---
title: "Period of 3sin(7πn/9 − 2) + cos(πn/13 + 2)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:fundamental-period-two-sinusoids"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/fundamental-period-two-sinusoids
site: BCT Engineering
---

# Period of 3sin(7πn/9 − 2) + cos(πn/13 + 2)

Signals and Systems · Chapter 1: Signals

- Asked in: 2078 Baisakh, 5 marks
- Video: [Periodic Discrete Time Signals (Solved Problems)](https://www.youtube.com/watch?v=dD_67ZmfPTc) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/fundamental-period-two-sinusoids
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Determine the fundamental period and fundamental frequency of the periodic signal x[n] = 3 sin((7π/9)n − 2) + cos((π/13)n + 2)

## Short answer

First term has angular frequency 7π/9 giving N1=18; second term ω=π/13 gives N2=26. Fundamental period N=lcm(18,26)=234 samples, fundamental frequency ω0=2π/234=π/117 rad/sample.

## Answer

A sum of DT sinusoids is periodic if each $\omega_i/2\pi$ is rational; $N = \text{LCM}(N_1, N_2)$. Phase shifts do not change the period.

**Term 1:** $3\sin\left(\frac{7\pi}{9}n - 2\right)$, $\omega_1 = \frac{7\pi}{9}$

$$
\frac{\omega_1}{2\pi} = \frac{7}{18} \;\Rightarrow\; N_1 = \frac{2\pi m}{7\pi/9} = \frac{18m}{7} \;\Rightarrow\; N_1 = 18\ (m=7)
$$

**Term 2:** $\cos\left(\frac{\pi}{13}n + 2\right)$, $\omega_2 = \frac{\pi}{13}$

$$
\frac{\omega_2}{2\pi} = \frac{1}{26} \;\Rightarrow\; N_2 = 26\ (m=1)
$$

**Fundamental period:**

$$
N = \text{LCM}(18, 26) = \text{LCM}(2\cdot 3^2,\ 2\cdot 13) = 2\cdot 9\cdot 13 = 234
$$

**Fundamental frequency:**

$$
\omega_0 = \frac{2\pi}{N} = \frac{2\pi}{234} = \frac{\pi}{117}\ \text{rad/sample}, \qquad f_0 = \frac{1}{234}\ \text{cycles/sample}
$$

**Answer:** $N = 234$ samples; $\omega_0 = \pi/117$ rad/sample ($f_0 = 1/234 \approx 0.00427$ cycles/sample).

## Check yourself

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

1. The period of 3 sin((7 pi/9)n - 2) is:
   A. 9
   B. 18
   C. 7
   D. 36
   Answer: B. w/2pi = 7/18 in lowest terms, so N1 = 18. (Key point: Period of 7 pi/9)

2. The period of cos((pi/13)n + 2) is:
   A. 13
   B. 26
   C. 52
   D. 2 pi
   Answer: B. w/2pi = 1/26, so N2 = 26. (Key point: Period of pi/13)

3. LCM(18, 26) equals:
   A. 468
   B. 234
   C. 117
   D. 78
   Answer: B. 18 = 2 x 3^2 and 26 = 2 x 13, so LCM = 2 x 9 x 13 = 234. (Key point: LCM)

4. The fundamental frequency is:
   A. pi/117 rad/sample
   B. pi/234 rad/sample
   C. 2 pi/18 rad/sample
   D. pi/13 rad/sample
   Answer: A. w0 = 2 pi/234 = pi/117 rad/sample. (Key point: Fundamental frequency)

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