Funciones análogicas de transferencia

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

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 (o función del sistema) de un sistema analógico se define como la "transformada de Laplace de la respuesta al impulso". Si "h(t)" es la respuesta al impulso, entonces la función de transferencia es la transformada de Laplace.

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 }_{-\infty }{h}\left(t\right)e^{-st}dt\;\;\;\ \mathrm{where}\;\;\;\; s=\sigma +i\omega. \end{align}

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La región de convergencia es una franja vertical 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/":): {\sigma }_{1}<\sigma <{\sigma }_{2} . Dentro de esta franja, H(s) no tiene polos ni otras singularidades (Figura 5). Para las funciones causales, utilizamos la transformada de Laplace unilateral.

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} F\left(s\right)=\int\limits^{\infty }_0{f}\left(t\right)e^{-st}dt=1\left\{f\left(t\right)\right\} \mathrm\;\;\;\; {where}\;\;\;\;\; s=\sigma +i\omega. \end{align}

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La región de convergencia de F(s) es el semiplano $ \sigma >{\sigma }_{1} $. El símbolo L se utiliza para indicar la transformada de Laplace unilateral.

El sistema ARMA analógico viene dado por la ecuación diferencial

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} y^{\left(p\right)}+{\alpha }_{1}y^{\left(p-1\right)}+\ldots +{\alpha }_py={\beta }_{{\rm o}} u^{\left(q\right)}+{\beta }_{1}u^{\left(q-1\right)}+\ldots +{\beta }_qu, \end{align}

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donde 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/":): y^{\left(n\right)} denota 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/":): d^ny/dt^n y 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/":): u^{\left(n\right)} denota 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/":): d^ny/dt^n . Supongamos ahora que la entrada es causal —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/":): u\left(t\right)=0 para t < 0 — de modo que la salida también es causal. Supongamos también que la entrada y la salida tienen condiciones iniciales cero; 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/":): \begin{align} u\left(0\right){\rm =0,\ }u^{\left(1\right)}\left(0\right){\rm =0,\ }u^{\left(2\right)}\left(0\right){\rm =0,\ldots ,\ }u^{\left(q-1\right)}\left(0\right)=0\\ y\left(0\right){\rm =0,\ }y^{\left(1\right)}\left(0\right){\rm =0,\ }y^{\left(2\right)}\left(0\right){\rm =0,\ldots ,\ }y^{\left(p-1\right)}\left(0\right)=0. \end{align}

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Si tomamos la transformada de Laplace (unilateral) de la ecuación diferencial anterior, obtenemos

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} \left(s^p+{\alpha }_{1}s^{p-1}+\ldots +{\alpha }_p\right){Y}\left(s\right)=\left({\beta }_0s^q+{\beta }_{1}s^{q-1}+\ldots +{\beta }_q\right)U\left(s\right) , \end{align}

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donde U(s) es la transformada de Laplace de la entrada e Y(s) es la transformada de Laplace de la salida. En el caso en el que la entrada es la función delta de Dirac 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/":): \delta \left(t\right) , que tiene una transformada de Laplace igual a 1, entonces la salida es la función de respuesta al impulso h(t), con la transformada de Laplace H(s). En tal caso, la ecuación anterior se convierte en

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} \left(s^p+{\alpha }_{1}s^{p-1}+\ldots +{\alpha }_p\right)H\left(s\right)\;\;\; \\ =\left({\beta }_0s^q+{\beta }_{1}s^{q-1}+\ldots +{\beta }_q\right) . \end{align}

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Si definimos los polinomios 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/":): \alpha \left(s\right) y 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/":): \beta \left(s\right) 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/":): \begin{align} \alpha \left(s\right)=s^p+{\alpha }_{1}s^{p-1}+\ldots +{\alpha }_p\;\;\;\; \\ \beta \left(s\right)={\beta }_0s^q+{\beta }_{1}s^{q-1}+\ldots +{\beta }_q, \end{align}

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Figure 5.  Franja vertical de convergencia de una transformada de Laplace.

Entonces vemos que H(s) 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)=\frac{\beta \left(s\right)}{\alpha \left(s\right)}. \end{align}

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La función H(s), que es la transformada de Laplace de la respuesta al impulso h(t), es la función de transferencia. Si factorizamos los polinomios 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/":): \alpha \left(s\right) y 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/":): \beta \left(s\right) , podemos escribir H(s) en su forma factorizada 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/":): \begin{align} H\left(s\right)=\frac{{\beta }_0\left(s-b_{1}\right)\left(s-b_{2}\right) ... \left(s-b_q\right)}{\left(s-a_{1}\right)\left(s-a_{2}\right)...\left(s-a_p\right)}. \end{align}

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Las constantes 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_{1} , 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_{2} , ..., 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_p son los polos de H(s), y las constantes $ b_{1} $, 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/":): b_{2} , ..., 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/":): b_q son los ceros. En el caso en el que los coeficientes 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/":): {\alpha }_k y 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/":): \beta _k sean reales, se deduce que todos los complejos 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_i y 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/":): b_i deben aparecer en pares complejos conjugados. Como s corresponde a la diferenciación, se deduce que 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/":): s^{-1} corresponde a la integración. Por lo tanto, podemos escribir H(s) 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/":): \begin{align} H\left(s\right)=\frac{{\beta }_0+{\beta }_{1}s^{-1}+{\beta }_{2}s^{-2}+\ldots +{\beta }_qs^{-q}} {1+{\alpha }_{1}s^{-1}+{\alpha }_{2}s^{-2}+\ldots +{\alpha }_ps^{-p}}s^{-\left(p-q\right)} \end{align}

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cuando queremos implementar el sistema mediante circuitos integradores.

Sigue leyendo

Sección previa	Siguiente sección
Funciones digitales de transferencia	Causalidad y estabilidad de sistemas digitales
Capítulo previo	Siguiente capítulo
Absorción	nada
Tabla de contenido

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
• Causalidad y estabilidad de sistemas digitales
• Causalidad y estabilidad de sistemas analógicos
• Respuesta en frecuencia de un sistema digital
• Predicción digital
• Predicción digital del error
• Predicción analógica del error

Vínculos externos

find literature about Analog transfer functions/es