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Difference equation

  1. Jun 10, 2007 #1
    This is some math from "Discrete-Time Signal Processing" by Oppenheim:

    We have the homogeneous difference equation:

    [tex] \sum_{k=0}^N a_k y_h [n-k] = 0 [/tex]

    "The sequence [tex] y_h[n] [/tex] is in fact a member of a family of solutions of the form:

    [tex] y_h[n] = \sum_{m=1}^ N A_m z^n_m [/tex]"

    So what is Oppenheim saying here? I suppose it is something like the solutions to differential equations. Right? But what is the z? I suppose it is related to the z transform, right?

    He then says: Substituting the second equation for the first shows that the complex numbers [tex] z_m [/tex] must be roots of the polynomial:

    [tex] \sum_{k=0}^N a_k z^{-k} = 0 [/tex]

    Could some one explain this to me?

    Thanks!
     
  2. jcsd
  3. Jun 10, 2007 #2
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