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| Both sides previous revision Previous revision Next revision | Previous revision | ||
| kb:z-transform [2021-05-05 01:02] – ↷ Page moved from kb:ee:z-transform to kb:z-transform jaeyoung | kb:z-transform [2024-04-30 04:03] (current) – external edit 127.0.0.1 | ||
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| The Z-transform converts a discrete-time signal into a complex frequency-domain representation. | The Z-transform converts a discrete-time signal into a complex frequency-domain representation. | ||
| - | It is the discrete-time equivalent of the [[kb:ee: | + | It is the discrete-time equivalent of the [[kb: |
| ===== Definition ===== | ===== Definition ===== | ||
| Line 19: | Line 19: | ||
| Usually, we will compute the inverse Z-transform by inspection, not by using the explicit formula. | Usually, we will compute the inverse Z-transform by inspection, not by using the explicit formula. | ||
| - | For a rational Z-domain transfer function, this can be done by partial fractions. Separate the fraction into multiple terms, each of which corresponds to a single pole. Then, each of these terms can be transformed to the time domain. Keep in mind that the time domain function depends on the [[kb:ee: | + | For a rational Z-domain transfer function, this can be done by partial fractions. Separate the fraction into multiple terms, each of which corresponds to a single pole. Then, each of these terms can be transformed to the time domain. Keep in mind that the time domain function depends on the [[kb: |
| ^ Z-domain representation ^ Region of convergence ^ Time-domain representation ^ | ^ Z-domain representation ^ Region of convergence ^ Time-domain representation ^ | ||
| | $H(z)=\frac{1}{z-p}$ | $|z| > p$ | $h[n]=p^{n-1}u[n-1]$ | | | $H(z)=\frac{1}{z-p}$ | $|z| > p$ | $h[n]=p^{n-1}u[n-1]$ | | ||
| | $H(z)=\frac{1}{z-p}$ | $|z| < p$ | $h[n]=-p^{n-1}u[-n]$ | | | $H(z)=\frac{1}{z-p}$ | $|z| < p$ | $h[n]=-p^{n-1}u[-n]$ | | ||