Name: convolution rules for width and area
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 17 s
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As propriedades de convolução importantes são as propriedades de largura, área, diferenciação e integração.
A propriedade width indica que, se as durações dos sinais de entrada forem T1 e T2, respectivamente, a largura da resposta de saída será igual à soma de ambas as durações, independentemente das formas das duas funções. Por exemplo, convolução de dois pulsos retangulares com durações de 2s e 1s resulta em uma função com largura de 3s.
A propriedade area afirma que a área sob a convolução de duas funções é igual ao produto das áreas sob cada função.
A propriedade de diferenciação afirma que a derivada de uma convolução é igual à convolução da derivada do sinal de entrada e da resposta ao impulso ou vice-versa.
Combinando as equações e generalizando para derivadas de ordem superior, obtém-se a relação geral de diferenciação.
A propriedade de integração para uma determinada convolução produz a convolução da resposta ao impulso e da resposta de entrada. Isso também pode ser declarado na sequência oposta.
Se uma convolução for realizada com uma função de degrau, o sistema LTI se comporta como um integrador ideal.

propriedades de convolução-ii

As propriedades importantes da convolução incluem largura, área, diferenciação e propriedades de integração.
A propriedade de largura indica que se as durações dos sinais de entrada forem T_1 e T_2, então a largura da resposta de saída será igual à soma de ambas as durações, independentemente das formas das duas funções. Por exemplo, convoluir dois pulsos retangulares com durações de 2 segundos e 1 segundo resulta em uma função com largura de 3 segundos.
A propriedade de área afirma que a área sob a convolução de duas funções é igual ao produto das áreas sob cada função. Matematicamente, se x(t) e h(t) são duas funções, então a expressão para a propriedade de área pode ser dada como,

<img alt="Equation1" src="/files/ftp_upload/16051/16051_PT%20EQUATION_1.svg" style="height: 16px;">
A propriedade de diferenciação afirma que a derivada de uma convolução é igual à convolução da derivada do sinal de entrada e da resposta ao impulso, ou vice-versa. Isso pode ser expresso como,
<img alt="Equation2" src="/files/ftp_upload/16051/16051_PT%20EQUATION_2.svg" style="height: 34px;">
Combinar essas equações e generalizar para derivadas de ordem superior fornece a relação de diferenciação geral.
A propriedade de integração indica que a integral de uma convolução produz a convolução da resposta ao impulso e a integral do sinal de entrada, ou vice-versa. Isso é matematicamente representado como:
<img alt="Equation3" src="/files/ftp_upload/16051/16051_PT%20EQUATION_3.svg" style="height: 42px;">
Se uma convolução for realizada com uma função degrau, o sistema LTI se comporta como um integrador ideal.
Essas propriedades — largura, área, diferenciação e integração — são cruciais para simplificar e entender as operações de convolução em sistemas LTI, facilitando a análise e o projeto de tarefas complexas de processamento de sinais.

Propriedades de Convolução-II

As propriedades de convolução importantes são as propriedades de largura, área, diferenciação e integração.
A propriedade width indica que, se as durações de ...

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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 seconds and 1 second gives an output with a width of 3 seconds.

The area property states that the area under the convolution of two functions equals the product of their individual areas. In other words, if x(t) and h(t) are convolved, the total area of the result depends on the area of each input. The differentiation property says that the derivative of a convolution is the convolution of the derivative of the input signal and the impulse response, or the other way around.

The integration property gives the matching rule for integrals. It says that integrating a convolution produces the convolution of the impulse response and the integral of the input signal, or vice versa. If a convolution is performed with a step function, the LTI system acts like an ideal integrator. Together, these width, area, differentiation, and integration rules help simplify convolution in LTI systems and support the analysis and design of more complex signal processing tasks.

Convolution Rules for Width and Area

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Convolution Rules for Width and Area

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