# Periodicity of the summation of two functions

1. Jul 7, 2011

### Jncik

1. The problem statement, all variables and given/known data

show whether or not $$x(t) = cos(t) + sin(\sqrt{2}*t)$$

is periodic

3. The attempt at a solution

the first period

is

T1 = $$2\pi$$

the second period is

T2 = $$\frac{2\pi}{\sqrt{2}}$$

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

2. Jul 7, 2011

### micromass

Hi Jncik!

If a function is periodic, then

$$f(t)=f(t+T)$$

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

$$f^{\prime\prime}(t)=f^{\prime\prime}(t+T)$$

What is the second derivative of our function here?? And do we get something nice?

3. Jul 7, 2011

### Jncik

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. Jul 7, 2011

### micromass

So we have

$$\cos(t)+\sin(\sqrt{2}t)=\cos(t+T)+\sin(\sqrt{2}(t+T))$$

and

$$-\cos(t)-2\sin(\sqrt{2}t)=-\cos(t+T)-\sin(\sqrt{2}(t+T))$$

What happens if you add both equations?

5. Jul 7, 2011

### Jncik

they become -sin(sqrt(2)t) = 0

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

6. Jul 7, 2011

### Ecthelion

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. Jul 7, 2011

### micromass

No, you end up with

$$\sin(\sqrt{2}t)=\sin(\sqrt{2}(t+T))$$

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

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. Jul 7, 2011

### Jncik

for sqrt(2) * T = k*pi ?

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

as for the ratio, is it T1/T2 or T2/T1?

9. Jul 7, 2011

### SammyS

Staff Emeritus
Small but important typo in the second equation (the one for the second derivative.)

Should be: $$-\cos(t)-2\sin(\sqrt{2}t)=-\cos(t+T)-\underset{\uparrow}{2}\sin(\sqrt{2}(t+T))$$