In the inverse Z-transform, which method is commonly used to compute the time-domain sequence by evaluating residues at the poles of X(z)?

In the inverse Z-transform, which method is commonly used to compute the time-domain sequence by evaluating residues at the poles of X(z)?

• A. Substituting z = e^{jω}
• B. Taking the limit as z → ∞
• C. Taking the inverse Z-transform term-by-term
• D. Evaluating residues at the poles of X(z)

Answer: (D)

The time-domain sequence from an inverse Z-transform is found most directly by using the residue theorem from complex analysis. When you form the inverse Z-transform integral, x[n] = 1/(2πj) ∮ X(z) z^{n-1} dz, you evaluate it by summing the residues at the poles of the integrand inside the chosen contour (the poles of X(z) inside the ROC). Each pole p_k contributes a term that, for simple poles, looks like Res{ X(z), z = p_k } times p_k^{n-1}. Altogether, x[n] becomes a sum of exponential terms tied to the pole locations (and, for complex-conjugate poles, yields damped sinusoidal components). This method directly links the pole structure of X(z) to the time-domain behavior of the sequence.

Other approaches don’t place the computation on the residue path. Substituting z = e^{jω} relates X(z) to the frequency-domain DTFT on the unit circle and doesn’t automatically produce the time-domain sequence via residues. Taking the inverse transform term-by-term or taking a limit as z → ∞ are useful techniques in other contexts, but they don’t leverage the residue contributions from the poles to build x[n] as cleanly and generally as the contour/residue method.