Periodicity of the summation of two functions

In summary: To answer the question, you get 0 = 0, which is technically true but doesn't seem to lead anywhere.In summary, the function x(t) = cos(t) + sin(\sqrt{2}*t) is not periodic. This can be shown by checking the ratio of the periods, which yields a rational number. Additionally, we can also show that the second derivative of the function is not periodic, which further supports the conclusion that the function is not periodic.
  • #1
Jncik
103
0

Homework Statement



show whether or not [tex]x(t) = cos(t) + sin(\sqrt{2}*t)[/tex]

is periodic

The Attempt at a Solution



the first period

is

T1 = [tex] 2\pi[/tex]

the second period is

T2 = [tex]\frac{2\pi}{\sqrt{2}}[/tex]

how can I determine whether it is periodic or not?

in my book(about signals and systems) it doesn't explain how to check this out
 
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  • #2
Hi Jncik! :smile:

If a function is periodic, then

[tex]f(t)=f(t+T)[/tex]

But if a function is periodic, then so is it's first and second derivative!, So

[tex]f^{\prime\prime}(t)=f^{\prime\prime}(t+T)[/tex]

What is the second derivative of our function here?? And do we get something nice?
 
  • #3
I get -cos(t) - 2*sin(sqrt(2)t)

I don't see how this helps me...

can we say that because T2/T1 is rational, then it's periodic?
 
  • #4
Jncik said:
I get -cos(t) - 2*sin(sqrt(2)t)

I don't see how this helps me...

can we say that because T2/T1 is rational, then it's periodic?

So we have

[tex]\cos(t)+\sin(\sqrt{2}t)=\cos(t+T)+\sin(\sqrt{2}(t+T))[/tex]

and

[tex]-\cos(t)-2\sin(\sqrt{2}t)=-\cos(t+T)-\sin(\sqrt{2}(t+T))[/tex]

What happens if you add both equations?
 
  • #5
they become -sin(sqrt(2)t) = 0

and I get t = pi*n/sqrt(2) where n is integer
 
  • #6
The way I've always approached this problem is by finding the respective Periods, checking the ratio, and if it yields a rational number than the signal is periodic.

Micromass - is your method a more elegant way (or simply a calculus based approach given where this question was posted...) to show that the 'n' found in these last steps, in order to make the period rational, must be irrational?
 
  • #7
Jncik said:
they become -sin(sqrt(2)t) = 0

and I get t = pi*n/sqrt(2) where n is integer

No, you end up with

[tex]\sin(\sqrt{2}t)=\sin(\sqrt{2}(t+T))[/tex]

and this must hold for all possible t! What are the only values of T that satisfy this?

Ecthelion said:
The way I've always approached this problem is by finding the respective Periods, checking the ratio, and if it yields a rational number than the signal is periodic.

Micromass - is your method a more elegant way (or simply a calculus based approach given where this question was posted...) to show that the 'n' found in these last steps, in order to make the period rational, must be irrational?

Yes, you are correct that you just need to check if the ratio of the periods is rational. So I'm trying to prove that it isn't periodic. I'm just trying to explain here where it comes from that you can just check the ratio of the periods...
 
  • #8
for sqrt(2) * T = k*pi ?

hence T = k*pi/(sqrt(2))?

as for the ratio, is it T1/T2 or T2/T1?
 
  • #9
micromass said:
So we have

[tex]\cos(t)+\sin(\sqrt{2}t)=\cos(t+T)+\sin(\sqrt{2}(t+T))[/tex]

and

[tex]-\cos(t)-2\sin(\sqrt{2}t)=-\cos(t+T)-\sin(\sqrt{2}(t+T))[/tex]

What happens if you add both equations?
Small but important typo in the second equation (the one for the second derivative.)

Should be: [tex]-\cos(t)-2\sin(\sqrt{2}t)=-\cos(t+T)-\underset{\uparrow}{2}\sin(\sqrt{2}(t+T))[/tex]
 

Related to Periodicity of the summation of two functions

1. What is the definition of periodicity of the summation of two functions?

The periodicity of the summation of two functions refers to the property of the combined function to repeat itself at regular intervals. In other words, the combined function will have the same values at certain points, known as periods, after a fixed interval of time or distance.

2. How is periodicity of the summation of two functions calculated?

The periodicity of the summation of two functions is calculated by finding the least common multiple (LCM) of the individual functions' periods. The LCM is the smallest positive value that both periods can divide into evenly, and it represents the period of the combined function.

3. Can the periodicity of the summation of two functions be infinite?

Yes, the periodicity of the summation of two functions can be infinite if the individual functions have infinite periods. In this case, the combined function will also have an infinite period and will repeat itself indefinitely.

4. What happens when the periods of the two functions are not whole numbers?

If the periods of the two functions are not whole numbers, the periodicity of the summation of the two functions can still be calculated. The periods will be expressed in terms of fractions, and the LCM of these fractions will be the period of the combined function.

5. How does the amplitude of the individual functions affect the periodicity of the summation?

The amplitude of the individual functions does not affect the periodicity of the summation. The combined function will still have the same period as calculated using the LCM of the individual periods, regardless of the amplitudes of the functions.

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