Z-Plane
A representation of the plane of the z-transform in both real and imaginary parts as a [see page 9, circle].
You can visualise the z-plane as a unit-circle. We can place Poles & Zeros around the circle and use them to define a filter function. The displacement of the pole/zero from the origin determines how wide the filter is (the range of frequencies at that point it lets through or blocks) and rotation around it determines the frequency-response (the position of the band) of the filter.
For example a pole at these locations has the following affects:
| Location | Angle \( \Omega \) | Affect |
|---|---|---|
| (1, 0) Far right | \( 2 \pi = 0 \) | Low pass filter |
| (1, 1) Top | \( \frac{\pi}{2} \) | Band pass filter |
| (-1, 0) Far left | \( \pi = \frac{f_s}{2} \) | High pass filter |
Observe that because it's a pole there's an increase in amplitude at these points which is why it's a pass filter. If it was sink it would minimise the magnitude at these points.
Frequencies in Play
We define \( f_s \) as the sampling-frequency which by the nyquist-theorem must be twice as big as the largest frequency we can encounter. This means that the top half of the circle is sufficient to hold the smallest to largest frequencies we can measure.
Note: The bottom half of the circle exists to hold the negative frequencies we may encounter.
We define \(f\) as the technical-frequency. It's the frequency of some point in the unit circle between the range of \( -\frac{f_s}{2} \leq f \leq \frac{f_s}{2} \).
The angular-frequency is the angle of a frequency \(f\) along the unit circle.
We define \( \Omega \) as the normalised-frequency. It's the angular rotation of the current frequency (angular-frequency) scaled by the sampling frequency. Essentially it's just the frequency matching the angle between a point (pole/sink) and the origin in the unit circle.