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I am reading a paper on frequency warping and I need to do a little manipulation of the Z-transform. Can somebody help me on how can I go about deriving equations $(3)$ and $(4)$ from equations $(1)$ and $(2)$.

$$ \tag{1} \bar{A}(z) = \sum_{k=0}^{p}\bar{a}_k\bar{z}^{-k}(z), \ \ \ \ \ \ \bar{a}_0=1 $$

$$ \tag{2} \bar{z}^{-1} = \dfrac{(1-a^2)z^{-1}}{1-az^{-1}}-a $$

putting $(2)$ in $(1)$ and using binomial formulas:

$$ \tag{3} \bar{A}(z) = \sum_{k=0}^{p}b_k\tilde{z}^{-k}(1-az^{-1})^{-k} $$

$$ \tag{4} b_k=\sum_{n=k}^{p}C_{kn}\bar{a}_n, \ \ \ \ \ \ C_{kn}= {n \choose k}(1-a^2)^k(-a)^{n-k} $$

Thanks in advance!

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    $\begingroup$ Every time someone is citing part of some paper it is highly appreciated to include a link to it... $\endgroup$ – jojek May 31 '14 at 18:14

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