Z-Transform

Why it matters

The Z-Transform is a powerful mathematical tool used in the analysis and design of discrete-time control systems and digital signal processing. It helps in understanding system behavior and stability, making it essential for engineers working with digital systems.

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. It is crucial for determining the stability and causality of systems.
• Poles and Zeros: The values of z that make the Z-Transform go to infinity (poles) or zero (zeros) are important for system analysis.
• Inverse Z-Transform: Used to convert back from the frequency domain to the time domain.
• Properties: Linearity, time-shifting, scaling in the z-domain, convolution, and initial value theorem are some key properties.

Formulas

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

Worked example

Given: A discrete-time signal x[n] = (0.5)^n u[n], where u[n] is the unit step function.

1. Find the Z-Transform:

   • Formula: X(z) = Σ (x[n] · z^(-n))
   • Substitute: X(z) = Σ ((0.5)^n · z^(-n)) for n = 0 to ∞
   • This is a geometric series with a = 1 and r = 0.5/z
   • Sum of series: X(z) = 1 / (1 - 0.5/z)
   • Simplify: X(z) = z / (z - 0.5)

2. Determine the ROC:

   • For convergence, |0.5/z| < 1
   • Therefore, |z| > 0.5

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

The same rational expression with ROC |z| < 0.5 corresponds to −(0.5)^n u[−n−1]. The ROC is therefore essential for unique bilateral inversion. The inverse contour must lie within the ROC and encircle the origin.

Common mistakes

• Confusing the ROC with the region of stability.
• Incorrectly applying the properties of the Z-Transform, such as time-shifting.
• Forgetting to check the ROC when determining system stability.

For GATE EC

Questions often involve finding the Z-Transform of a given sequence, determining the ROC, and analyzing system stability. Practice problems on inverse Z-Transform and properties like convolution in the z-domain.

Quick check

1. What is the Z-Transform of δ[n], the unit impulse function?
2. How does the ROC affect system stability?
3. What is the inverse Z-transform of X(z) = z/(z − 0.5) with ROC |z| > 0.5?

Answers: 1. 1, 2. The ROC must include the unit circle for stability, 3. x[n] = (0.5)^n u[n]