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Difference Equations

  1. Dec 11, 2007 #1
    1. The problem statement, all variables and given/known data
    For the discrete-time system:


    Find the impulse response.

    2. Relevant equations

    3. The attempt at a solution

    Alright so I started like this:




    Now this is where I'm stuck. I know I need to get the equation for [tex]h_0[k][/tex], but I don't know how. The equation they got is:


    Can anyone tell me how they got there?
  2. jcsd
  3. Dec 11, 2007 #2
    It is the homogeneous solution of the difference equation:

    You can take the characteristic equation which is a quadratic:

    [tex]m^2 + \frac{1}{6}m-\frac{1}{6}=0[/tex]

    and then take the roots. You will find the roots to be -1/2 and 1/3. The
    [tex]C_1 [/tex] and [tex]C_1 [/tex] are constants made necessary because the ambiguity in the solution (same as differential equations). The answer is then just the roots taken to the power of k. k is just the value in a sequence.
    Last edited: Dec 11, 2007
  4. Dec 11, 2007 #3
    Thank you so much, this clears everything up for me!
  5. Dec 11, 2007 #4
    you are welcome
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