Determining periodic/aperiodic signals

[ga14]

How do you determine if a function is periodic or not without graphing it?

For example, cos^2(2pit) is periodic, as is sin^3(2t). But e^(-2t)cos(2pit) is nonperiodic, as is the discrete signal x[n]=cos(2n).

Does this have to do with Fourier series somehow? Is there an easier way to determine it? Thanks for any help.

[KalTorak]

Sure, Fourier series would be one way to look at it. A periodic function has a Fourier sequence (not series, since it's only a countably infinite number of Fourier coefficients) with zero coefficients everywhere BUT at a fundamental frequency f_0, and harmonics 2f_0, 3f_0, ... (It could be that there are some k for which kf_0 is zero, but all the nonzero coefficients HAVE to be at integer multiples of the fundamental.)

if it doesn't behave that way, it's not periodic.

[ga14]

I'm not really knowledgable about Fourier series yet, is there another way to look at it?

[DrPizza]

Without graphing it....

quickest way is: experience, and knowing the general shapes of "common" functions

[kevinthenerd]

Originally posted by: DrPizza
Without graphing it....

quickest way is: experience, and knowing the general shapes of "common" functions

Agreed. Trigonometric functions usually do (unless they have messed-up arguments).

[kevinthenerd]

Originally posted by: ga14
How do you determine if a function is periodic or not without graphing it?

For example, cos^2(2pit) is periodic, as is sin^3(2t). But e^(-2t)cos(2pit) is nonperiodic, as is the discrete signal x[n]=cos(2n).

Does this have to do with Fourier series somehow? Is there an easier way to determine it? Thanks for any help.

I just came up with a little something that you might find interesting. I thought about it while I was working on some math (before my quiz tomorrow).

In the case y = a sin b (where a and b are functions of x, not just constants), the function y is periodic if the derivative of both of the functions is equal to zero and a constant, respectively. In your aforementioned example, the derivative of 2pit with respect to t would indeed be a constant (2pi), but, using the chain rule, the derivative of e^(-2t) is -2e^(-2t), which has obvious fluctuations, no matter how many times you take the derivative. (In fact, each time you take the derivative, you would be messing up your function by a factor of 2 each time.)


...so that's one way to tell if a trigonometric function is periodic, I suppose.

[kevinthenerd]

and elliptic functions are doubly periodic:

http://mathworld.wolfram.com/EllipticFunction.html


General info here:
http://mathworld.wolfram.com/PeriodicFunction.html

Edit: 2:30 A.M..... Back to studying math 8-)