Nuhertz

Pole Zero Plots


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Features

Filter Solutions pole zero plots have zoom in, zoom out, and screen drag capability using the mouse keys and cursor.

Poles and zeros may be moved by the left mouse key and cursor. Poles and zeros may be added or deleted with the right mouse key and Cursor. All pass poles and zeros may be added by right clicking the mouse while simultaneously depressing the left mouse key.


General Information

Pole Zero plots allow you to gather information about the filter you have designed. Poles are defined as the complex frequencies that makes the overall gain of the filter transfer function infinite. Zeros are are complex frequencies that make the overall gain of the transfer function zero. When the pole zero plot button is selected, Filter Solutions draws the complex plane and depicts each pole as an "X" and each zero as an "O". Left click the mouse to display the exact location of any selected pole or zero.

 

Continuous Domain plots

In the continuous domain, or S domain, filter zeros are placed up and down the imaginary axis and tell you what frequency to expect the transmission zeros to occur in your filter. These zero frequencies in the pole zero plot should match the transmission zero frequencies on the frequency trace, and should match the LC tank frequency displayed on the circuit if you have a circuits license.  Alternatively, poles and zeros may be displayed on polar coordinates, with Wo as the circular axis and Q as the radial

The poles locations offer other information about the filter. If you are familiar with filter theory, they will help you to validate the filters generated by Filter Solutions. In general, the poles located close to the imaginary axis produce a real time ringing response, and poles placed near the real axis produce a real time exponential response. Poles placed to the right side of the imaginary axis produce an unstable response. For a filter to be stable, all poles MUST be left of the imaginary axis.

This example is from a 5th order low pass Elliptic filter. The zeros are clearly seen at the correct frequencies along the imaginary axis. All poles are placed properly on the left side of the imaginary axis.

 

Modifying Pole Zero Plots

Poles and zeros on or added to the horizontal or vertical axis always stay on the horizontal or vertical axis when moved with the cursor. Moving a complex poles and zero automatically updates the location of the conjugate pole or zero. Moving a pole or zero in an all pass automatically updates the location of all other poles and zeros in the all pass.

Continuous Pole Zero Plot in Cartesian Coordinates
Continuous Pole Zero Plot in Cartesian Coordinates
Continuous Pole Zero Plot in Polar Coordinates
Continuous Pole Zero Plot in Polar Coordinates

Continuous Pole Zero Plot in Squared Up Polar Coordinates

 

Discrete Domain Plots

In the discrete domain, or Z domain, what was the imaginary axis of the continuous domain now becomes a unit circle. The continuous frequency of 0+J0 maps to the discrete frequency 1+J0. The frequency of -1+J0 represents the aliasing frequency. Stable systems contain all poles inside the unit circle. Transmission zeros are generally around the side of the unit circle, but may be off the circle due to imperfect frequency mapping or quantization error.

For a filter to be stable, all poles MUST be inside the unit circle. Large order filters have poles that are VERY close the edge of the circle. It takes only slight quantization error to bump the pole outside the unit circle. This is one cause of large order filters going unstable when they are implemented into a target with finite precision arithmetic. In general, the more complex filter forms, such as the Cascade and Parallel forms, cause the poles to wander less than the simple, Standard form. This is one reason the complex forms produce more stable filters.

The following examples are from the same low pass 5th order Elliptic filter digitized with a Bilinear transformation used in the continuous example. The second picture is merely a close-up view of the right side poles and zeros. It is seen that the transmission zeros are on the unit circle, a transmission zero is at continuous infinity (-1+J0 in the discrete domain for Bilinear transformations), and the poles are all inside the unit circle. Note that two poles are very close to the edge of the unit circle. Small arithmetic execution errors may push these poles outside the circle producing an unstable filter.

 

Modifying Pole Zero Plots

Poles and zeros on or added to the horizontal axis always stay on the horizontal axis when moved withe cursor. Poles and zeros on or added to the unit circle always stay on the unit circle when moved with the cursor. Moving a complex poles and zero automatically updates the location of the conjugate pole or zero. Moving a pole or zero in an all pass automatically updates the location of all other poles and zeros in the all pass.

Discrete Pole Zero Plot
Discrete Pole Zero Plot

Discrete Pole Zero Plot
Discrete Pole Zero Plot, Close-up View

 

All Pass Poles and Zeros

Right clicking the mouse while depressing the left mouse key will add an all pass to the pole zero plane. This useful when it is necessary to adjust the phase of the filter without effecting the magnitude.Continuous all passes have zeros equal and opposite the vertical axis from the poles. Discrete all passes have inversly proportional poles and zeros.First order all passes are have one pole and one zero on the horizontal axis. Second order all passes have one pole conjugate pair and one zero conjugate pair.

Continuous All Pass Discrete All PAss

Second Order All Pass Example

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