Inverse Z-Transform

Introduction

The inverse Z-Transform recovers the discrete-time sequence $x[n]$ from its z-domain representation $X(z)$. In practice, you will rarely need to evaluate the formal contour integral definition — instead, three practical methods handle virtually all cases you encounter in engineering: partial fraction expansion, long division (power series), and the residue method. Of these, partial fraction expansion is by far the most commonly used for rational Z-Transforms, while long division serves as both an alternative method and a verification tool.

Understanding inverse Z-Transform techniques is essential because system analysis typically proceeds as: formulate difference equation → transform to z-domain → solve algebraically for $Y(z)$ → invert to get $y[n]$. The final step requires the methods described here.

Method 1: Partial Fraction Expansion

For rational Z-Transforms (ratio of polynomials in $z$), partial fraction expansion is the standard approach. The key technique is to work with $X(z)/z$ rather than $X(z)$ directly, because the standard transform pair $\frac{z}{z-a} \leftrightarrow a^n u[n]$ has $z$ in the numerator.

Systematic Procedure

Step 1: Form the ratio $\frac{X(z)}{z}$

Step 2: Decompose $\frac{X(z)}{z}$ into partial fractions: $$\frac{X(z)}{z} = \frac{A_1}{z-p_1} + \frac{A_2}{z-p_2} + \cdots$$

Step 3: Multiply both sides by $z$: $$X(z) = \frac{A_1 z}{z-p_1} + \frac{A_2 z}{z-p_2} + \cdots$$

Step 4: Invert term by term using the standard pair:

• If ROC is $|z| > |p_i|$ (causal): $\frac{A_i z}{z - p_i} \leftrightarrow A_i p_i^n u[n]$
• If ROC is $|z| < |p_i|$ (anti-causal): $\frac{A_i z}{z - p_i} \leftrightarrow -A_i p_i^n u[-n-1]$

Detailed Example

Find $x[n]$ for $X(z) = \frac{2z^2 - z}{z^2 - 3z + 2}$ with ROC: $|z| > 2$.

Step 1: Factor the denominator: $z^2 - 3z + 2 = (z-1)(z-2)$

Step 2: Form $X(z)/z$: $$\frac{X(z)}{z} = \frac{2z-1}{(z-1)(z-2)}$$

Step 3: Partial fractions using the cover-up method: $$A = \left.\frac{2z-1}{z-2}\right|_{z=1} = \frac{2-1}{1-2} = \frac{1}{-1} = -1$$

$$B = \left.\frac{2z-1}{z-1}\right|_{z=2} = \frac{4-1}{2-1} = \frac{3}{1} = 3$$

Step 4: Reconstruct and invert: $$X(z) = \frac{-z}{z-1} + \frac{3z}{z-2}$$

Since ROC is $|z| > 2$ (outside both poles), both terms are causal: $$x[n] = [-(1)^n + 3(2)^n]u[n] = (3 \cdot 2^n - 1)u[n]$$

Verification: $x[0] = 3-1 = 2$. Using the initial value theorem: $\lim_{z\to\infty}X(z) = \lim_{z\to\infty}\frac{2z^2-z}{z^2-3z+2} = 2$ ✓

Handling Repeated Poles

When $X(z)/z$ has a repeated pole at $z = p$ of order $m$:

$$\frac{X(z)}{z} = \frac{A_1}{z-p} + \frac{A_2}{(z-p)^2} + \cdots + \frac{A_m}{(z-p)^m} + \text{other terms}$$

The coefficients are found by: $$A_k = \frac{1}{(m-k)!}\frac{d^{m-k}}{dz^{m-k}}\left[(z-p)^m \frac{X(z)}{z}\right]_{z=p}$$

The inverse of a second-order pole term $\frac{z}{(z-a)^2}$ gives $n \cdot a^{n-1} u[n]$.

Method 2: Long Division (Power Series Expansion)

Long division provides the sequence values directly as coefficients in a power series in $z^{-1}$:

$$X(z) = x[0] + x[1]z^{-1} + x[2]z^{-2} + x[3]z^{-3} + \cdots$$

Procedure

Arrange $X(z)$ as a ratio of polynomials in $z^{-1}$, then perform polynomial long division of the numerator by the denominator.

Example

Find the first few values of $x[n]$ for $X(z) = \frac{1}{1-0.5z^{-1}}$, ROC: $|z| > 0.5$.

Perform the division: $$\frac{1}{1-0.5z^{-1}} = 1 + 0.5z^{-1} + 0.25z^{-2} + 0.125z^{-3} + \cdots$$

Reading off coefficients: $x[0] = 1, \; x[1] = 0.5, \; x[2] = 0.25, \; x[3] = 0.125$

Pattern: $x[n] = (0.5)^n u[n]$ ✓

When Long Division is Preferred

• Verifying results from partial fractions
• Computing initial sequence values quickly without full inversion
• Handling non-rational transforms where partial fractions do not apply
• Computer implementation (polynomial division is algorithmic)

For anti-causal signals (ROC inside a circle), divide in powers of $z$ (positive powers) to obtain $x[-1], x[-2], \ldots$

Method 3: Contour Integration (Residue Method)

The formal inverse Z-Transform is defined by the contour integral:

$$x[n] = \frac{1}{2\pi j}\oint_C X(z) z^{n-1} dz$$

where $C$ is a counterclockwise contour within the ROC enclosing the origin.

Using the residue theorem: $$x[n] = \sum_k \text{Res}\left[X(z)z^{n-1}, z = p_k\right]$$

where the sum is over all poles enclosed by the contour.

For simple poles: $\text{Res}[F(z), z=p] = \lim_{z\to p}(z-p)F(z)$

This method is mathematically elegant but computationally equivalent to partial fractions for rational transforms. It becomes useful primarily for complex cases or when proving general results.

Critical Role of the ROC

The ROC determines which inverse applies to a given algebraic expression. Consider:

$$X(z) = \frac{z}{z-a}$$

Case 1 — ROC: $|z| > |a|$ (right-sided/causal): $$x[n] = a^n u[n]$$

Case 2 — ROC: $|z| < |a|$ (left-sided/anti-causal): $$x[n] = -a^n u[-n-1]$$

The same formula yields completely different sequences! In practice, if the problem states "causal system" or specifies ROC outside the outermost pole, use Case 1. If the ROC is inside a pole, use Case 2.

Complex Conjugate Poles

When poles occur in conjugate pairs $p = re^{j\theta}$ and $p^* = re^{-j\theta}$, the combined contribution gives a damped sinusoid:

$$x[n] = 2|A|r^n \cos(\theta n + \angle A) \cdot u[n]$$

where $A$ is the residue at $p$. This produces oscillatory behavior with frequency determined by $\theta$ and decay rate determined by $r$.

Key Takeaways

• Partial fractions via $X(z)/z$ is the standard and most efficient method for rational transforms
• Long division gives sequence values directly as power series coefficients — excellent for verification
• The cover-up method efficiently finds partial fraction coefficients for simple poles
• For causal signals (ROC outside poles): poles give $p_i^n u[n]$ terms
• For anti-causal signals (ROC inside poles): poles give $-p_i^n u[-n-1]$ terms
• Repeated poles produce terms like $n \cdot a^n u[n]$
• Complex conjugate poles produce damped discrete sinusoids
• Always specify or verify the ROC before performing the inversion