Signals and systems, determining if signals are periodic.

Discussion in 'Homework Help' started by chickwolf, Apr 6, 2016.

  1. chickwolf

    Thread Starter New Member

    Mar 9, 2014
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    Greetings everyone, studying some signals and systems at the moment and getting stumped by a part in one of the questions I was hoping you would be able to help me with. For the following question I am trying to determine whether or not it is periodic, and if it is, what its fundamental period is.

    I wasn't sure how to do use the AAC formula editor tool so have quickly knocked it up and attached it, apologies for the inconvenience.

    I have determined the fundamental period but am unsure how to progress to show if the signal if periodic or not mathematically. If anyone has any pointers as to what the next step may be I would be very grateful.
     
  2. Papabravo

    Expert

    Feb 24, 2006
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    The key to showing periodicity for either a continuous or discrete function is to demonstrate that the value of the function is the same for any constant offset equal to any integer multiple of the period. For example you can use the following identity to prove that a continuous sinewave is periodic,

    sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

    sin(x + 2n\pi) = sin(x)cos(2n\pi)+cos(x)sin(2n\pi) = sin(x)\cdot (1)+cos(x)\cdot (0)=sin(x)
     
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  3. chickwolf

    Thread Starter New Member

    Mar 9, 2014
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    So in terms of my questions, α = (2π/3)t and β = 3? Or have I got that completely wrong?
     
  4. Papabravo

    Expert

    Feb 24, 2006
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    Close. The argument to the sin function

    \text is \math\frac{2\pi}{3}\cdot(t+3)

    so

    \alpha=\frac{2\pi}{3}t,\;\text and\math\;\beta=\left(\frac{2\pi}{3}\cdot3\right)=2\pi
     
    Last edited: Apr 6, 2016
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  5. chickwolf

    Thread Starter New Member

    Mar 9, 2014
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    Awesome! Thank you very much, put that all in and got the starting expression back proving that it is indeed, periodic.
     
  6. Papabravo

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    Feb 24, 2006
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  7. dannyf

    Well-Known Member

    Sep 13, 2015
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    It is a sine function. How can it not be periodic?
     
  8. WBahn

    Moderator

    Mar 31, 2012
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    Because the sampling may make it a periodic.

    The same basic rule for periodicity applies to discrete-time systems:

    x[n + aP] = x[n]

    where P is the integer, in samples, and a is an integer.

    If the signal period is not rational, then no integer multiple of the sampling period will ever coincide with an integer multiple of the signal period and the sample stream will be aperiodic (remember, the samples are all that exist, the system neither knows nor cares what is going on between samples).
     
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