Causalidad y estabilidad de sistemas analógicos
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| Series | Geophysical References Series |
|---|---|
| Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |
| Author | Enders A. Robinson and Sven Treitel |
| Chapter | 15 |
| DOI | http://dx.doi.org/10.1190/1.9781560801610 |
| ISBN | 9781560801481 |
| Store | SEG Online Store |
La función de transferencia Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H\left(s\right) de un sistema analógico es la transformada de Laplace de la función impulso-respuesta Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h\left(t\right) . Un sistema causal es aquel para el cual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h\left(t\right) es unilateral, es decir, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h\left(t\right)=0 para $ t<0 $. Supongamos que observamos la respuesta al impulso de un sistema causal como Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t\to \infty . Si
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \int\limits^{\infty }_0{|}h\left(t\right){|}\ dt<\infty , \end{align} ()
Nosotros decimos que el sistema es estable.
En el Ejemplo A, la señal analógica causal prototipo es la señal exponencial causal.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h\left(t\right)=0 \mathrm{\;\;\; for\ } t{\rm <0,}\;\;\; h\left(t\right)=e^{at} \mathrm{\;\;\; for\ } t{\rm >0.} \end{align} ()
Para nuestros propósitos, asumimos que el parámetro "a" es un número real. La transformada de Laplace es
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} H\left(s\right)=\int\limits^{\infty }_0{e^{-st}} e^{at}dt=\frac{1}{s-a}. \end{align} ()
Esta integral converge para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm \ Re\ }\left(s\right)>a . Es decir, la región de convergencia es el semiplano a la derecha del punto a. El caso A1 es para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm |a|<0} . La región de convergencia incluye el eje Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i\omega (Figura 8a), y la señal causal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h\left(t\right) es estable. El caso A2 es para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a > 0 . La región de convergencia no incluye el eje Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i\omega (Figura 8b), y la señal causal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h\left(t\right) es inestable.
En el Ejemplo B, el prototipo de señal analógica anticausal es la señal exponencial anticausal.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} h\left(t\right)=-e^{at} \mathrm{\;\;\;\; for\;\;\;} t{\rm <0,} \\ h\left(t\right)=0 \mathrm{\;\;\;\; for\;\;\; } t\ge 0. \end{align} ()
Su transformada de Laplace es
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} H\left(s\right)=\int\limits^0_{-\propto }{e^{-st}} \left(-e^{a{\rm t}}\right)dt \;\;\;\;\;\;\; \\ =-\int\limits^{\infty }_0{e^{\left(s-a\right)t}}dt=\frac{1}{s-a}, \end{align} ()
que converge para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm \ Re\ }\left(s\right)<a . Es decir, la región de convergencia es el semiplano a la izquierda del punto a. El caso B1 es para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a<0 . La región de convergencia incluye el eje Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i\omega (Figura 9a), y la señal anticausal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h\left(t\right) es inestable. El caso B2 es para Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a>0 . La región de convergencia incluye el eje Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): i\omega (Figura 9b), y la señal anticausal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h\left(t\right) es estable. Como es el caso con las señales digitales, normalmente elegiríamos la estabilidad a expensas de la causalidad en el caso de las señales analógicas.
Sigue leyendo
| Sección previa | Siguiente sección |
|---|---|
| Causalidad y estabilidad de sistemas digitales | Respuesta en frecuencia de un sistema digital |
| Capítulo previo | Siguiente capítulo |
| Absorción | nada |
También en este capítulo
- Introducción - Capítulo 15
- Sistemas digitales lineales invariantes en el tiempo
- Sistemas analógicos lineales invariantes en el tiempo
- Funciones digitales de transferencia
- Funciones análogicas de transferencia
- Causalidad y estabilidad de sistemas digitales
- Respuesta en frecuencia de un sistema digital
- Predicción digital
- Predicción digital del error
- Predicción analógica del error
Vínculos externos
| find literature about Causality and stability of analog systems/es |
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