Inverse Z-Transform

This process reconstructs a discrete time signal from its frequency domain equation. Digital control systems depend on it to convert calculated responses back into physical timing commands.

Core Concept

You need the inverse transform to see how a system behaves over discrete time steps. A complex algebraic expression in the z domain does not tell you the actual sample values directly.

Several mathematical methods exist to perform this inversion. The partial fraction expansion method works best for rational functions with distinct roots. You break a complex fraction into simple terms that match standard transform pairs.

Power series expansion offers another valid approach. You divide the numerator by the denominator to generate a sequence of coefficients representing the time domain samples. This method shines when you only need the first few values.

Key Formula

The formal definition involves a contour integral in the complex plane along a closed path.

x[n] = (1 / 2 pi j) * integral( X(z) * z^(n-1) dz ). The integral path must lie entirely within the region of convergence. Students mostly use partial fractions instead of direct integration.

Problem: Find inverse transform of X(z) = z / (z^2 - 3z + 2) for right-sided sequence.\nGiven: X(z) / z = 1 / ((z-1)(z-2)).\nSteps:\nFind partial fractions for X(z)/z: A/(z-1) + B/(z-2).\nSolve for A: A = 1 / (1-2) = -1.\nSolve for B: B = 1 / (2-1) = 1.\nWrite expanded form: X(z)/z = -1/(z-1) + 1/(z-2).\nMultiply back by z: X(z) = -z/(z-1) + z/(z-2).\nApply standard pairs: x[n] = (-1^n + 2^n) u[n].\nFinal Answer: x[n] = (2^n - 1) u[n].

Exam Tip: Always divide X(z) by z before performing the partial fraction expansion. This leaves a z in the numerator when you multiply it back, perfectly matching standard transform pair formulas.

Properties Summary

• Contour integration provides the strict mathematical definition.
• Partial fraction expansion requires a strictly proper rational function.
• Long division yields values for specific time indices directly.
• Residue theorem solves the contour integral algebraically.
• The region of convergence dictates whether sequences are right-sided or left-sided.

Quick Revision

• Inversion moves from complex plane back to discrete time.
• Standard transform pairs are your primary tool.
• Keep track of the region of convergence carefully.
• Right-sided regions correspond to causal sequences.
• Left-sided regions correspond to anti-causal sequences.
• Exam trap: Forgetting to check the region of convergence before writing the final time sequence.