---
title: "Behaviour of x[n] = cαⁿ"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:dt-real-exponential-behaviour"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dt-real-exponential-behaviour
site: BCT Engineering
---

# Behaviour of x[n] = cαⁿ

Signals and Systems · Chapter 1: Signals

- Asked in: 2076 Baisakh, 4 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/dt-real-exponential-behaviour
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Explain the behaviour of discrete time complex exponential x[n] = cαⁿ where c and α are real.

## Short answer

For x[n]=cα^n with real c,α (c>0): α>1 grows unbounded; 0<α<1 decays; α=1 constant; -1<α<0 alternates sign and decays; α<-1 alternates and grows; α=-1 alternates ±c. A negative c flips the sequence.

## Answer

With $c$ and $\alpha$ both real, $x[n] = c\alpha^n$ is a **real exponential sequence**. Its shape depends on the value of $\alpha$ (take $c > 0$):

| Value of $\alpha$ | Behaviour of $x[n]$ |
|---|---|
| $\alpha > 1$ | Grows exponentially with $n$ (unbounded) |
| $\alpha = 1$ | Constant, $x[n] = c$ |
| $0 < \alpha < 1$ | Decays exponentially towards 0 |
| $-1 < \alpha < 0$ | Alternates in sign and decays |
| $\alpha = -1$ | Alternates between $+c$ and $-c$ |
| $\alpha < -1$ | Alternates in sign and grows |

For negative $\alpha$, $\alpha^n$ is positive for even $n$ and negative for odd $n$, giving the alternating pattern. A negative $c$ simply flips the whole sequence.

```text
 0 < alpha < 1: decays
  o
  |
  |  o
  |  |  o  o
--+--+--+--+--o-> n
  0  1  2  3  4

 alpha > 1: grows
              o
              |
           o  |
        o  |  |
  o  o  |  |  |
--+--+--+--+--+-> n
  0  1  2  3  4

 -1 < alpha < 0: alternates, decays
  o
  |
  |
  |     o
--+--+--+--+--o-> n
     |     o
     o
  0  1  2  3  4

 alpha < -1: alternates, grows
        o
        |
  o     |
--+--+--+--+-> n
     |     |
     o     |
           |
           o
  0  1  2  3
```

**Examples:** $(0.5)^n u[n]$ decays (used for stable system impulse responses); $(-0.8)^n u[n]$ oscillates and decays; $2^n u[n]$ grows (unstable).

## Check yourself

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

1. For c > 0, how does x[n] = c α^n behave when 0 < α < 1?
   A. It grows without bound
   B. It stays constant
   C. It alternates in sign and decays
   D. It decays exponentially towards 0 without changing sign
   Answer: D. Powers of a number between 0 and 1 shrink and stay positive. (Key point: Decay condition)

2. Why does x[n] = c α^n alternate in sign for negative α?
   A. α^n is positive for even n and negative for odd n
   B. Because c becomes negative
   C. Because n becomes negative
   D. Because the magnitude oscillates
   Answer: A. A negative base raised to even and odd integer powers gives alternating signs. (Key point: Negative alpha)

3. What does x[n] = c(-1)^n look like?
   A. It decays to zero
   B. It alternates between +c and -c with constant magnitude
   C. It is constant at c
   D. It grows while alternating
   Answer: B. |α| = 1 keeps the magnitude fixed while the sign flips every sample. (Key point: Alpha = -1)

4. Which α gives a sequence that alternates in sign and grows?
   A. -1 < α < 0
   B. α > 1
   C. α < -1
   D. 0 < α < 1
   Answer: C. Negative α gives alternation, and |α| > 1 gives growth. 2^n u[n] (α = 2) grows without alternating. (Key point: Growth with alternation)

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