Discrete-time Systems and Pole-Zero Analysis

Discrete-time Systems

• The Z-transform is a powerful tool for analyzing discrete-time systems, which are systems that process discrete-time signals.
• Discrete-time systems can be represented by difference equations, which describe the relationship between input and output signals.
• The Z-transform is used to convert difference equations into algebraic equations, making it easier to analyze and design discrete-time systems.

Pole-Zero Analysis

• Pole-zero analysis is a method of analyzing the frequency response of a discrete-time system using the Z-transform.
• Poles:
  • Defined as the values of z that make the transfer function infinite.
  • Represent the resonant frequencies of the system.
  • A system with poles close to the unit circle will have a large response to certain frequencies.
• Zeros:
  • Defined as the values of z that make the transfer function zero.
  • Represent the frequencies that are completely attenuated by the system.
• Stability:
  • A system is stable if all poles are inside the unit circle.
  • A system is unstable if any pole is outside the unit circle.
• Frequency Response:
  • The Z-transform can be used to find the frequency response of a discrete-time system.
  • The frequency response is a plot of the magnitude and phase of the system's transfer function versus frequency.