Name: convolution rules for width and area
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 17 s
Channel: JoVE

Las propiedades de convolución importantes son las propiedades de ancho, área, diferenciación e integración.
La propiedad width indica que si las duraciones de las señales de entrada son T1 y T2, respectivamente, entonces el ancho de la respuesta de salida es igual a la suma de ambas duraciones, independientemente de las formas de las dos funciones. Por ejemplo, la convolucionación de dos pulsos rectangulares con duraciones de 2s y 1s da como resultado una función con un ancho de 3s.
La propiedad area afirma que el área bajo la convolución de dos funciones es igual al producto de las áreas bajo cada función.
La propiedad de diferenciación establece que la derivada de una convolución es igual a la convolución de la derivada y la respuesta al impulso de la señal de entrada o viceversa.
Combinando las ecuaciones y generalizando para derivadas de orden superior se obtiene la relación de diferenciación general.
La propiedad de integración para una convolución dada produce la convolución de la respuesta al impulso y la respuesta de entrada. Esto también se puede expresar en la secuencia opuesta.
Si se realiza una convolución con una función escalonada, el sistema LTI se comporta como un integrador ideal.

propiedades de convolución ii

Las propiedades de convolución importantes incluyen las propiedades de ancho, área, diferenciación e integración.
La propiedad de ancho indica que si las duraciones de las señales de entrada son T_1 y T_2, entonces el ancho de la respuesta de salida es igual a la suma de ambas duraciones, independientemente de las formas de las dos funciones. Por ejemplo, la convolución de dos pulsos rectangulares con duraciones de 2 segundos y 1 segundo da como resultado una función con un ancho de 3 segundos.
La propiedad de área afirma que el área bajo la convolución de dos funciones es igual al producto de las áreas bajo cada función. Matemáticamente, si x(t) y ℎ(t) son dos funciones, entonces la expresión para la propiedad del área puede darse como,
<img alt="Equation1" src="/files/ftp_upload/16051/16051_PT%20EQUATION_1.svg" style="text-align: center; height: 16px;">
La propiedad de diferenciación establece que la derivada de una convolución es igual a la convolución de la derivada de la señal de entrada y la respuesta al impulso, o viceversa. Esto puede expresarse como,
<img alt="Equation2" src="/files/ftp_upload/16051/16051_PT%20EQUATION_2.svg" style="text-align: center; height: 34px;">
Combinando estas ecuaciones y generalizando para derivadas de orden superior se obtiene la relación de diferenciación general.
La propiedad de integración indica que la integral de una convolución produce la convolución de la respuesta al impulso y la integral de la señal de entrada, o viceversa. Esto se representa matemáticamente como:
<img alt="Equation3" src="/files/ftp_upload/16051/16051_PT%20EQUATION_3.svg" style="text-align: center; height: 42px;">
Si se realiza una convolución con una función escalonada, el sistema LTI se comporta como un integrador ideal.
Estas propiedades (ancho, área, diferenciación e integración) son cruciales para simplificar y comprender las operaciones de convolución en sistemas LTI, lo que facilita el análisis y el diseño de tareas complejas de procesamiento de señales.

Propiedades de convolución II

Las propiedades de convolución importantes son las propiedades de ancho, área, diferenciación e integración.
La propiedad width indica que si las duraciones de ...

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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 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 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...

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Convolution Rules for Width and Area

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