Z Transform
  • One-sided z-transform

$X(z) = \Sigma_{n=0}^{\infty} x(n) z^{-n}$

  • Two-sided z-transform

$X(z) = \Sigma_{n=-\infty}^{\infty} x(n) z^{-n}$

  • Give a sequence, the set of values of z for which the z-transform converges, i.e., $|X(z)|< \infty$, is called the region of convergence.
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$|X(z)| \leq \Sigma_{n=-\infty}^{-1} |x(n)r^{-n}| + \Sigma_{n=0}^{\infty}|\frac{x(n)}{r^n}|$
$|X(z)| \leq \Sigma_{n=1}^{\infty} |x(-n)r^{n}| + \Sigma_{n=0}^{\infty}|\frac{x(n)}{r^n}|$
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