---
title: "Period of cos(πn/2)·cos(πn/4)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:period-product-of-cosines"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/period-product-of-cosines
site: BCT Engineering
---

# Period of cos(πn/2)·cos(πn/4)

Signals and Systems · Chapter 1: Signals

- Asked in: 2071 Bhadra, 3 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/period-product-of-cosines
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

If the signal is periodic, find the fundamental period of the signal x[n] = cos(πn/2)·cos(πn/4).

## Short answer

Both factors have rational frequencies with periods N1=4 and N2=8, so the product is periodic. The fundamental period is the LCM: N=8.

## Answer

A product (or sum) of discrete-time periodic signals is periodic if each part is periodic; the LCM of the individual periods is then a period (the fundamental period can be smaller, so it is worth checking).

**First factor:** $\cos(\pi n/2)$, $\omega_1 = \pi/2$:

$$
\frac{\omega_1}{2\pi} = \frac{1}{4} \Rightarrow N_1 = 4
$$

**Second factor:** $\cos(\pi n/4)$, $\omega_2 = \pi/4$:

$$
\frac{\omega_2}{2\pi} = \frac{1}{8} \Rightarrow N_2 = 8
$$

Both ratios are rational, so both factors are periodic and the product is periodic with

$$
N = \text{LCM}(4, 8) = 8
$$

**Check using a product-to-sum identity:**

$$
\cos\frac{\pi n}{2}\cos\frac{\pi n}{4} = \frac{1}{2}\left[\cos\frac{3\pi n}{4} + \cos\frac{\pi n}{4}\right]
$$

$\cos(3\pi n/4)$: $\frac{3}{8} \Rightarrow N = 8$; $\cos(\pi n/4)$: $N = 8$. LCM = 8, which agrees.

**Answer:** $x[n]$ is periodic with fundamental period $N = 8$.

## Check yourself

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

1. What are the periods of cos(πn/2) and cos(πn/4)?
   A. 2 and 4
   B. π/2 and π/4
   C. 1/4 and 1/8
   D. 4 and 8
   Answer: D. ω/2π = 1/4 and 1/8, so N = 4 and 8. (Key point: Individual periods)

2. What is the fundamental period of the product?
   A. 8
   B. 4
   C. 32
   D. 12
   Answer: A. N = LCM(4, 8) = 8. (Key point: Combined period)

3. How does the product-to-sum identity confirm the period?
   A. It gives cos(3πn/4) only, with N = 4
   B. It gives (1/2)[cos(3πn/4) + cos(πn/4)], both with N = 8
   C. It shows the signal is aperiodic
   D. It gives two terms with periods 3 and 4
   Answer: B. 3/8 and 1/8 both reduce to N = 8, matching the LCM. (Key point: Verification)

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