Z-Transform

Why it matters

The Z-Transform is crucial for analyzing and designing discrete-time control systems and digital signal processing applications. It allows engineers to work with complex signals and systems in the frequency domain, making it easier to understand system behavior and stability.

Key ideas

• Definition: The Z-Transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation.
• Region of Convergence (ROC): The set of values in the complex plane for which the Z-Transform converges. The ROC is essential for determining the stability and causality of a system.
• Properties: The Z-Transform has several properties such as linearity, time-shifting, scaling in the z-domain, and convolution, which simplify the analysis of discrete-time systems.
• Inverse Z-Transform: Used to convert back from the frequency domain to the time domain, allowing for the reconstruction of the original signal.

Formulas

• Z-Transform: X(z) = Σ (x[n] * z^(-n)) for n = -∞ to ∞
  • X(z): Z-Transform of the signal
  • x[n]: Discrete-time signal
  • z: Complex frequency variable
• Inverse Z-Transform: x[n] = (1/2πj) ∮ X(z) * z^(n-1) dz
  • x[n]: Original discrete-time signal
  • X(z): Z-Transform of the signal
  • z: Complex frequency variable

For inversion, choose a counterclockwise closed contour encircling the origin inside the ROC. The algebraic expression alone generally does not uniquely identify a sequence. For an LTI impulse response, BIBO stability requires absolute summability; for ordinary rational systems this corresponds to an ROC containing the unit circle.

Worked example

Given a discrete-time signal x[n] = (0.5)^n u[n], find its Z-Transform.

1. Identify the signal: x[n] = (0.5)^n u[n], where u[n] is the unit step function.
2. Apply the Z-Transform formula: X(z) = Σ (0.5)^n * z^(-n) for n = 0 to ∞
3. Simplify the series: This is a geometric series with a common ratio r = 0.5/z.
4. Use the geometric series formula: X(z) = 1 / (1 - 0.5/z)
5. Simplify: X(z) = z / (z - 0.5)
6. Determine the ROC: |z| > 0.5

Final Answer: X(z) = z / (z - 0.5), ROC: |z| > 0.5

Common mistakes

• Confusing the Z-Transform with the Laplace Transform, which is used for continuous-time signals.
• Incorrectly determining the Region of Convergence, leading to wrong conclusions about system stability.
• Forgetting to apply the unit step function u[n] when calculating the Z-Transform of causal signals.

For GATE EE

Questions on the Z-Transform often involve finding the Z-Transform of a given sequence, determining the ROC, and using properties of the Z-Transform to simplify expressions. Practice problems involving inverse Z-Transforms and stability analysis of discrete-time systems.

Quick check

1. What is the Z-Transform of x[n] = δ[n]?
2. How does the ROC affect system stability?
3. What is the inverse Z-Transform of X(z) = z / (z - 0.5)?

Answers: 1. X(z) = 1; 2. The ROC must include the unit circle for stability; 3. With |z| > 0.5, x[n] = (0.5)^n u[n]; with |z| < 0.5, x[n] = −(0.5)^n u[−n−1]