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I have a finite linear difference equation

$$y(n)=ax(n-1)+bx(n-2)+cx(n-3)+\ldots+fx(n-m)\text,$$

relating an input $x(n)$ to an output $y(n)$. If I assume periodicity of type $x(n-2)=x(n)$, the right hand side reduces to a summation over $x(n-1)$ and $x(n-2)$ (where $x(n-2)=x(n)$) and the collective coefficients say $A$ and $B$:

$$y(n)=AX(n-1)+Bx(n)$$

($y(n)$ then also has period $2$.)

Applying the $\mathcal Z$-transform to this and exploring the zeros (poles just at $z=0$) I would like to make comments on the coefficients predicting that a period 2 system is possible.

How can I do this regarding zeros inside/outside the unit circle ?

Gilles
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Michelle
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    Could you clarify this sentence: "I would like to make comments on the coefficients predicting that a period 2 system is possible"? I'm not sure I understand what it exactly is that you want to achieve. – Matt L. Dec 13 '17 at 15:37
  • Experimentally I observed that if the zeros (poles of the inverse transfer function) are all outside the unit circle, then the coefficients A and B support a period two input and output. For a different set a,b,c, and hence different A and B, where zeros are not all outside, the input and output do not have a period two pattern. I have tested this for other periodic scenarios and zeros of G(z) (or poles of 1/G(z)) all outside the unit circle for the new arrangement of coefficients confirmed that periodic prediction - how can I explain this? – Michelle Dec 13 '17 at 15:49
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    If the input is periodic with period 2 then the output must be periodic with period 2 as well (apart from transients). I think I still don't exactly understand what the question is. What is given and what would you like to compute? And what is a "period 2 system"? – Matt L. Dec 13 '17 at 16:26
  • Hi @Michelle, I just had to add a few empty lines to mark paragraphs, and a few $ to mark formulas, to make your question readable. Please feel encouraged to give future posts more structure! Furthermore, I also don't understand the question. "making comments" is not really something I know how to help you with. – Marcus Müller Dec 13 '17 at 16:30
  • rephrase - Is it possible (and are there criteria) that the impulse response (the coefficients of the difference equation) of a BIBO system has a part to play in creating periodic tendencies (not neccessarlly related to that of the input) observed in an output ? – Michelle Dec 13 '17 at 16:35
  • I have reasons to believe that one can predict the likelyness of a certain periodicity to be present in an output by evaluating the coefficients of the difference equation with no knowledge of the input except that it is bounded. – Michelle Dec 13 '17 at 17:04
  • You can have (decaying) transients with a certain frequency, but not with a non-recursive first order system as in your question. You would need a second order recursive system with poles away from the origin. Take a look at this answer to a related question. – Matt L. Dec 13 '17 at 17:10
  • I understand now - it answers my question! – Michelle Dec 13 '17 at 17:19
  • We don't :( ${}$ – Matt L. Dec 13 '17 at 17:21
  • Yes it helped me :) go in piece! – Michelle Dec 13 '17 at 17:22
  • Sorry, that comment reminded me of this silly song. https://www.youtube.com/watch?v=_vPWE2Ebz48 – Peter K. Dec 14 '17 at 13:14
  • @Michelle Please post your own answer to your question when the system lets you! It may help someone in the future (and you can use it for reference). – Peter K. Dec 14 '17 at 13:15

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