There is a simple graphical relationship between the
z-plane and the frequency response of a filter.
For example, if the transfer function of a system is:
$$H(z)=\frac1{1 - 2r \cos(\theta)z^{-1} + r^2 z^{-2}}$$
with \(r=0.97\) and \(\theta=\frac23 \pi\).
Multiply top and bottom by \(z^2\) to get:
$$H(z)=\frac{z^2}{z^2 - 2r \cos(\theta)z + r^2}$$
What are the poles and zeros of this \(H(z)\)?
Will \(r\) and \(\theta\) be the radius and angle of a pole?
Note: the transfer function above can be written with the symbolic toolbox in
MATLAB:
The two poles have a magnitude of 0.97 and
their angles are +120 degrees and -120 degrees.
Figure 1 is a plot of the pole locations and zero locations in the
complex Z-plane.
Figure 1. Pole-Zero plot of \(H(z)=\frac1{1+0.97z^{-1}+0.9409z^{-2}}\)
Figure 2. 3D surface plot of \(H(z)=\frac1{1+0.97z^{-1}+0.9409z^{-2}}\)
Are the poles and zeros where they should be, based on factoring the
polynomials?
Figure 2 is a 3D plot of \(H(z)\) over the entire complex Z-plane.
You can see the two peaks caused by the poles and the
valley in between formed by the zeros at \(z=0\).
The frequency response is is found by evaluating
\(H(z)\) along the contour defined by
\(z\) equal \(e^{j\hat\omega}\).
In Figure 1, the unit circle is given for reference;
the two poles lie just inside the unit circle.
Figure 2 shows a blue line that traces out the unit circle.
The frequency response is the curve you get if you ride a bike along
the unit circle and plot your altitute. Figure 3 is the frequency
response of \(H(z)\).
Figure 3. Frequency Response plot of
\(H(z)=\frac1{1+0.97z^{-1}+0.9409z^{-2}}\)
Figure 4 is a movie showing how the frequency response is found by tracing
around the unit circle.
Figure 4. Movie
Here is another movie to help you visualize what is happening