Poles & Zeros
Refers to the [see page 10, points] at which the numerator or denominator of the z-plane transfer function equals \(0\).
The transfer function can be rewritten as the ratio of two polynomials in terms of \( z \). \[ H[z] = \frac{P[z]}{Q[z]} \]
Given this definition anywhere we encounter:
| Condition | Consequence | Corresponds to frequencies where the filter function tends to: |
|---|---|---|
| \( P[z] = 0 \) | Zeros | Zero (I.E. Anti-Resonances) |
| \( Q[z] = 0 \) | Poles | Infinity (I.E. Resonances) |
We can visualise these tendencies on the z-plane at [see page 11, here].
Deriving a z-plane Transfer Function
Given the formulation above and the definition of poles and zeroes, we can now define the z-plane-transfer-function in terms of poles-and-zeros. The function must be 0 wherever there's a zero and must be infinity wherever there is a pole. See [see page 11, decaying] example.
From a pole/zero plot we can quickly a filter and find it's transfer function by using the equation:
\begin{align*} H[z] &=& \frac^{p}(z - z_i)}^{q}(z - z_i)} \\
&=& \frac{\prod \text{"distance of zeros from unit circle"}}{\prod \text{"distance of poles from unit circle"}} \\\\\end{align*}
The [see page 12, proximity] of a point to a zero or a pole on the z-plane circle influences it's magnitude at that point.
| Pole/Zero | Location | Magnitude |
|---|---|---|
| Zero | Close to zero | Small |
| Pole | Close to pole | Large |
| Zero | On the unit circle | Zero |
| Pole | On the unit circle | Infinity |
See [see page 12, filter-response] derivation.