) = sgn(t) - 1
   B. u(t) = 1 + sgn(t)
   C. u(t) = (1/2)[1 + sgn(t)]
   D. u(t) = 2 sgn(t)
   Answer: C. 1 + sgn(t) is 2 for t > 0 and 0 for t < 0; halving gives 1 and 0, with value 1/2 at t = 0. (Key point: Step in terms of signum)

2. What is the energy of u(t)?
   A. 1 J
   B. Zero
   C. 1/2 J
   D. Infinite, since the integral of 1 from 0 to infinity diverges
   Answer: D. E = integral of u^2 dt = integral of 1 from 0 to infinity, which is infinite, so it is not an energy signal. (Key point: Energy of the unit step)

3. What is the average power of u(t), and its type?
   A. 1/2 W, power signal
   B. 1 W, energy signal
   C. Zero, energy signal
   D. Infinite, neither
   Answer: A. P = lim (1/T)(T/2) = 1/2, finite and non-zero, so u(t) is a power signal. (Key point: Power of the unit step)

## Related in Communication Systems

- [Hilbert transform properties; energy and power of unit step](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-properties-unit-step-energy) — Communication Systems, chapter 2
- [Signum signal; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asignum-distortionless) — Communication Systems, chapter 1
- [Energy and power signal; bandwidth of a system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenergy-power-bandwidth-system) — Communication Systems, chapter 1
- [Shannon-Hartley theorem; energy/power of e^(−2t)u(t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-hartley-energy-e2t) — Communication Systems, chapter 1
- [Energy or power: x(t) = A cos2πft](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenergy-power-acos) — Communication Systems, chapter 1, asked 2×
- [PSD of a power signal and its interpretation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsdf-power-signal-interpretation) — Communication Systems, chapter 1, asked 2×

## Related in other subjects

- [Energy or power: unit step, impulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aenergy-power-check-signals) — Data Communication, chapter 2, asked 2×
- [Unit step signal and relation to impulse](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aunit-step-relation-impulse) — Digital Signal Analysis and Processing, chapter 1
- [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
- [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×