---
title: "Behaviour of DT complex exponential"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:dt-complex-exponential-behaviour"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dt-complex-exponential-behaviour
site: BCT Engineering
---

# Behaviour of DT complex exponential

Signals and Systems · Chapter 1: Signals

- Asked in: 2073 Bhadra, 2+3 marks
- Video: [Exponential Signals (Real and Complex)](https://www.youtube.com/watch?v=IFaliBPD1UY) — 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/dt-complex-exponential-behaviour
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define discrete time complex exponential signal and its different types of behavior.

## Short answer

A DT complex exponential is x[n]=Cα^n. For real α, behavior depends on |α|: grows (>1), decays (0<α<1), or alternates and decays/grows. For α=e^(jω0) it is sinusoidal, periodic only if ω0/(2π) is rational. General complex exponential combines exponential envelope with oscillation.

## Answer

A **discrete-time complex exponential** is

$$
x[n] = C\alpha^{n}
$$

where $C$ and $\alpha$ are in general complex numbers. (With $\alpha = e^{\beta}$ it is also written $Ce^{\beta n}$.) Its behaviour depends on $C$ and $\alpha$:

#### 1. Real exponential ($C$, $\alpha$ real)
- $\alpha > 1$: grows exponentially.
- $0 < \alpha < 1$: decays exponentially.
- $-1 < \alpha < 0$: alternates in sign and decays.
- $\alpha < -1$: alternates in sign and grows.
- $\alpha = 1$: constant; $\alpha = -1$: alternates between $+C$ and $-C$.

#### 2. Purely imaginary exponent ($|\alpha| = 1$), sinusoidal
With $\alpha = e^{j\omega_0}$: $x[n] = Ce^{j\omega_0 n} = C(\cos\omega_0 n + j\sin\omega_0 n)$. The magnitude is constant; real and imaginary parts are sampled sinusoids. It is periodic only if $\omega_0/2\pi$ is rational, and frequencies $\omega_0$ and $\omega_0 + 2\pi$ give identical signals.

#### 3. General complex exponential
With $C = |C|e^{j\theta}$ and $\alpha = |\alpha|e^{j\omega_0}$:

$$
x[n] = |C||\alpha|^n\cos(\omega_0 n+\theta) + j|C||\alpha|^n\sin(\omega_0 n+\theta)
$$

- $|\alpha| = 1$: constant-amplitude sinusoid.
- $|\alpha| < 1$: sinusoid with **decaying** envelope.
- $|\alpha| > 1$: sinusoid with **growing** envelope.

```text
 0<a<1 decay        -1<a<0 alternating decay
 o                    o
 | o                  |   o
 | | o o .            | . | . .
-+-+-+-+-+-> n     ---+-+-+-+-+-> n
                        o
```

## Check yourself

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

1. A DT complex exponential is written:
   A. x[n] = C alpha^n
   B. x[n] = C n alpha
   C. x[n] = C + alpha n
   D. x[n] = alpha C^n only for real C
   Answer: A. x[n] = C alpha^n with C and alpha generally complex. (Key point: General form)

2. For real alpha with -1 < alpha < 0, x[n] behaves as:
   A. Decays monotonically
   B. Grows monotonically
   C. Alternates in sign and decays
   D. Alternates in sign and grows
   Answer: C. Negative alpha flips the sign each step and |alpha| < 1 shrinks the magnitude. (Key point: Real alpha behaviour)

3. When |alpha| = 1, alpha = e^(j w0), the signal:
   A. Has a growing envelope
   B. Is a constant-amplitude sinusoid, periodic only if w0/2pi is rational
   C. Always decays
   D. Is always periodic
   Answer: B. Magnitude stays constant, and DT periodicity needs rational w0/2 pi. (Key point: Unit magnitude case)

4. For |alpha| > 1 in the general complex case, x[n] is:
   A. A sinusoid with decaying envelope
   B. A sinusoid with growing envelope
   C. A constant
   D. Purely real
   Answer: B. The factor |alpha|^n multiplies the sinusoid and grows with n. (Key point: Envelope behaviour)

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