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

The important convolution properties are width, area, differentiation, and integration properties.
The width property indicates that if the durations of input signals are T1 and T2, respectively, then the width of the output response equals the sum of both durations, irrespective of the shapes of the two functions. For instance, convolving two rectangular pulses with durations of 2s and 1s results in a function with a width of 3s.
The area property asserts that the area under the convolution of two functions equals the product of the areas under each function.
The differentiation property states that the derivative of a convolution equals the convolution of the input signal&#39;s derivative and impulse response or vice versa.
Combining the equations and generalizing for higher order derivatives gives the general differentiation relation.
The integration property for a given convolution yields the convolution of the impulse response and input response. This can also be stated in the opposite sequence.
If a convolution is performed with a step function, the LTI system behaves like an ideal integrator.

convolutie-eigenschappen-ii

De belangrijke convolutie-eigenschappen omvatten breedte-, oppervlakte-, differentiatie- en integratie-eigenschappen.

De breedte-eigenschap geeft aan dat als de duur van de invoersignalen T_1 en T_2 zijn, de breedte van de uitvoer respons gelijk is aan de som van beide duurtijden, ongeacht de vorm van de twee functies. Bijvoorbeeld, het convolueren van twee rechthoekige pulsen met een duur van 2 seconden en 1 seconde resulteert in een functie met een breedte van 3 seconden.

De oppervlakte-eigenschap beweert dat het oppervlak onder de convolutie van twee functies gelijk is aan het product van de oppervlakken onder elke functie. Wiskundig gezien, als x(t) en h(t) twee functies zijn, dan kan de uitdrukking voor de oppervlakte-eigenschap worden gegeven als,

<img alt="Equation1" src="/files/ftp_upload/16051/16051_PT EQUATION_1.svg" style="height: 16px;">

De differentiatie-eigenschap stelt dat de afgeleide van een convolutie gelijk is aan de convolutie van de afgeleide en impulsrespons van het ingangssignaal, of omgekeerd. Dit kan worden uitgedrukt als,

<img alt="Equation2" src="/files/ftp_upload/16051/16051_PT EQUATION_2.svg" style="height: 34px;">

Door deze vergelijkingen te combineren en te generaliseren voor afgeleiden van hogere orde, ontstaat de algemene differentiatie relatie.

De integratie-eigenschap geeft aan dat de integraal van een convolutie de convolutie van de impulsrespons en de integraal van het ingangssignaal oplevert, of omgekeerd. Dit wordt wiskundig weergegeven als:

<img alt="Equation3" src="/files/ftp_upload/16051/16051_PT EQUATION_3.svg" style="height: 42px;">

Als een convolutie wordt uitgevoerd met een stapfunctie, gedraagt ​​het LTI-systeem zich als een ideale integrator.

Deze eigenschappen, breedte, oppervlakte, differentiatie en integratie — zijn cruciaal voor het vereenvoudigen en begrijpen van de convolutie bewerkingen in LTI-systemen, waardoor het eenvoudiger wordt om complexe signaalverwerking taken te analyseren en ontwerpen.

Convolutie-eigenschappen-II

The important convolution properties are width, area, differentiation, and integration properties.
The width property indicates that if the durations of ...

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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 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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Learn the key convolution rules for width, area, differentiation, and integration in LTI systems, with a simple rectangular pulse example and step-function behavior.

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Convolution Rules for Width and Area

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