---
title: "Behaviour of x(t) = c e^(at)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:ct-complex-exponential-behaviour"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-complex-exponential-behaviour
site: BCT Engineering
---

# Behaviour of x(t) = c e^(at)

Signals and Systems · Chapter 1: Signals

- Asked in: 2075 Bhadra, 3 marks
- 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-complex-exponential-behaviour
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Explain the behaviour of continuous time complex exponential signal x(t) = c e^(at).

## Short answer

The signal x(t) = C e^(at) encompasses three behaviours: real exponential (a real: grows if a>0, decays if a<0, constant if a=0); purely imaginary (a=jω₀: constant-amplitude periodic sinusoid); and general complex (a=r+jω₀: damped sinusoid if r<0, growing if r>0). Each case depends on the real and imaginary parts of a.

## Answer

The CT complex exponential is $x(t) = Ce^{at}$, where $C$ and $a$ may be complex. Its behaviour has three cases.

#### 1. Real exponential ($C$, $a$ real)
- $a > 0$: grows exponentially (e.g. chain reactions, unstable systems).
- $a < 0$: decays exponentially (e.g. RC discharge, damped systems).
- $a = 0$: constant, $x(t) = C$.

#### 2. Periodic complex exponential ($a = j\omega_0$, purely imaginary)
$x(t) = Ce^{j\omega_0 t} = C(\cos\omega_0 t + j\sin\omega_0 t)$. Magnitude is constant; real and imaginary parts are sinusoids. It is periodic for every $\omega_0 \neq 0$, with $T_0 = 2\pi/|\omega_0|$.

#### 3. General complex exponential ($a = r + j\omega_0$, $C = |C|e^{j\theta}$)

$$
x(t) = |C|e^{rt}\cos(\omega_0 t+\theta) + j|C|e^{rt}\sin(\omega_0 t+\theta)
$$

- $r = 0$: sinusoid of constant amplitude.
- $r < 0$: **damped** sinusoid (decaying envelope $|C|e^{rt}$), e.g. RLC circuit response.
- $r > 0$: **growing** sinusoid.

The envelope $\pm|C|e^{rt}$ bounds the oscillation: it shrinks for $r<0$ and expands for $r>0$.

## Check yourself

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

1. For real C and a < 0, how does x(t) = C e^(at) behave?
   A. It grows exponentially
   B. It is constant
   C. It oscillates
   D. It decays exponentially
   Answer: D. A negative real exponent gives decay, e.g. RC discharge. (Key point: Real exponent cases)

2. When a = jω0, what is true of x(t) = C e^(jω0 t)?
   A. Constant magnitude and periodic for every ω0 ≠ 0 with T0 = 2π/|ω0|
   B. Magnitude decays with time
   C. It is periodic only if ω0/2π is rational
   D. It is a real growing exponential
   Answer: A. The magnitude |C| is constant and the real and imaginary parts are sinusoids. (Key point: Imaginary exponent)

3. For a = r + jω0, which case gives a damped sinusoid?
   A. r > 0
   B. r < 0
   C. r = 0
   D. ω0 = 0
   Answer: B. The envelope |C|e^(rt) shrinks when r < 0, as in an RLC response. (Key point: Damped sinusoid)

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