---
title: "CT unit step; periodicity of e^(jωt)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:unit-step-ct-periodicity"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/unit-step-ct-periodicity
site: BCT Engineering
---

# CT unit step; periodicity of e^(jωt)

Signals and Systems · Chapter 1: Signals

- Asked in: 2081 Chaitra, 1+3 marks
- Video: [Unit Step Signal](https://www.youtube.com/watch?v=tI7sEfdnYdE) — 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/unit-step-ct-periodicity
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define continuous time unit step signal. Derive the necessary condition for the signal x(t) = e^(jωt) to be periodic.

## Short answer

The CT unit step u(t)=1 for t≥0, 0 for t<0. For e^(jωt) to be periodic, e^(jωT)=1 ⇒ ωT=2πm (m integer). For ω≠0, fundamental period T0=2π/|ω|. Hence a CT complex exponential is always periodic for any non-zero ω.

## Answer

#### CT unit step
$$
u(t) = \begin{cases} 1, & t \ge 0 \\ 0, & t < 0 \end{cases}
$$
It switches from 0 to 1 at $t = 0$; $u(t) = \int_{-\infty}^{t}\delta(\tau)d\tau$.

#### Periodicity of $x(t) = e^{j\omega t}$

$x(t)$ is periodic with period $T$ if $x(t+T) = x(t)$ for all $t$:

$$
e^{j\omega(t+T)} = e^{j\omega t}e^{j\omega T} = e^{j\omega t} \;\Rightarrow\; e^{j\omega T} = 1
$$

$$
\cos\omega T + j\sin\omega T = 1 \;\Rightarrow\; \omega T = 2\pi m, \quad m = \pm1, \pm2, \ldots
$$

So the necessary condition is $\omega T = 2\pi m$. For $\omega \ne 0$ the smallest positive $T$ (with $m = 1$) is

$$
T_0 = \frac{2\pi}{|\omega|}
$$

Since $T$ can be any real number, this is satisfied for **every** non-zero $\omega$: a CT complex exponential is always periodic (for $\omega = 0$ it is a constant, periodic with any $T$). This differs from DT, where $\omega/2\pi$ must also be rational.

## Check yourself

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

1. The CT unit step relates to the impulse as:
   A. u(t) = d delta/dt
   B. u(t) = integral of delta from -infinity to t
   C. u(t) = delta(t)^2
   D. u(t) = t delta(t)
   Answer: B. The step is the running integral of the impulse. (Key point: Step as integral)

2. For e^(j w t) to have period T we need:
   A. w T = pi
   B. w T = 2 pi m
   C. w = 2 pi m
   D. T = w
   Answer: B. e^(jwT) = 1 requires wT to be an integer multiple of 2 pi. (Key point: Periodicity condition)

3. The fundamental period of e^(j w t), w != 0, is:
   A. 2 pi |w|
   B. 2 pi / |w|
   C. pi / w
   D. 1/w
   Answer: B. The smallest positive T (m = 1) is 2 pi/|w|. (Key point: Fundamental period)

4. Why is CT e^(jwt) periodic for all w != 0?
   A. T may be any real number
   B. w is always rational
   C. t is an integer
   D. Because it is real
   Answer: A. No integer restriction on T, so no rationality condition on w is needed. (Key point: Compared with DT)

## Close variants set in other exams

- [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×

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