---
title: "Periodicity of 5sin(3πn/8 − π/2) − 2cos(7πn/12)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:periodicity-sum-of-sinusoids"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodicity-sum-of-sinusoids
site: BCT Engineering
---

# Periodicity of 5sin(3πn/8 − π/2) − 2cos(7πn/12)

Signals and Systems · Chapter 1: Signals

- Asked in: 2080 Asoj, 4 marks; 2070 Magh, 4 marks (asked 2 times)
- Video: [Find whether the following signals are periodic or not, If periodic find the fundamental period.](https://www.youtube.com/watch?v=9b-yKbDVNUo) — Engg-Course-Made-Easy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/periodicity-sum-of-sinusoids
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Determine whether the signal x[n] = 5 sin[(3π/8)n − π/2] − 2 cos[(7π/12)n] is periodic or not. If the signal is periodic, calculate its fundamental period and fundamental frequency.

## Short answer

Both sinusoids are periodic because 3π/8 and 7π/12 divided by 2π give rational fractions 3/16 and 7/24. Their periods are 16 and 24 samples; LCM(16,24)=48. So x[n] is periodic with fundamental period N=48 samples and fundamental frequency ω0=π/24 rad/sample.

## Answer

A DT sinusoid $\sin(\omega_0 n+\theta)$ is periodic if $\omega_0/2\pi$ is rational; its period is the smallest integer $N = 2\pi m/\omega_0$. A sum is periodic if each term is periodic, and its period is the LCM of the individual periods.

**Term 1:** $5\sin\left(\frac{3\pi}{8}n - \frac{\pi}{2}\right)$, $\omega_1 = \frac{3\pi}{8}$

$$
\frac{\omega_1}{2\pi} = \frac{3}{16}\ (\text{rational}) \;\Rightarrow\; N_1 = \frac{2\pi m}{3\pi/8} = \frac{16m}{3} \;\Rightarrow\; N_1 = 16\ (m = 3)
$$

**Term 2:** $2\cos\left(\frac{7\pi}{12}n\right)$, $\omega_2 = \frac{7\pi}{12}$

$$
\frac{\omega_2}{2\pi} = \frac{7}{24}\ (\text{rational}) \;\Rightarrow\; N_2 = \frac{24m}{7} \;\Rightarrow\; N_2 = 24\ (m = 7)
$$

Both terms are periodic, so $x[n]$ is periodic with

$$
N = \text{LCM}(16, 24) = 48
$$

The phase $-\pi/2$ does not affect the period.

**Fundamental frequency:**

$$
\omega_0 = \frac{2\pi}{N} = \frac{2\pi}{48} = \frac{\pi}{24}\ \text{rad/sample}, \qquad f_0 = \frac{1}{48}\ \text{cycles/sample}
$$

**Answer:** $x[n]$ is periodic; $N = 48$ samples, $\omega_0 = \pi/24$ rad/sample ($f_0 = 1/48$).

## Check yourself

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

1. For 5 sin((3 pi/8)n - pi/2), what is the period N1?
   A. 8
   B. 16
   C. 3
   D. 32
   Answer: B. w/2pi = 3/16 in lowest terms, so N1 = 16. (Key point: Period of first term)

2. For 2 cos((7 pi/12)n), what is N2?
   A. 12
   B. 7
   C. 24
   D. 14
   Answer: C. w/2pi = 7/24, so the smallest integer period is 24. (Key point: Period of second term)

3. How is the period of the sum found, and what is it here?
   A. Sum of periods, 40
   B. Product, 384
   C. LCM of 16 and 24, giving 48
   D. GCD of 16 and 24, giving 8
   Answer: C. A sum of periodic terms repeats at the LCM of the individual periods: LCM(16, 24) = 48. (Key point: Period of the sum)

4. With N = 48, the fundamental frequency w0 is:
   A. pi/24 rad/sample
   B. pi/48 rad/sample
   C. pi/12 rad/sample
   D. 48 rad/sample
   Answer: A. w0 = 2 pi/N = 2 pi/48 = pi/24 rad/sample; the phase -pi/2 does not affect it. (Key point: Fundamental frequency)

## Close variants set in other exams

- [Period of 3sin(7πn/9 − 2) + cos(πn/13 + 2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afundamental-period-two-sinusoids) — Signals and Systems, chapter 1, asked 3×

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