---
title: "cos(πn/3) with frequency increased by 2π"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:frequency-increased-by-2pi"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/frequency-increased-by-2pi
site: BCT Engineering
---

# cos(πn/3) with frequency increased by 2π

Signals and Systems · Chapter 1: Signals

- Asked in: 2072 Magh, 2 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/frequency-increased-by-2pi
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

The signal x[n] = cos(πn/3) is a periodic signal. What happens if the fundamental frequency of this signal is increased by 2π?

## Short answer

Increasing the frequency by 2π gives cos((π/3+2π)n)=cos(πn/3) because 2πn is an integer multiple of 2π. The signal is unchanged; DT sinusoids whose frequencies differ by 2π are identical.

## Answer

$x[n] = \cos\left(\frac{\pi n}{3}\right)$ has $\omega_0 = \pi/3$ and period $N = 6$. Increase the frequency by $2\pi$:

$$
\begin{aligned}
y[n] &= \cos\left[\left(\frac{\pi}{3} + 2\pi\right)n\right] = \cos\left(\frac{\pi n}{3} + 2\pi n\right) \\
&= \cos\left(\frac{\pi n}{3}\right) = x[n]
\end{aligned}
$$

because $2\pi n$ is an integer multiple of $2\pi$ for every integer $n$.

**Nothing changes:** the new signal is identical to the original, with the same samples and the same period $N = 6$. DT sinusoids whose frequencies differ by $2\pi$ (or any multiple of $2\pi$) are the same signal. So the frequency of a DT signal is unique only over an interval of length $2\pi$ (e.g. $-\pi < \omega \le \pi$ or $0 \le \omega < 2\pi$), unlike CT signals, where a higher frequency always oscillates faster. This is the basis of **aliasing** in sampling.

## Check yourself

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

1. What happens to cos(πn/3) if its frequency is increased by 2π?
   A. It oscillates faster
   B. The signal is unchanged
   C. Its period doubles
   D. It becomes a sine
   Answer: B. cos((π/3 + 2π)n) = cos(πn/3 + 2πn) = cos(πn/3) because 2πn is a multiple of 2π. (Key point: Effect of adding 2π)

2. Over what range is a DT frequency unique?
   A. Any interval of length π
   B. Only above zero
   C. Any interval of length 2π
   D. The whole real line
   Answer: C. Frequencies differing by multiples of 2π give identical sequences. (Key point: Frequency uniqueness)

3. What is the period of cos(πn/3)?
   A. 3
   B. 12
   C. 2π/3
   D. 6
   Answer: D. ω0/2π = 1/6, so N = 6. (Key point: Original period)

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