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 key idea here is that inverting a Z-transform often comes down to contour integration and residues. For the inverse Z-transform, the time-domain sequence x[n] is given by an integral around a closed loop in the region of convergence: x[n] = (1/2πj) ∮ X(z) z^{n-1} dz. By the residue theorem, this integral equals the sum of the residues of the integrand X(z) z^{n-1} at the poles of X(z) that lie inside the chosen contour. In other words, the entire time-domain sequence is built from the contributions of each pole, with each pole contributing a term whose form is an exponential in n (a geometric-type sequence), scaled by the corresponding residue.

This method is particularly convenient because it directly ties the shape of x[n] to the pole structure of X(z). For a causal sequence, the ROC is outside the outermost pole and the sum of residues inside the contour gives x[n] as a sum of terms like constant times (pole)^{n} for n ≥ 0. The exact coefficients come from the residues, so each pole’s influence is captured cleanly in one calculation.

So, evaluating residues at the poles of X(z) is the standard way to obtain the time-domain sequence from its Z-transform, because it uses the fundamental link between poles, residues, and the exponential terms that appear in x[n].