Name: testing system stability with convolution
Uploaded: 2024-09-26T13:16:56.000Z
Duration: 1 min 24 s
Channel: JoVE

The stability of systems can be ascertained using convolution.
If an input signal has a constant that ensures that the signal&#39;s modulus never exceeds it at any point in time, it is considered bounded.
A system is bounded-input bounded-output, also known as BIBO stable if any given bounded input signal invariably produces a bounded output. The figure illustrates examples of BIBO-stable and non-BIBO-stable systems.
The convolution integral can be used to assess BIBO stability when a bounded continuous-time input is applied to a Linear Time-Invariant system.
The boundedness of the input signal may be represented through a constant, and the convolution integral formulations determine the bounded output.
If the integrand is finite, the output is finite, meaning a continuous-time system with an absolutely integrable impulse response is BIBO stable.
This same process applies to discrete-time systems; a bounded output is achieved through the convolution integral.
The output is finite if the summation term has a finite value, indicating that a discrete-time system is BIBO stable if its impulse response is absolutely summable.

bibo stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system. The boundedness of the input signal is represented by a constant, and the convolution integral helps determine whether the output remains bounded. Mathematically, this means that if the integrand of the convolution integral is finite, the output will also be finite. Specifically, a continuous-time system is BIBO stable if its impulse response is integrable, meaning the integral of the absolute value of the impulse response is finite and is represented as,
<img alt="Integral equation for signal processing, &#8747;|h(&#964;)|d&#964;&#60;&#8734;, stability condition, mathematical concept." src="/files/ftp_upload/16053/16053_PT%20EQUATION_1.svg" style="text-align: center; height: 36px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
This condition ensures that the output remains within bounds for any bounded input signal, thus confirming the system's stability.
The same principle applies to discrete-time systems. BIBO stability in discrete-time systems is determined through the summation of the convolution series. For a discrete-time system, the output is finite if the summation term has a finite value, indicating that the system is BIBO stable if its impulse response is summable as given in the expression below.
<img alt="Static equilibrium concept, equation &#931;|h[k]|&#60;&#8734; diagram, illustrating mathematical convergence." src="/files/ftp_upload/16053/16053_PT%20EQUATION_2.svg" style="text-align: center; height: 28px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
In other words, if the sum of the absolute values of the impulse response is finite, the system will produce a bounded output for any bounded input, confirming BIBO stability.
The importance of BIBO stability lies in its application to real-world systems, where ensuring that outputs remain within acceptable limits in response to bounded inputs is crucial. Understanding and applying the concepts of convolution and impulse response integrals are vital for designing and analyzing stable systems in both continuous and discrete-time domains.

BIBO stability of continuous and discrete -time systems

The stability of systems can be ascertained using convolution.
If an input signal has a constant that ensures that the signal&#39;s modulus never exceeds ...

bibo-stability-of-continuous-and-discrete-time-systems

Education

JoVE Core

Electrical Engineering

Unit 1

Linear Time- Invariant Systems

KEY TERMS AND CONCEPTS

testing system stability with convolution

System stability in signal processing is often tested with convolution. A system is BIBO stable, or bounded-input bounded-output stable, when every bounded input signal produces a bounded output signal. A bounded input signal has a modulus that never exceeds a set constant at any time.

For a continuous-time Linear Time-Invariant, or LTI, system, BIBO stability is checked with the convolution integral. The input is bounded by a constant, and the convolution integral shows whether the output stays bounded too. If the integrand in the convolution integral remains finite, the output remains finite as well. A continuous-time system is BIBO stable when its impulse response is integrable, which means the integral of the absolute value of the impulse response is finite.

The same idea applies to discrete-time systems. Here, stability is checked through the convolution series rather than the convolution integral. A discrete-time system is BIBO stable if the summation is finite, which means its impulse response is summable. In other words, if the sum of the absolute values of the impulse response is finite, any bounded input will still produce a bounded output.

BIBO stability matters in real-world system design and analysis. It helps ensure that outputs stay within acceptable limits when inputs are bounded. Understanding convolution and impulse response conditions is key for studying stable continuous-time and discrete-time systems.

Testing System Stability with Convolution

testing-system-stability-with-convolution

linear time-invariant systems

Impulse Response in LTI Systems

Linear time-invariant systems are defined by linear elements and constant parameters. A system is linear when it shows homogeneity and additivity. These two ideas together are called the superposition property. Superposition is a key rule for analyzing linear systems. For a linear time-invariant system, the response to an impulse at the input can describe the whole input-output behavior. An impulse is a very short, sudden input. Once the impulse response is known, the response to other inputs...

impulse response

RC Circuit Impulse Response

RC circuit impulse response describes how the capacitor voltage changes after a brief input signal. In this system, the voltage source is the input, and the capacitor voltage is the output. The state of the circuit before and after excitation is defined separately so the response can be tracked clearly. Kirchhoff’s law is used to write the input signal equation for the RC circuit. The capacitor current and voltage provide the output relation. After substituting the current expression and...

convolution: math, graphics, and discrete signals

convolution-math-graphics-and-discrete-signals

Convolution in LTI Systems and Signal Response

Convolution describes how a linear time-invariant, or LTI, system responds to an input signal. In an LTI system, the convolution of two signals is written with a convolution operator. The method assumes that all initial conditions are zero. The convolution integral can be split into two parts. These are the zero-input response, also called the natural response, and the zero-state response, also called the forced response. The initial time is marked as t0. To make the integral easier to work...

convolution properties i

LTI System Tricks for Easier Convolution

LTI system properties can make convolution easier to work with. LTI stands for linear time-invariant. In these systems, the input signal and the impulse response can be handled in helpful ways that simplify the final output. The commutative property means the input and the impulse response can be switched without changing the result. The associative property says that when three functions are convolved, the order of grouping does not change the answer. For two LTI systems in series, this lets...

convolution properties ii

Convolution Rules for Width and Area

Convolution rules help describe how an output signal changes when two signals are combined in an LTI system. The main rules here are width, area, differentiation, and integration. These rules make convolution easier to analyze and use in signal processing. The width property says that if two input signals last for T1 and T2, the output response lasts for T1 plus T2. This is true no matter what shape the two functions have. For example, convolving two rectangular pulses with durations of 2...

deconvolution

Finding an Impulse Response from Signals

Deconvolution is a method for finding a system’s impulse response from known input and output signals. It is also called inverse filtering. This technique is useful when the system’s characteristics are not known directly and must be inferred from the signals that can be observed. One common way to do deconvolution is polynomial division. In this approach, the input and output sequences are treated as coefficients of polynomials written in descending order. Long division is then used to divide...

System stability in signal processing is often tested with convolution. A system is BIBO stable, or bounded-input bounded-output stable, when every bounded input signal produces a bounded output signal. A bounded input signal has a modulus that never exceeds a set constant at any time. For a continuous-time Linear Time-Invariant, or LTI, system, BIBO stability is checked with the convolution integral. The input is bounded by a constant, and the convolution integral shows whether the output...

key terms and concepts

Learn how convolution is used to test BIBO stability in continuous-time and discrete-time systems by checking impulse response integrals and sums.

Testing System Stability with Convolution | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

Testing System Stability with Convolution

Description

Transcript

Tags