Does sum of two Periodic signals give Periodic signal ?

Signals is an electromagnetic or electrical current that carries data from one system or network to another. In simple words, Information converted into an electrical form suitable for transmission is called as signal.

The sum of two periodic signals may or may not result in a periodic signal. Whether the sum is periodic depends on the frequencies of the individual periodic signals and the relationship between those frequencies.

If the frequencies of the two signals are incommensurate (not rational multiples of each other), then the sum of the signals is generally not periodic. In this case, the combined waveform does not repeat itself over time, and the resulting signal is aperiodic.

However, if the frequencies of the two signals are rational multiples of each other (meaning one frequency is an integer multiple of the other), then the sum of the signals can be periodic.

The periodicity of the resulting signal depends on the relationship between the frequencies. For example, if the frequencies have a common fundamental frequency or if their ratios are simple fractions (such as 1:2, 2:3, etc.), the resulting sum could exhibit periodic behaviour.

In this article, we are going to see whether the sum of two periodic signals is a Periodic signal or not.

You can watch a quick video or read along.

What is Periodic signal?

Any signal which has a particular pattern and which will repeat continuously after a particular interval of time is called as periodic signal.

Ex- sinusoidal wave, co-sine wave, square wave, triangular wave, etc comes under periodic wave.

Many students have a question in their mind, that when we add two periodic signal, whether we will get a periodic signal or non-periodic signal?

Let us find answer to this question in detail.

Condition to check Periodicity

Consider an example, f(t) = sint

g(t) = cos 2 t

Both are the periodic signals.

When we add this two signals, we get

h(t) = f(t) + g(t) = sin t + cos 2t

For the signal to be periodic signals must satisfy the following conditions,

h(t)= h(t\pm T)= h(t+T)

T- fundamental time period

If any signal will satisfy this condition, then that signal is called as periodic signal.

If the signal is not following this condition, then that signal is not periodic signal.

As we are going to add two signals, so we will take ‘+’ sign.

Let us take signal individually,

f(t) = sin t

g(t) = cos 2t

Let us find the value of T₁ for first signal.

T_{1}=\frac{2\pi }{\omega }

ω=1

Putting value of ω, we get

T_{1}=\frac{2\pi }{1 }= 2\pi

Let us find the value of T₂ for second signal.

T_{2}=\frac{2\pi }{\omega }

ω=2

Putting value of ω, we get

T_{1}=\frac{2\pi }{2 }= \pi

For finding the value of T we will take ratios

\frac{T_{1}}{T_{2}}=\frac{2\pi }{\pi }=\frac{2}{1}=\frac{m}{n}

nT₁ = mT₂ = T

T= nT₁ = 1 x 2π = 2π

T= 2π

Time period will be 2π.

For T= 2π

h(t+2π) = sin (t+2π)+ cos(2(t+2π))

= sin (t+2π) + cos(2t+4π)

h(t+2π) = sin t + cos 2t

h(t+2π) = h(t)

Condition is satisfied, that means ṭhe addition of two periodic signal is a periodic signal.

Hence, proved.

Therefore, from these derivations we come to the conclusion that ‘ Sum of two periodic functions can be Periodic.

Hope, so ! You have well understood that how to identify the Periodic and non-periodic signal.