The Z-Transform
Similar to the Laplace Transform, the Z-Transform is an extension of the Fourier Transform, in this case the DT Fourier Transform. As previously defined, the response, $ y[n]\! $, of a DT LTI system is $ y[n] = H(z)z^n\! $, where $ H(z) = \sum^{\infty}_{n = -\infty} h[n]z^{-n}\! $. When $ z = e^{j\omega}\! $ with $ \omega\! $ real, this summation equals the Fourier Transform of $ h[n]\! $. When $ z\! $ is not restricted to this value, the summation is know as the Z-Transform of $ h[n]\! $. To be exact,
where $ z\! $ is a complex variable. This is sometimes denoted as $ X(z) = Z(x[n])\! $
