Poles and Zeros

Poles and Zeros are concepts, used in computing analog and digital signal filters. They are also important in other areas of engineering. Poles and Zeros describe the transfer function of linear system well enough for many practical purposes, such as finding the needed values for resistor and capacitors in the filter electronics.

Signal filter (can also be some other devices) has input and output, and transfers input to the output through some function. As filter contains elements with "memory" (inductors and capacitors), the output depends on how the input value has previously been changing over time. The direct analysis of these dependencies would be very complex, but the mathematical method known as Laplace transform can transform this function into another function that is more easy to understand.

Transfer function

In its simplest form for continuous-time input signal and output , the transfer function is the linear mapping of the Laplace transform of the input, , to the output :

where is the transfer function. If multiple filters are connected sequentially, the computed output Y(s) becomes the input of the second filter, and only needs to be multiplied by its own transfer function to compute the output of the whole chain. This can be used to improve the quality of the filtering.

The transfer function is a function of the complex number, so it can be viewed as a 2D surface, one horizontal axis representing the real and another the imaginary part of the number. The [1] contains very nice plots of these surfaces.

Poles and zeros

A simple lowpass RC filter
The 2D surface of the transfer function often contains "hills" that rise up till infinity. These infinite peaks are called poles. For instance, the transfer function of the simple lowpass RC filter is

.

This expression is infinite at the single point

This is the real part of the transfer function for this circuit. The imaginary part is zero. Hence the circuit has a single pole, located at the real (horizontal) axis. Sophisticated filters have more poles that often contain non zero imaginary part as well. However poles are normally symmetric over horizontal (real) axis. For instance, if the inductor would be connected sequentially to the resistor, such circuit would also be a filter, having two poles with non zero complex component, symmetrically placed on boths side of the real axis.

The pole of the transfer function, computed for the low pass RC filter with C = 50 μF and R = 10 kΩ

Places where the transfer function is equal to zero are similarly called zeros. See [2] for diagrams, how poles and zeros are placed in the most commonly used filters (Butterworth, Chebyshev, and Bessel). This reference also contains the actual wiring diagrams of the possible filters.

The applet (when active) allows to explore transfer functions with poles and zeros at various locations. The transfer function is defined in the graph at the top left where you can add and drag poles or zeros, the bottom part representing various characteristics of the obtained filter in the real time.

The filter analysis plots

• The most obvious plot for the filter is magnitude over frequency, the "usual" filter diagram. Experimenting with the applet you will see that the steepness of the cutoff improves with more poles that means more complex electronic design of the filter. Filter also changes phase, delaying the signal, and this delay depends on the frequency. The second (bottom) plot in the Bode set shows how the phase shift depends on the frequency. This pair, default for the applet, is called Bode plots.
• A Nyquist plot is is represented by a graph in polar coordinates in which the gain and phase of a frequency response are plotted. It shows the phase as the angle and the magnitude as the distance from the origin. This plot combines the both parts of Bode plot — magnitude and phase — on a single graph, with frequency as a parameter along the curve. See[3] for more on this plot.
• In the Nichols plot the logarithm of the magnitude is plotted against the phase of a frequency response on orthogonal axes. Same as Nyquist plot, this plot combines both types of Bode plot — magnitude and phase — on a single graph, with frequency as a parameter along the curve.
• The step response plot shows how the filter would respond when the input level rapidly jumps up. It is an important reaction to explore as it reflects the system stability to the sudden changes of the signal. A function with the single "step" over time is called Heaviside function.

References

Acknowledgements

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