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 01-19-2004, 10:29 AM #1 ga14 Member   Join Date: Nov 2002 Posts: 42 Determining periodic/aperiodic signals 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.
 01-19-2004, 02:31 PM #2 KalTorak Member   Join Date: Jun 2001 Posts: 55 RE:Determining periodic/aperiodic signals 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.
 01-19-2004, 02:36 PM #3 ga14 Member   Join Date: Nov 2002 Posts: 42 RE:Determining periodic/aperiodic signals I'm not really knowledgable about Fourier series yet, is there another way to look at it?
 01-21-2004, 04:11 PM #4 DrPizza AdministratorElite MemberGoat Whisperer     Join Date: Mar 2001 Location: Western NY Posts: 47,510 RE:Determining periodic/aperiodic signals Without graphing it.... quickest way is: experience, and knowing the general shapes of "common" functions __________________ Fainting Goats
01-23-2004, 10:44 PM   #5
kevinthenerd
Platinum Member

Join Date: Jun 2002
Posts: 2,914
RE:Determining periodic/aperiodic signals

Quote:
 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).
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KLD

02-06-2004, 02:20 AM   #6
kevinthenerd
Platinum Member

Join Date: Jun 2002
Posts: 2,914
RE:Determining periodic/aperiodic signals

Quote:
 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.
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KLD

 02-06-2004, 02:27 AM #7 kevinthenerd Platinum Member   Join Date: Jun 2002 Posts: 2,914 RE: Determining periodic/aperiodic signals 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-) __________________ KLD