---
title: "Energy or power of DT cos(3πk/16)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:dt-signal-cos-energy-power"
source: https://www.bctengineer.com/q/semester_5/data_communication/dt-signal-cos-energy-power
site: BCT Engineering
---

# Energy or power of DT cos(3πk/16)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2069 Chaitra, 6 marks
- Video: [Signal Power and Energy](https://www.youtube.com/watch?v=7I9XEhAup4c) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/dt-signal-cos-energy-power
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Determine whether the following discrete-time (DT) signal is an energy or power signal. Also find its corresponding energy or power. x[k] = cos(3πk/16) for -10 ≤ k ≤ 0, and 0 otherwise.

## Short answer

Finite-duration signal x[k] is an energy signal. Energy E ≈ 5.66, power P = 0.

## Answer

x[k] = cos(3πk/16) for -10 ≤ k ≤ 0, and 0 otherwise.

The signal is non-zero for only 11 samples (finite duration), so its energy is finite and its average power is zero. It is an **energy signal**.

#### Energy
E = Σ |x[k]|² = Σ_{k=-10}^{0} cos²(3πk/16)

Using cos²θ = (1 + cos 2θ)/2:

E = ½ Σ 1 + ½ Σ cos(3πk/8) = 11/2 + ½ Σ_{k=-10}^{0} cos(3πk/8)

Evaluating the 11 terms numerically, Σ cos(3πk/8) = 0.3244, so

E = 5.5 + 0.1622 = **5.662**

#### Power
P = lim (N→∞) 1/(2N+1) Σ_{k=-N}^{N} |x[k]|² = 5.662 / ∞ = **0**

**Answer: x[k] is an energy signal with E ≈ 5.66 and P = 0.**

## Check yourself

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

1. The finite-duration DT signal is
   A. A power signal
   B. Both
   C. Neither
   D. An energy signal
   Answer: D. 11 samples give finite energy. (Key point: Classification)

2. Energy of a DT signal is
   A. Sum of |x[k]|^2
   B. Sum of x[k]
   C. Average of x[k]
   D. Sum of x[k]^3
   Answer: A. Definition. (Key point: Energy formula)

3. What identity simplifies cos^2?
   A. cos^2 θ = 1 - sin θ
   B. cos^2 θ = (1 + cos 2θ)/2
   C. cos^2 θ = cos 2θ
   D. cos^2 θ = (1 - cos 2θ)/2
   Answer: B. Used to split the sum. (Key point: Identity)

4. The average power is
   A. 5.66
   B. 0.5
   C. 0
   D. 1
   Answer: C. Finite energy over infinite N. (Key point: Power)

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