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

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In summary, if x_1(t) and x_2(t) have period T, then x_3(t)=ax_1(t)+bx_2(t) (a, b constant) also has period T. However, it is not always true that if T is the period of both x_1(t) and x_2(t), it is also the period of x_3(t).
  • #1
VinnyCee
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Homework Statement



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



Homework Equations



[tex]x_1\left(t\,+\,T\right)\,=\,x_1(t)[/tex]

[tex]x_2\left(t\,+\,T\right)\,=\,x_2(t)[/tex]



The Attempt at a Solution



[tex]x_3(t)\,=\,a\,x_1(t)\,+\,b\,x_2(t)[/tex]

[tex]x_3(t\,+\,T)\,=\,a\,x_1(t\,+\,T)\,+\,b\,x_2(t\,+\,T)[/tex]

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

[tex]x_3\left(t\,+\,T\right)\,=\,a\,x_1(t)\,+\,b\,x_2(t)[/tex]

[tex]x_3(t)\,=\,a\,x_1(t)\,+\,b\,x_2(t)[/tex]

[tex]\therefore\,x_3\left(t\,+\,T\right)\,=\,x_3(t)\,\forall\,t\,\in\,\mathbb{R}[/tex]



Does this look right?
 
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  • #2
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 [itex]2\pi[/itex] is the period of both x1 and x2 but x1-x2= 2sin(2x) which has fundamental period [itex]\pi[/itex]. (But [itex]2\pi[/itex] is still a period.)
 
  • #3


Yes, your solution is correct. You have correctly used the given equations to show that x_3(t) has the same period T as x_1(t) and x_2(t). This means that x_3(t) will repeat itself after every T time units, just like x_1(t) and x_2(t). This is an important property in periodic functions and is useful in many applications, including signal processing and oscillatory systems. Well done!
 

1. How can I prove that x3 has the same period as x1 and x2?

The period of a function is the smallest positive number T such that f(x+T) = f(x) for all x. To prove that x3 has the same period as x1 and x2, you can show that x3(x+T) = x3(x) for all x. This would prove that x3 also has a period of T.

2. Can x3 have a different period than x1 and x2?

No, if x1 and x2 have the same period T, then x3 will also have the same period T. This is because x3 is a linear combination of x1 and x2, and the period of a linear combination is always the same as the period of the individual functions.

3. How does the coefficient a affect the period of x3?

The coefficient a does not affect the period of x3. As long as x1 and x2 have the same period T, x3 will also have a period of T, regardless of the value of a. However, changing the value of a will affect the amplitude and phase of x3.

4. Can x3 have a period of 0?

No, x3 cannot have a period of 0. This is because x3 is a linear combination of x1 and x2, and the period of a linear combination is always the same as the period of the individual functions. Therefore, if x1 and x2 have a period of 0, x3 will also have a period of 0.

5. How does the phase difference between x1 and x2 affect the period of x3?

The phase difference between x1 and x2 does not affect the period of x3. This is because x3 is a linear combination of x1 and x2, and the period of a linear combination is always the same as the period of the individual functions. However, the phase difference will affect the shape and position of the graph of x3.

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