| Line 8: | Line 8: | ||
== Relationship between Z-Transform and Fourier Transform == | == Relationship between Z-Transform and Fourier Transform == | ||
Fourier | Fourier | ||
| + | |||
<math>X(\omega) = X(e^{j\omega})\!</math> | <math>X(\omega) = X(e^{j\omega})\!</math> | ||
Revision as of 16:25, 3 December 2008
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])\! $.
Relationship between Z-Transform and Fourier Transform
Fourier
$ X(\omega) = X(e^{j\omega})\! $
