SHOW: if x1 & x2 have a period T then x3 = a*x1 + b*x2 also has period T

[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.)