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Closed-form expression

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  1. Feb 12, 2017 #1
    • Thread moved from the technical Mth forums, so no Homework Template is shown
    1. The problem statement, all variables and given/known data

    The assignment is to find a closed-form expression for the FIR least squares inverse filter of length N for each of the following systens
    2. Relevant equations

    $$
    1.G( z ) = \frac{1}{1 - \alpha z^{-1}}; | \alpha | < 1 \\
    2. G(z) = 1 - z^{-1} \\
    3. G(z) = \frac{\alpha - z^{-1}}{1 - \alpha z^{-1}}; |\alpha| < 1
    $$
    3. The attempt at a solution


    Anybody have any ideas, I can't really understand what is meant by close-form either from the book or from wikipedia. My guess is:
    $$
    1. G( z ) = \frac{z}{z - \alpha} \\
    2. G( z ) = z-1 \\
    3. G( z ) = \frac{z - 1 }{z - \alpha}
    $$
    Every helping hand is welcome
     
    Last edited: Feb 12, 2017
  2. jcsd
  3. Feb 13, 2017 #2

    haruspex

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    Your attempted solution does not seem to bear any relationship to that clause. Not an area I know anything about, but it is clear that you are not merely being asked to present G(z) in a closed form (those expressions already are).
    Closed form means an equation of the form function = (some combination of standard functions).
    That is, the right hand side cannot contain any references back to the function being expressed, nor integrals, nor sums, nor any special functions defined for the purpose. There are some grey areas.
     
  4. Feb 13, 2017 #3
    Yes, you are right. I realized that I have misunderstood the quetion, I'm supposed to first tale the inverse of $G(z)$
    $$
    1. G^{-1}(z)=\frac{1}{G(z)}=1-\alpha z^{-1} \\
    2. G^{-1}(z)=\frac{1}{1-z^{-1}} \\
    3. G^{-1}(z)=\frac{1-\alpha z^{-1}}{\alpha-z^{-1}}
    $$
    Now, I only need to make the inverse z-transform
    $$
    1. g(n) = -\alpha \delta(n-1) \\
    2. g(n) = -u(n-1) \\
    3.
    $$
    I'm not sure about the third
     
    Last edited: Feb 13, 2017
  5. Feb 13, 2017 #4

    haruspex

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    Expand as constant+constant/(α-z-1)?
     
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