Name: convolution in lti systems and signal response
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 24 s
Channel: JoVE

En cualquier sistema LTI, la convolución de dos señales se denota utilizando un operador de convolución, suponiendo que todas las condiciones iniciales son cero.
La integral de convolución se divide en dos partes: la entrada cero o respuesta natural y la respuesta de estado cero o forzada del sistema, con t0 indicando el tiempo inicial.
La integral de convolución se puede simplificar suponiendo que tanto la señal de entrada como la respuesta al impulso son cero para valores de tiempo negativos.
La convolución gráfica implica cuatro pasos: plegado, desplazamiento, multiplicación e integración. Considere un circuito RC con una señal de pulso de entrada y una respuesta de salida especificadas. Inicialmente, el plegado se realiza creando una imagen especular de la señal de entrada a lo largo del eje Y. A esto le sigue un ligero cambio, multiplicación e integración de las señales resultantes. La convolución resultante se representa gráficamente.
En la convolución de tiempo discreto, la respuesta se determina aplicando una entrada a un sistema de tiempo discreto y utilizando la respuesta al impulso y la integral de convolución.
La convolución de la señal de entrada discreta y la respuesta al impulso forman la suma de convolución para la respuesta del sistema.

convolución: matemáticas, gráficos y señales discretas

En cualquier sistema LTI (Linear Time-Invariant, invariante en el tiempo lineal), la convolución de dos señales se denota mediante un operador de convolución, suponiendo que todas las condiciones iniciales son cero. La integral de convolución se puede dividir en dos partes: la respuesta de entrada cero o natural y la respuesta de estado cero o forzada, donde t_0 indica el tiempo inicial.
Para simplificar la integral de convolución, se supone que tanto la señal de entrada como la respuesta al impulso son cero para valores de tiempo negativos. El proceso de convolución gráfica implica cuatro pasos: plegado, desplazamiento, multiplicación e integración.
Considere un circuito RC con una señal de pulso de entrada y una respuesta de salida especificadas. Inicialmente, el plegado se realiza creando una imagen reflejada de la señal de entrada a lo largo del eje y. A esto le sigue el desplazamiento, donde la señal plegada se desliza a lo largo del eje de tiempo. A continuación, la multiplicación de las señales plegadas y desplazadas se realiza punto por punto. Finalmente, la integración de la señal resultante a lo largo del tiempo proporciona el resultado de la convolución. Este proceso se puede representar gráficamente.

En la convolución de tiempo discreto, la respuesta del sistema se determina aplicando una entrada a un sistema de tiempo discreto y utilizando la respuesta al impulso y la suma de convolución. La convolución de la señal de entrada discreta x[n] y la respuesta al impulso h[n] forman la suma de convolución para la respuesta del sistema:

<img alt="Equation1" src="/files/ftp_upload/16049/16049_PT%20EQUATION_1.svg" style="text-align: center; height: 28px;">
Esta suma calcula la señal de salida y[n] en cada paso de tiempo discreto n. Comprender tanto la convolución continua como la discreta es esencial para analizar los sistemas LTI y predecir su comportamiento en respuesta a varias entradas.

Convolución: Matemáticas, gráficos y señales discretas

En cualquier sistema LTI, la convolución de dos señales se denota utilizando un operador de convolución, suponiendo que todas las condiciones iniciales son cero.
La convolución ...

convolution-math-graphics-and-discrete-signals

Education

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JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Linear Time- Invariant Systems

Sistemas Lineales e Invariantes en el Tiempo

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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 with, both the input signal and the impulse response are taken to be zero for negative time values.

Graphical convolution follows four steps: folding, shifting, multiplication, and integration. First, folding makes a mirror image of the input signal along the y-axis. Then the folded signal is shifted, or slid, along the time axis. After that, the folded and shifted signals are multiplied point by point.

The last step is integration over time, which gives the convolution result. This process can be shown graphically for an RC circuit with a given input pulse signal and output response. The same idea also appears in discrete-time systems. There, the input signal x[n] and the impulse response h[n] are combined with the convolution sum to find the output y[n] at each discrete time step n. Knowing both continuous and discrete convolution helps describe LTI system behavior for different inputs.

Convolution in LTI Systems and Signal Response

convolution-in-lti-systems-and-signal-response

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 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...

bibo stability of continuous and discrete -time systems

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

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...

key terms and concepts

Learn how convolution predicts LTI system response using folding, shifting, multiplication, and integration, plus the discrete convolution sum for x[n] and h[n].

Convolution in LTI Systems and Signal Response | 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.

electrical engineering

Convolution in LTI Systems and Signal Response

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