Z Transform

Discrete analog of Laplace Transform

$ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} $

    Where z is a complex variable.

Relationship between Z-Transform and F.T.

• $ X(\omega) = X(e^{j\omega}) $

• $ X(z)=X(re^{j\omega}) $

Then $ X(z) = F(x[n]r^-n) $

    $ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty x[n](re^{j\omega})^{-n} = \sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n} $