Proving a linear combination of periodic functions is periodic

[VinnyCee]

Homework Statement

Show that if $x_1(t)$ and $x_2(t)$ have period T, then $x_3(t)\,=\,ax_1(t)\,+\,bx_2(t)$ (a, b constant) has the same period T.

Homework Equations

$$x_1\left(t\,+\,T\right)\,=\,x_1(t)$$

$$x_2\left(t\,+\,T\right)\,=\,x_2(t)$$

The Attempt at a Solution

$$x_3(t)\,=\,a\,x_1(t)\,+\,b\,x_2(t)$$

$$x_3(t\,+\,T)\,=\,a\,x_1(t\,+\,T)\,+\,b\,x_2(t\,+\,T)$$

Since the relevant equations (above) are true...

$$x_3\left(t\,+\,T\right)\,=\,a\,x_1(t)\,+\,b\,x_2(t)$$

$$x_3(t)\,=\,a\,x_1(t)\,+\,b\,x_2(t)$$

$$\therefore\,x_3\left(t\,+\,T\right)\,=\,x_3(t)\,\forall\,t\,\in\,\mathbb{R}$$



Does this look right?

 

[HallsofIvy]

Yes, that's fine. That shows that if T is a period of both x₁ and x₂ then it is a period of ax₁+ bx₂ for any a and b. It is NOT always true that if T is the period (i.e. smallest period) of both x₁ and x₂ then it is the period of ax₁+ bx₂. Example: x₁= sin(x)+ sin(2x), x₂= -sin(x)+ sin(2x). Then $2\pi$ is the period of both x₁ and x₂ but x₁-x₂= 2sin(2x) which has fundamental period $\pi$. (But $2\pi$ is still a period.)