Final answer:
The student's question relates to Mathematics, specifically to the partial fraction expansion of a z-transform, a technique used in signal processing and control systems. It requires the separation of a complex fraction into simpler terms.
Step-by-step explanation:
The student's question involves conducting a partial fraction expansion of the given z-transform, H(z). We need to express the complex fraction as a sum of simpler fractions, which will allow us to invert the transform more easily in other contexts such as solving difference equations or analyzing signal processing systems.
To perform a partial fraction expansion, we assume H(z) = A/(1-0.5z) + B/(1-0.9z) + C/(1-2z) and find constants A, B, and C that satisfy this equation. The original expression for H(z) must be equated to this assumed form, and through various algebraic manipulations, including potentially clearing denominators and equating coefficients, we can solve for A, B, and C.
It's important to note transformation techniques such as z-transforms and partial fractions are fundamental tools in digital signal processing and control system analysis, understanding them allows students to analyze and synthesize discrete-time systems.