Z-Plane Transfer Function
Is a [see page 7, reordering] of the z-transform function to find z-domain-transfer-function which characterises the behaviour of the filter in terms of it's output and input when \(H[Z]\) is equivalent to the fourier-transform.
\[ H[z] = \frac{Y[z]}{X[z]} \]
| Term | Meaning |
|---|---|
| \( H[z] \) | The z-domain transfer function |
| \( Y[z] \) | The output of a linear time domain filter |
| \( X[z] \) | The input to the filter to get \( Y[z] \) |
See [see page 8, common Z-transform pairs], I.E. The affect of different transfer-functions on a signal.
| \( H(z) \) | Name | Description |
|---|---|---|
| \( {z}^{0} = 1 \) | Unit Impulse | \( \delta[n] \) is \( 1 \) only when \( n = 0 \) |
| \( {z}^{-m} \) | Shifted Impulse | We've shiften \( n \) from Unit Impulse |
| \( \frac{1}{1 - {z}^{-1}} \) | Unit Step | |
| \( \frac{1}{1 - a {z}^{-1}} \) | Decaying Exponential |
Note: \[ \delta[n] = \begin{cases} 1, & \text{if $n=0$} \\ 0, & \text{otherwise}\end{cases} \]
Warn: The decaying exponential has \(a^n\) because \(n\) is decreasing, it's still decaying.