Z-Transform Definition

The Z-transform is the discrete-time counterpart of the Laplace transform. It converts sequences defined at integer time indices into functions of a complex variable z, enabling algebraic analysis of discrete-time systems. Understanding the Z-transform is essential for digital signal processing, digital filter design, and the GATE signals and systems syllabus.

Core Concept Explanation

The Z-transform of a discrete-time sequence x[n] is defined as X(z) = sum from n = -infinity to +infinity of x[n] times z^(-n), where z is a complex variable. This is the bilateral or two-sided Z-transform. For causal sequences where x[n] = 0 for n less than 0, the sum starts from n = 0, giving the unilateral Z-transform. Most practical causal system analysis uses the unilateral form.

The variable z can be written in polar form as z = r * e^(j*Omega), where r is the magnitude and Omega is the discrete-time angular frequency. When r equals 1, the Z-transform reduces to the Discrete-Time Fourier Transform (DTFT), evaluated on the unit circle. This is the direct analog of evaluating the Laplace transform on the imaginary axis to get the continuous-time Fourier transform.

The Region of Convergence (ROC) is the set of values of z for which the sum defining X(z) converges. The ROC must always be specified along with X(z) because different sequences can have the same rational expression for X(z) but different ROCs, corresponding to different time-domain signals. For a right-sided (causal) sequence, the ROC is the exterior of a circle. For a left-sided sequence, the ROC is the interior of a circle.

Mathematical Expression

The relationship between the Z-transform and the Laplace transform is established through sampling. If x(t) is sampled at rate 1/T to give x[n] = x(nT), then substituting z = e^(sT) into the Z-transform gives the sampled Laplace transform. This mapping sends the left half s-plane to the interior of the unit circle in the z-plane, the right half s-plane to the exterior, and the imaginary axis to the unit circle itself.

Common Z-transform pairs that are essential for GATE include: the unit impulse delta[n] transforming to 1 with ROC as all z; the unit step u[n] transforming to z/(z-1) with ROC |z| greater than 1; and the exponential sequence a^n u[n] transforming to z/(z-a) with ROC |z| greater than |a|. The inverse Z-transform recovers x[n] from X(z) using partial fractions similar to the Laplace inverse method.

Practical Understanding

Digital filters, such as IIR and FIR filters implemented on DSP processors, are described by difference equations. Taking the Z-transform of the difference equation converts it to an algebraic equation in z, giving the system function H(z). The poles and zeros of H(z) in the z-plane determine the filter's frequency response. Poles near the unit circle at a particular angle create a peak in the frequency response at that discrete frequency.

For a stable discrete-time system, all poles of H(z) must lie strictly inside the unit circle, since this corresponds to poles in the left half s-plane under the mapping z = e^(sT). A pole exactly on the unit circle gives sustained oscillation, and a pole outside gives exponential growth. This is the discrete-time stability criterion analogous to requiring LHP poles in continuous time.

Given:
x[n] = (0.5)^n * u[n]
a = 0.5 (causal sequence)

Why this formula applies:
This is a geometric series with ratio z^(-1)*a, converges for |z| > |a|

Formula:
X(z) = sum_{n=0}^{inf} a^n * z^{-n} = sum_{n=0}^{inf} (a/z)^n = 1/(1 - a*z^{-1})

Substitution:
a = 0.5
X(z) = 1/(1 - 0.5*z^{-1}) = z/(z - 0.5)

Calculation:
ROC: |z| > 0.5  (exterior of circle of radius 0.5)
Pole at z = 0.5 (inside unit circle → stable system)
Zero at z = 0

Final Answer with units:
X(z) = z/(z - 0.5),  ROC: |z| > 0.5
System stable since pole |0.5| < 1 (inside unit circle)

Exam Tip: For GATE, the ROC of a causal sequence is always the exterior of a circle: |z| > r. For an anti-causal (left-sided) sequence, ROC is the interior: |z| < r. If the ROC includes the unit circle, the DTFT exists. Always state the ROC when writing a Z-transform; X(z) without ROC is incomplete.

Quick Revision

• Z-transform: X(z) = sum x[n] z^(-n). Bilateral for general sequences, unilateral for causal.
• Key pairs: delta[n] ↔ 1; u[n] ↔ z/(z-1), |z|>1; a^n u[n] ↔ z/(z-a), |z|>|a|.
• ROC is mandatory: causal → |z| > r; anti-causal → |z| < r; two-sided → annular ring.
• Relation to Laplace: z = e^(sT). LHP in s ↔ inside unit circle in z.
• DTFT = X(z) evaluated on unit circle (|z|=1), valid only if ROC includes unit circle.
• Stability of discrete-time system: all poles of H(z) must satisfy |pole| < 1.
• Common trap: forgetting to specify ROC; different ROCs for same X(z) give completely different x[n].