This tutorial discusses pole-zero analysis of digital filters. Every digital filter can be specified by its poles and zeros (plus a gain factor). Poles and zeros give useful insights into a filter's response, and can be used as the basis for digital filter design. Since it is quite difficult to qualitatively analyse the Laplace transform and Z-transform of a system, as mappings of their magnitude and phase or real part and imaginary part result in multiple mappings of 2-dimensional surfaces in 3-dimensional space so we commonly examine plots of a transfer function's poles and zeros to try to gain a qualitative idea of what a system does. Once the Z-transform of a system has been determined, we can use the information contained in the polynomials to plot the function and more easily observe its defining characteristics. The Z-transform may be defined as:

The two polynomials, P (z) and Q(z), allow us to find the poles and zeros of the Z-Transform.

1. The value(s) for z where P (z) = 0.

2. The complex frequencies that make the overall gain of the filter transfer function zero.

1. The value(s) for z where Q(z) = 0.

2. The complex frequencies that make the overall gain of the filter transfer function infinite.

Below is a simple transfer function:

Find the poles and zeroes of this function.

2. The Z-plane 

Once the poles and zeroes have been found for a given Z-transform, they can be plotted onto the Z-plane. The Z-plane is a complex plane with an imaginary and real axis referring to the complex-valued variable, z. The position on the complex plane is given by re^iω and the angle from the positive, real axis around the plane is denoted by ω. When mapping poles and zeroes onto the plane, poles are denoted by an “x” and zeroes by an “o”. The figure below shows the Z-plane and zeroes onto the plane.

Figure 2. The circle represents the unit circle.  Poles and zeroes plotted onto the complex plane for the transfer function:

Example 2:

Plot the poles and zeroes of the transfer function given in Example 1 onto the complex plane.

Example 3:

The poles and zeroes of a complex transfer function are somewhat more difficult to solve. Find the poles and zeroes of the following complex transfer functions, and plot them on the Z-plane. Remember: complex solutions will plot along the imaginary axis.

3. Frequency response and the Z-plane 

The frequency response may be written in terms of the system poles and zeros by substituting iω for z directly into the factored form of the transfer function. Based on the location of the poles and zeroes, we can quickly determine the frequency response. This is a result of the correspondence between the frequency response and the transfer function evaluated on the unit circle in the pole/zero plots. The frequency response of the system is defined as:

factoring the transfer function into poles and zeroes and multiplying both the numerator and denominator by  e^iω gives:

Now we have the frequency response in a form that can be used to interpret the physical characteristics of the filters frequency response. The numerator and denominator contain a term of the form (e^iω -h), where h is either a zero, denoted by  c_k or a pole by d_k. As previously stated, the pole or zero represents a vector from the origin of the Z-plane to its location within the plane,  e^iω represents a vector from the origin of the Z-plane to its location on the unit circle. Thus, the vector (e^iω - h) connects the pole or zero to a place on the unit circle and will allow us to plot the frequency response of the system. From this, we can deduce that;

The reason it is helpful to understand and create pole/zero plots in terms of frequency response is due to their ability to help us easily design a filter. This is particularly useful in seismic processing as we often wish to recover the true earth response and remove any filtering effects that may be caused by the seismometer.

Example 4.