---
title: "Sum of periodic signals can be aperiodic"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:sum-periodic-signals-aperiodic"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/sum-periodic-signals-aperiodic
site: BCT Engineering
---

# Sum of periodic signals can be aperiodic

Signals and Systems · Chapter 1: Signals

- Asked in: 2078 Poush, 4 marks
- Video: [EE 207, 11- Sum of Periodic Signals](https://www.youtube.com/watch?v=CaWN6CJyAx4) — Wessam Mesbah
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/sum-periodic-signals-aperiodic
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

"Combination of two continuous-time periodic signal might result in an aperiodic signal." Justify this statement using any numerical example.

## Short answer

Example: x(t)=cos(2t)+cos(πt) has component periods π and 2 s; ratio π/2 is irrational, so no common period exists. Thus the sum is aperiodic despite both components being periodic.

## Answer

If $x_1(t)$ has period $T_1$ and $x_2(t)$ has period $T_2$, the sum $x_1(t)+x_2(t)$ is periodic only if there is a common period $T = mT_1 = kT_2$ for some integers $m, k$, i.e. only if

$$
\frac{T_1}{T_2} = \frac{k}{m} = \text{rational number}
$$

If $T_1/T_2$ is **irrational**, there is no common period and the sum is **aperiodic**, even though each part is periodic.

#### Numerical example
$x(t) = \cos 2t + \cos \pi t$

- $\cos 2t$: $T_1 = \frac{2\pi}{2} = \pi$ s
- $\cos \pi t$: $T_2 = \frac{2\pi}{\pi} = 2$ s

$$
\frac{T_1}{T_2} = \frac{\pi}{2} \quad\text{(irrational)}
$$

No integers $m, k$ satisfy $m\pi = 2k$, so the peaks of the two cosines never line up again after $t = 0$ (at $t=0$ the sum is 2, but it never returns to exactly 2). Hence **$x(t)$ is aperiodic**.

**Contrast (periodic case):** $\cos 2t + \cos 3t$ has $T_1 = \pi$, $T_2 = 2\pi/3$, ratio $3/2$ (rational), so the sum is periodic with $T = \text{LCM} = 2\pi$ s.

This justifies the statement: a combination of two CT periodic signals may be aperiodic when the ratio of their periods is irrational.

## Check yourself

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

1. The sum of two CT periodic signals is periodic only if:
   A. T1 = T2
   B. T1/T2 is rational
   C. T1/T2 is irrational
   D. T1 and T2 are integers
   Answer: B. A common period mT1 = kT2 exists only when the ratio of periods is rational. (Key point: Condition for periodic sum)

2. For cos 2t + cos(pi t), the periods are:
   A. pi and 2
   B. 2 and pi
   C. 1 and 2
   D. 2 pi and pi
   Answer: A. T1 = 2 pi/2 = pi and T2 = 2 pi/pi = 2. (Key point: Periods in the example)

3. T1/T2 = pi/2 is irrational, so the sum is:
   A. Periodic with T = 2 pi
   B. Aperiodic
   C. Periodic with T = pi
   D. An energy signal
   Answer: B. No integers m, k satisfy m pi = 2k, so there is no common period. (Key point: Verdict)

4. cos 2t + cos 3t is periodic with period:
   A. pi
   B. 2 pi
   C. 6
   D. 2 pi/3
   Answer: B. Periods pi and 2 pi/3 have ratio 3/2 (rational), LCM = 2 pi. (Key point: Contrast case)

## Related in Signals and Systems

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