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Math Help - help me with inverse z transform

  1. #1
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    help me with inverse z transform

    Find the causal signal x(n) for the ff. z transform
    X(z) = \frac{0.5z}{z^2-z+0.5}
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  2. #2
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    Dividing top and bottom by z^{2} yields

    X(z)=\dfrac{0.5z^{-1}}{1-z^{-1}+0.5z^{-2}}.

    Comparing this expression to # 20 in this list, we see that we must have, if possible,

    0.5=a\sin(\omega_{0}),

    1=2a\cos(\omega_{0}), and

    0.5=a^{2}.

    You can make this work out. What do you get for a and \sin(\omega_{0}) and \cos(\omega_{0})? What is your radius of convergence?
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by TechnicianEngineer View Post
    Find the causal signal x(n) for the ff. z transform
    X(z) = \frac{0.5z}{z^2-z+0.5}
    The 'rexcursive relation' corresponding the the z-transform...

    \displaystyle X(z)= \frac{1}{2}\ \frac{z^{-1}}{1-z^{-1} + \frac{z^{-2}}{2}} (1)

    ... is...

    \displaystyle x(n)= x(n-1)- \frac{x(n-2)}{2} + \frac{\delta(n-1)}{2}\\,\\ x(-1)=0\\,\\ x(-2)=0 (2)

    ... that has to be solved...

    Kind regards

    \chi \sigma
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  4. #4
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    the region of convergence is |z| > 0.707106

    solving for a
    \sqrt(0.5) = a
    a = 0.707106

    solving for \omega_o
    \sin^{-1}(\frac{1}{2*0.707106}}) = \omega_o
    \omega_o = 45 degrees

    the inverse z transform is
    x[n] = (0.707107)^n \sin(\frac{\pi}{4}n)u[n]

    is this correct?
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  5. #5
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    You should check that the values you obtained check out with the third equation (three equations and two unknowns renders your system overdetermined, but the third equation must be satisfied). If it does, then I'd say you have the correct answer.
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