Bilateral Z-Transform

Introduction

The bilateral Z-Transform is the most general form of the Z-Transform, summing over all integer indices from $-\infty$ to $+\infty$. While the unilateral version handles causal signals and initial-value problems, the bilateral form is essential when dealing with signals that extend in both directions along the time axis. Non-causal filters, theoretical analysis of signal properties, and rigorous stability theory all require the bilateral formulation.

Think of it this way: if a signal has energy or information content at negative time indices, the unilateral transform simply ignores that portion. The bilateral transform captures the complete picture, making it indispensable for theoretical signal processing and for understanding how the Region of Convergence truly determines signal characteristics.

Formal Definition

The bilateral Z-Transform of a discrete-time sequence $x[n]$ is defined as:

$$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$

This infinite sum converges only for specific values of the complex variable $z$. The set of all $z$ values for which this sum converges absolutely forms the Region of Convergence (ROC). Unlike the unilateral version, the bilateral transform can produce an annular ROC — a ring-shaped region in the z-plane.

Decomposition into Causal and Anti-Causal Parts

Any two-sided sequence can be decomposed into a right-sided (causal) part and a left-sided (anti-causal) part:

$$x[n] = x[n]u[n] + x[n]u[-n-1] = x_R[n] + x_L[n]$$

The Z-Transform of each part converges in a different region:

• Right-sided part $x_R[n]$: converges for $|z| > r_1$ (outside a circle)
• Left-sided part $x_L[n]$: converges for $|z| < r_2$ (inside a circle)

The bilateral Z-Transform converges only in the intersection of these two regions, which forms an annulus: $r_1 < |z| < r_2$.

Worked Example: Two-Sided Exponential Sequence

Consider the sequence:

$$x[n] = (0.5)^n u[n] - (2)^n u[-n-1]$$

Right-sided part: $(0.5)^n u[n]$ has Z-Transform $\frac{z}{z-0.5}$ with ROC $|z| > 0.5$

Left-sided part: $-(2)^n u[-n-1]$ has Z-Transform $\frac{z}{z-2}$ with ROC $|z| < 2$

By linearity, the bilateral Z-Transform is:

$$X(z) = \frac{z}{z-0.5} + \frac{z}{z-2}$$

The ROC is the intersection of the individual ROCs: $0.5 < |z| < 2$

This is an annular region bounded by the pole at $z = 0.5$ on the inside and the pole at $z = 2$ on the outside. Since the unit circle ($|z| = 1$) lies within this annulus, the DTFT of this signal exists.

The Annular ROC: Properties and Interpretation

The annular nature of the bilateral ROC has several important implications:

Existence condition: For the bilateral Z-Transform to exist, the inner radius must be strictly less than the outer radius ($r_1 < r_2$). If the right-sided part has poles at larger radii than the left-sided part's poles, the annulus collapses and the transform does not exist.

DTFT existence: The Discrete-Time Fourier Transform exists if and only if the unit circle ($|z| = 1$) lies within the annular ROC. This provides a simple geometric test for whether a two-sided signal has a well-defined frequency spectrum.

Multiple valid ROCs: The same algebraic expression for $X(z)$ can correspond to different sequences depending on which annular region is chosen as the ROC. Consider:

$$X(z) = \frac{z}{(z-0.5)(z-2)}$$

This expression has three possible ROCs:

• $|z| > 2$: both parts are causal → $x[n] = [2^n - (0.5)^n] \cdot (\text{some scaling}) \cdot u[n]$
• $|z| < 0.5$: both parts are anti-causal
• $0.5 < |z| < 2$: right-sided from pole at 0.5, left-sided from pole at 2

Each ROC yields a completely different time-domain sequence.

Convergence Analysis

For a general two-sided sequence, absolute convergence requires:

$$\sum_{n=-\infty}^{\infty} |x[n]| \cdot |z|^{-n} < \infty$$

Splitting into positive and negative index sums:

$$\sum_{n=0}^{\infty} |x[n]| \cdot |z|^{-n} + \sum_{n=1}^{\infty} |x[-n]| \cdot |z|^{n} < \infty$$

The first sum converges for $|z|$ sufficiently large (outside some circle). The second sum converges for $|z|$ sufficiently small (inside some circle). The bilateral ROC exists only when these two constraints are simultaneously satisfiable.

Example: Signal with No Bilateral Z-Transform

Consider $x[n] = 3^{|n|}$. The right-sided part $(3^n u[n])$ requires $|z| > 3$ for convergence. The left-sided part $(3^{-n} u[-n-1])$ requires $|z| < 1/3$. Since $3 > 1/3$, there is no overlap — the annulus is empty. Therefore, this signal has no bilateral Z-Transform and no DTFT.

Relationship to DTFT and Frequency Response

When the unit circle lies within the annular ROC, evaluating $X(z)$ on the unit circle gives the DTFT:

$$X(e^{j\Omega}) = X(z)\big|_{z=e^{j\Omega}} = \sum_{n=-\infty}^{\infty} x[n] e^{-j\Omega n}$$

This connects the z-domain analysis to frequency-domain characterization. For system analysis, if the transfer function $H(z)$ has its ROC containing the unit circle, then $H(e^{j\Omega})$ gives the system's frequency response.

Comparison with Unilateral Z-Transform

Applications in System Theory

The bilateral Z-Transform is essential in several theoretical contexts:

1. Stability analysis: Determining whether all poles of $H(z)$ have the unit circle in their ROC
2. Non-causal filter design: Wiener filters and other optimal filters that are inherently two-sided
3. Signal classification: Distinguishing between finite-energy and infinite-energy signals based on ROC properties
4. Spectral factorization: Decomposing power spectra into causal and anti-causal components

Key Takeaways

• The bilateral Z-Transform sums from $-\infty$ to $+\infty$, handling complete two-sided sequences
• The ROC is an annulus $r_1 < |z| < r_2$, bounded by poles on each side
• The DTFT exists only if the unit circle lies within the annular ROC
• The same algebraic $X(z)$ can correspond to entirely different sequences depending on the ROC
• If the annulus collapses (inner radius exceeds outer radius), the transform does not exist
• The bilateral form is necessary for theoretical completeness; the unilateral suffices for most practical implementations