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!