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Difference Equations

  1. Sep 4, 2006 #1

    I'm a bit rusty on difference eqns. Here's the problem:

    [tex] y_n -0.5 y_{n-1} = x_n [/tex]

    Here's what I can get out of it so far:

    [tex] y_1 -0.5 y_0 = x_1 [/tex]
    [tex] y_2 -0.5 y_1 = x_2 [/tex]
    [tex] y_3 -0.5 y_2 = x_3 [/tex]

    I just need some sense of direction, not a solution. It seems to me that this kind of problem requires that some boundary conditions be given, but that is not the case. That's all I have! Maybe it's something very simple I can't see right now.

    Any help is highly appreciated.
  2. jcsd
  3. Sep 4, 2006 #2


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    so you have a difference equation. what are you trying to do with it?

    are you trying to figure out what [itex]y_n[/itex] is? do you know what [itex]x_n[/itex] is?
  4. Sep 4, 2006 #3
    I assume the goal is to get [tex]y_n[/tex] since that was not explicitly stated. The directions are simply "solve the difference equation". That doesn't help much, does it? I do not have [tex]x_n[/tex]
  5. Sep 4, 2006 #4


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    Science Advisor

    It's already about as solved as it can get if you don't know the x's.
  6. Sep 4, 2006 #5
    Yes, but the answer wouldn't be that obviuous; it sounds like there is information missing, as rbj pointed out. I'll go talk to my instructor. As soon as I find out the answer or have another question, I'll get back to you guys. Thanks for all the help!
  7. Sep 4, 2006 #6
    Have you been learning about z-transforms?

    y[n] -> Y(z)
    y[n-1] -> Y(z)/z

    then solve for Y(z) = f(X(z)), then transform back to y[n] = f(x[n]) and you will have eliminated the y[n-1].
  8. Sep 5, 2006 #7
    Z-transforms really work on this kind of problem. I've found a book with a very easy-to-follow introduction. Thanks for the hint. Here is my answer:

    [tex]h[n] = -2^{-n}u[n-1][/tex]
  9. Sep 5, 2006 #8
    One thing to be careful about with z-transforms is that some functions, such as [tex] H(z) = \frac{1}{1-az^{-1}}[/tex], have multiple inverse transforms. Which to use depends on the region of convergence of H(z), which affects (or is defined by) the stability and causality of the system.
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