Determining periodic/aperiodic signals

Discussion in 'Highly Technical' started by ga14, Jan 19, 2004.

  1. ga14

    ga14 Member

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    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.
     
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  3. KalTorak

    KalTorak Member

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    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.
     
  4. ga14

    ga14 Member

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

    DrPizza Administrator Elite Member Goat Whisperer
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    Without graphing it....

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

     
  6. kevinthenerd

    kevinthenerd Platinum Member

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

    kevinthenerd Platinum Member

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    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.
     
  8. kevinthenerd

    kevinthenerd Platinum Member

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