Lecture

Region of Convergence and Stability

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture covers the concept of Region of Convergence (ROC) in signal processing, explaining how to determine the ROC for different types of signals and transfer functions. It also delves into the stability criterion for systems, clarifying the relationship between ROC and system stability.

Official source

Ontological neighbourhood

Information engineering

Control theory and engineering: Control theory

Related concepts (32)

Nyquist stability criterion

In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker at Siemens in 1930 and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, is a graphical technique for determining the stability of a dynamical system.

BIBO stability

In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for signals and systems that take inputs. If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded. A signal is bounded if there is a finite value such that the signal magnitude never exceeds , that is For discrete-time signals: For continuous-time signals: For a continuous time linear time-invariant (LTI) system, the condition for BIBO stability is that the impulse response, , be absolutely integrable, i.

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

Transfer function

In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models the system's output for each possible input. They are widely used in electronic engineering tools like circuit simulators and control systems. In some simple cases, this function can be represented as two-dimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve.

Related lectures (64)

Nyquist Introduction

Introduces the Nyquist criterion to determine system stability and covers BIBO stability and stability conditions.

System Identification and Stability

Explores system identification, stability criteria, and challenges in stabilizing cameras on moving platforms.

Stability in Feedback Systems

Explores BIBO stability in feedback systems, covering criteria, Nyquist criterion, and system sampling impact.

LTI Systems: Analysis and Properties

Covers the analysis and properties of Linear Time-Invariant (LTI) systems.

Signals & Systems II: Difference Equations and Inverse Operators