---
title: "Energy or power: u[n] and δ[n]"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:energy-power-unit-step-impulse"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/energy-power-unit-step-impulse
site: BCT Engineering
---

# Energy or power: u[n] and δ[n]

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2072 Kartik, 2+3 marks
- Video: [Energy And Power Signal ,Unit Step and Unit Impluse Signal](https://www.youtube.com/watch?v=d2yL36RMuZc) — Nucleus : GATE ESE NIC ISRO
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/energy-power-unit-step-impulse
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Define energy and power signal. Check whether the signals x[n] = u[n] and x[n] = δ[n] are energy or power type.

## Short answer

Energy signal has finite energy; power signal has finite nonzero average power. Unit step u[n]: energy infinite, power=½, so power signal. Unit impulse δ[n]: energy=1 (finite), power=0, so energy signal.

## Answer

#### Definitions
- **Energy signal:** E = Σ |x[n]|² is finite (0 < E < ∞); then average power P = 0.
- **Power signal:** E = ∞ but P = lim(N→∞) 1/(2N+1) Σ(n=−N to N) |x[n]|² is finite and non-zero.

#### Unit step x[n] = u[n]
E = Σ(n=0 to ∞) 1² = ∞, so it is not an energy signal.

P = lim 1/(2N+1) Σ(n=0 to N) 1 = lim (N+1)/(2N+1) = **½**

P is finite, so u[n] is a **power signal**.

#### Unit impulse x[n] = δ[n]
E = Σ δ²[n] = 1 (only the sample at n = 0 is non-zero). It is finite, so δ[n] is an **energy signal**.

Check: P = lim 1/(2N+1) · 1 = 0, as expected for an energy signal.

| Signal | Energy | Power | Type |
|---|---|---|---|
| u[n] | ∞ | ½ | Power |
| δ[n] | 1 | 0 | Energy |

## Check yourself

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

1. What is the energy of u[n]?
   A. 1
   B. Infinity
   C. 1/2
   D. 0
   Answer: B. E = sum of 1^2 for n >= 0 diverges, so E is infinite. (Key point: Energy of unit step)

2. What is the average power of u[n]?
   A. 1/2
   B. 1
   C. 0
   D. Infinity
   Answer: A. P = lim (N+1)/(2N+1) = 1/2, finite and non-zero, so u[n] is a power signal. (Key point: Power of unit step)

3. Why is delta[n] an energy signal?
   A. Its average power is 1/2
   B. It has infinite duration
   C. Its energy is zero
   D. Only one sample is non-zero, so E = 1 is finite
   Answer: D. E = sum of delta^2[n] = 1, finite, and its power is 0. (Key point: Impulse classification)

4. For an energy signal such as delta[n], the average power is
   A. 1
   B. 1/2
   C. 0
   D. infinite
   Answer: C. Finite energy divided by an ever-growing 2N+1 tends to 0. (Key point: Energy-power relation)

## Related in Digital Signal Analysis and Processing

- [Energy/power signals; periodicity of complex exponential](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-periodic-1) — Digital Signal Analysis and Processing, chapter 1
- [Energy/power signals; periodicity of cos + sin](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-periodic-2) — Digital Signal Analysis and Processing, chapter 1
- [Periodic signals and fundamental period](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aperiodic-signals-fundamental-period) — Digital Signal Analysis and Processing, chapter 1

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