Funciones análogicas de transferencia

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
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

The transfer function (or system function) of an analog system is defined as the Laplace transform of the impulse response. If h(t) is the impulse response, then the transfer function is the Laplace transform


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/":): {\displaystyle \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}} (38)

The region of convergence is a vertical strip 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/":): {\displaystyle {\sigma }_{1}<\sigma <{\sigma }_{2}} . Within this strip, H(s) has no poles or other singularities (Figure 5). For causal functions, we use the one-sided Laplace transform


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/":): {\displaystyle \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}} (39)

The region of convergence of F(s) is the half-plane 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/":): {\displaystyle \sigma >{\sigma }_{1}} . The symbol L is used to denote the one-sided Laplace transform.

The analog ARMA system is given by the differential equation


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/":): {\displaystyle \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}} (40)

where 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/":): {\displaystyle y^{\left(n\right)}} denotes 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/":): {\displaystyle d^ny/dt^n} and 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/":): {\displaystyle u^{\left(n\right)}} denotes 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/":): {\displaystyle d^ny/dt^n} . Let us now assume that the input is causal — that is, 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/":): {\displaystyle u\left(t\right)=0} for t < 0 — so that the output also is causal. We also assume that the input and output have zero initial conditions; that is,


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/":): {\displaystyle \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}} (41)

If we take the (one-sided) Laplace transform of the above differential equation, we obtain


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/":): {\displaystyle \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}} (42)

where U(s) is the Laplace transform of the input and Y(s) is the Laplace transform of the output. In the case in which the input is the Dirac delta function 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/":): {\displaystyle \delta \left(t\right)} , which has a Laplace transform equal to 1, then the output is the impulse response function h(t), with the Laplace transform H(s). In such a case, the above equation becomes


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/":): {\displaystyle \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}} (43)

If we define the polynomials 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/":): {\displaystyle \alpha \left(s\right)} and 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/":): {\displaystyle \beta \left(s\right)} as


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/":): {\displaystyle \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}} (44)
Figure 5.  Vertical strip of convergence of a Laplace transform.

then we see that H(s) is


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/":): {\displaystyle \begin{align} H\left(s\right)=\frac{\beta \left(s\right)}{\alpha \left(s\right)}. \end{align}} (45)

The function H(s), which is the Laplace transform of the impulse response h(t), is the transfer function. If we factor the polynomials 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/":): {\displaystyle \alpha \left(s\right)} and 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/":): {\displaystyle \beta \left(s\right)} , we can write H(s) in its factored form as


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/":): {\displaystyle \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}} (46)

The constants 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/":): {\displaystyle 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/":): {\displaystyle 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/":): {\displaystyle a_p} are the poles of H(s), and the constants 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/":): {\displaystyle 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/":): {\displaystyle 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/":): {\displaystyle b_q} are the zeros. In the case in which the 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/":): {\displaystyle {\alpha }_k} and 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/":): {\displaystyle \beta _k} coefficients are real, it follows that all complex 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/":): {\displaystyle a_i} and 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/":): {\displaystyle b_i} must occur in complex-conjugate pairs. Because s corresponds to differentiation, it follows that 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/":): {\displaystyle s^{-1}} corresponds to integration. Thus, we can write H(s) as


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/":): {\displaystyle \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}} (47)

when we want to implement the system by means of integrating circuits.


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