Condition for periodicity of linear combination of signals

[the_amateur]

What is the condition for the continuous time signal x(t) to be periodic if it is the linear combination of n periodic signals.

where

x(t) = a_{1}x_{1}(t)+a_{2}x_{2}(t)+a_{3}x_{3}(t)+......a_{n}x_{n}(t)

where
x_{i}(t) is periodic with fundamental period T_{i} \forall i, where i \in [1,n]Also provide the fundamental period of x(t) with a proof. thanks.

 

[jashua]

Let the fundamental period of x(t) be T such that x(t)=x(t+T). Then

x(t+T)=a_1x_1(t+T) + a_2x_2(t+T) + \ldots + a_nx_n(t+T)
=a_1x_1(t+k_1T_1) + a_2x_2(t+k_2T_2) + \ldots + a_nx_n(t+k_nT_n) = x(t).

Hence, x(t) is periodic with the period T only if k_1, k_2, \ldots k_n are some integers. Then, since we have

T=k_1T_1 = k_2T_2 = \ldots = k_nT_n,

the fundamental period of x(t), T, is the least common multiple of T_1, T_2, \ldots, T_n.

 

[the_amateur]

Thanks for the answer!

I guess here k_{i} indicates the number of wavelengths of x_{i}(t) in the time period T of x(t).So it has to be an integer as only then x(t) can be periodic.

Please correct me if i am wrong.

 

[jashua]

Your guess is correct.

 

[the_amateur]

thanks again!