---
title: "Periodic signals and fundamental period"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:periodic-signals-fundamental-period"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/periodic-signals-fundamental-period
site: BCT Engineering
---

# Periodic signals and fundamental period

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2070 Chaitra, 3 marks
- 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_7/digital_signal_analysis_and_processing/periodic-signals-fundamental-period
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

Determine which of the following signals are periodic and compute their fundamental period: i) cos(πn²/8)  ii) cos(n/2) cos(πn/4).

## Short answer

i) cos(πn²/8) is periodic: N=8 satisfies phase shift condition for all n. ii) cos(n/2)cos(πn/4) simplifies to sum of cosines with frequencies (½±π/4)/2π which are irrational, so non-periodic.

## Answer

A discrete signal is periodic if x[n + N] = x[n] for all n, for some smallest positive integer N (the fundamental period).

#### i) x[n] = cos(πn²/8)
Need cos(π(n + N)²/8) = cos(πn²/8) for every n. Expanding, the phase changes by

π(2nN + N²)/8 = nNπ/4 + N²π/8.

This must be a multiple of 2π for all n:
- nNπ/4 = 2πk for every n requires N to be a multiple of 8.
- Try N = 8: N²π/8 = 64π/8 = 8π, a multiple of 2π. Satisfied.

So the signal is **periodic with fundamental period N = 8**. Verified numerically: cos(πn²/8) for n = 0 … 7 repeats for n = 8 … 15.

#### ii) x[n] = cos(n/2) cos(πn/4)
Use cos A cos B = ½[cos(A + B) + cos(A − B)]:

x[n] = ½ cos((½ + π/4)n) + ½ cos((½ − π/4)n)

Frequencies: ω₁ = ½ + π/4 and ω₂ = ½ − π/4. For periodicity, ω/2π must be rational:

ω₁/2π = 1/(4π) + 1/8, which is irrational (because of π).

Similarly cos(n/2) alone has f = 1/(4π), which is irrational. So no integer N exists.

The signal is **not periodic**.

## Check yourself

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

1. A discrete signal is periodic with period N if
   A. x[n+N] = x[n] only for n = 0
   B. x[n+N] = x[n] for all n, N a positive integer
   C. x[n+N] = -x[n]
   D. N may be any positive real number
   Answer: B. The period must be an integer and the repeat must hold for every n. (Key point: Definition of period)

2. What is the fundamental period of cos(pi n^2/8)?
   A. 8
   B. 4
   C. 16
   D. It is not periodic
   Answer: A. The phase change n N pi/4 + N^2 pi/8 is a multiple of 2 pi for all n when N = 8 (N^2 pi/8 = 8 pi). (Key point: cos(pi n^2/8) period)

3. Why is cos(n/2) cos(pi n/4) not periodic?
   A. Product of cosines is never periodic
   B. Its period is infinite because pi/4 is small
   C. Because it has two different frequencies
   D. Frequencies 1/2 +- pi/4 give w/2pi irrational, so no integer N exists
   Answer: D. The product splits into cosines with w = 1/2 +- pi/4; w/2pi involves pi and is irrational. (Key point: Reason for non-periodicity)

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