Which method is commonly used for inverse Z-transform when the denominator degree is higher than the numerator?

• A. Partial fraction method
• B. Residue method
• C. Long division method
• D. State-space method

Answer: (C)

The long division method is commonly used for inverse Z-transform when the denominator degree is higher than the numerator. In this scenario, the expression can be rewritten as a power series which allows for easier manipulation in terms of finding the coefficients corresponding to each power of ( z^{-1} ). Specifically, when you perform long division, you systematically divide the numerator by the denominator, breaking down the terms until you can express the original function as an infinite series.

This method effectively converts a rational function into a form that can be expanded into a series, revealing the time-domain sequence directly related to the Z-transform. This is especially useful when the degree of the denominator exceeds that of the numerator because it assures that the resulting series converges and captures the sequence accurately.

In contrast, other methods like the partial fraction method and residue method are generally employed when dealing with polynomial fractions where the degrees are comparable or for specific types of poles in the Z-domain. The state-space method is a more general approach that does not specifically cater to the needs of inverse Z-transformation in cases of degree discrepancies. Thus, the long division method stands out as the optimal choice for scenarios where the numerator's degree does not match or exceed that of the denominator.