The given function is:

$G(z) = \frac{(1 - 9z)^2}{z^2 + 1}$

Let's expand and simplify:

1. Expand the numerator:

$(1 - 9z)^2 = 1 - 18z + 81z^2$

2. The function now looks like:

$G(z) = \frac{1 - 18z + 81z^2}{z^2 + 1}$

3. The function is already in its simplest form. However, if needed, you can analyze it further, such as finding poles, zeros, or simplifying for specific purposes.

Would you like more details on a specific aspect, like finding poles, zeros, partial fractions, or perhaps plotting this function?

Related Questions:

1. How do you find the poles of a rational function?
2. What are the zeros of $G(z)$ and how do you calculate them?
3. How can this function be represented using partial fraction decomposition?
4. What is the behavior of this function as $z$ approaches infinity?
5. How would this function behave if used in a discrete-time system analysis?

Tip:

For stability analysis in control systems, the poles of the transfer function (values of $z$ that make the denominator zero) are critical.