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<center><math>X(z) = \sum^{\infty}_{n = -\infty} x[n]z^{-n}\!</math></center> | <center><math>X(z) = \sum^{\infty}_{n = -\infty} x[n]z^{-n}\!</math></center> | ||
| − | where <math>z\!</math> is a complex variable. This is sometimes denoted as <math>X(z) = Z(x[n])\!</math> | + | where <math>z\!</math> is a complex variable. This is sometimes denoted as <math>X(z) = Z(x[n])\!</math>. |
| + | |||
| + | == Relationship between Z-Transform and Fourier Transform == | ||
| + | Fourier | ||
| + | <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})\! $
