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== The Z-Transform == | == 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, <math>y[n]\!</math>, of a DT LTI system is <math>y[n] = H(z)z^n\!</math>, where <math>H(z) = \sum^{\infty}_{n = -\infty} h[n]z^{-n}\!</math>. When <math>z = e^{j\omega}\!</math> with <math>\omega\!</math> real, this summation equals the Fourier Transform of <math>h[n]\!</math>. When <math>z\!</math> is not restricted to this value, the summation is know as the Z-Transform of <math>h[n]\!</math>. To be exact, | 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, <math>y[n]\!</math>, of a DT LTI system is <math>y[n] = H(z)z^n\!</math>, where <math>H(z) = \sum^{\infty}_{n = -\infty} h[n]z^{-n}\!</math>. When <math>z = e^{j\omega}\!</math> with <math>\omega\!</math> real, this summation equals the Fourier Transform of <math>h[n]\!</math>. When <math>z\!</math> is not restricted to this value, the summation is know as the Z-Transform of <math>h[n]\!</math>. To be exact, | ||
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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> | ||
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| + | where <math>z\!</math> is a complex variable. This is sometimes denoted as <math>X(z) = Z(x[n])\!</math> | ||
Revision as of 16:23, 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])\! $
