Apéndice N: El teorema del retraso de energía

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This page is a translated version of the page Appendix N: The energy-delay theorem and the translation is 100% complete.
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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 13
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

Supongamos que la respuesta de un filtro pasa-todo 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/":): p_n a una entrada arbitraria $ x_{n} $ 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/":): y_n :


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_n= x_n*p_n. \end{align} (N1)

Debido a que el espectro de amplitud de un filtro pasa todo es la unidad, la energía de la entrada es igual a la energía de la salida; 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} \sum^{\infty }_{-\infty }{y^{2}_n}= \sum^{\infty }_{-\infty }{x^{2}_n}. \end{align} (N2)

En otras palabras, la salida de un filtro pasa todo tiene la misma energía total que la entrada. Ahora definamos la entrada truncada 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{array}{l} u_n = x_n \;\;\;{\rm for}\;\;n \le N \\ u_n \; = \;0\;\;\;\;\;{\rm for}\;\;n > N. \\ \end{array} (N3)

En otras palabras, 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_n se obtiene a partir de 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/":): x_n truncándolo para obtener 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/":): n\le N . Si pasamos la entrada truncada por el filtro de paso total, obtenemos la salida dada por


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} v_n= \sum^n_{{j=}-\infty }{p_j}u_{n-j}. \end{align} (N4)

El límite superior que aparece en la convolución anterior 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} v_n= \sum^N_{j= -\propto }{p_j}u_{n-j}= \sum^N_{j= -\propto }{p_j}x_{n-j}= y_n \end{align} (N5)

para cada 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/":): n\le N . La energía total de la entrada truncada es


$ {\begin{aligned}\sum _{-{\rm {\infty }}}^{\infty }{u_{n}^{2}}=\sum _{-\infty }^{N}{u_{n}^{2}}.\end{aligned}} $ (N6)

La energía total de la salida resultante de la entrada truncada 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} \sum^{\infty }_{-\infty }{v^{2}_n}= \sum^N_{-\infty }{v^{2}_n}+\sum^{\infty }_{N+1}{v^{2}_n}. \end{align} (N7)

Las ecuaciones N-6 y N-7 para la energía total son iguales, por lo que tenemos


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} \sum^N_{-\propto }{u^{2}_n}= \sum^N_{-\infty }{y^{2}_n}+\sum^{\infty }_{N+1}{v^{2}_n}, \end{align} (N8)

Dado 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/":): x_n= u_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/":): y_n= v_n 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/":): n\le N , 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} \sum^N_{-\propto }{x^{2}_n}= \sum^N_{-\propto }{y^{2}_n}+\sum^{\infty }_{N+1}{v^{2}_n}, \end{align} (N9)

De la ecuación N-9, concluimos que


$ {\begin{aligned}\sum _{-\infty }^{N}{X_{n}^{2}}\geq \sum _{-\propto }^{N}{y_{n}^{2}}.\end{aligned}} $ (N10)

La ecuación N-10 es el primer resultado. Indica que la energía parcial (es decir, la acumulación de energía) de la entrada es mayor que la energía parcial (es decir, la acumulación de energía) de la salida de un filtro de paso total (Robinson, 1962[1]).

Ahora estamos listos para establecer el teorema de retardo de energía. La representación canónica dice que cualquier filtro causal estable 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_n puede expresarse como un filtro de retardo mínimo 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/":): g_n en cascada con un filtro de paso total 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/":): p_n . Si la entrada 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/":): z_n se aplica al filtro de retardo mínimo 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/":): g_n , la salida 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} x_n= g_n*z_n. \end{align} (N11)

Si se aplica la misma entrada 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/":): z_n al filtro 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_n , la salida 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} y_n= h_n*z_n= \left[g_{n}*z_n\right]*p_n. \end{align} (N12)

Las ecuaciones N-11 y N-12 muestran 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/":): \begin{align} y_n= x_n*p_n. \end{align} (N13)

Del primer resultado 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/":): \begin{align} \sum^N_{-\infty }{x^{2}_n}\ge \sum^N_{-\propto }{y^{2}_n}. \end{align} (N14)

Esta ecuación es el teorema del retardo de energía (Robinson, 1963a[2], 1963b[3]). Dice que si se aplica la misma entrada tanto a un filtro de retardo mínimo como a cualquier otro filtro causal estable con el mismo espectro de amplitud, entonces la energía parcial de la salida del filtro de retardo mínimo es mayor o igual a la energía parcial de la salida del otro filtro. En otras palabras, un filtro de retardo mínimo produce menos retardo de energía que cualquier otro filtro causal con el mismo espectro de amplitud. Este teorema es la razón por la que se utiliza el término "retardo mínimo". , en referencia al hecho de que un filtro de retardo mínimo retrasa menos la energía.


Referencias

  1. ↑ Robinson, E. A., 1962, Random wavelets and cybernetic systems: Charles Griffin and Co. and Macmillan.
  2. ↑ Robinson, E. A., 1963a, Nekotorye svoystva razlozheniya vol'da stasionarnykh sluchaynykh protsessov [Propiedades de la descomposición de Wold de procesos estocásticos estacionarios (en ruso)]: Teoriya Veroyatnostei ee Primememiya, Akademiya Nauk SSSR, 7, no 2, 201–211.
  3. ↑ Robinson, E. A., 1963b, Propiedades extremas de la descomposición de Wold: Diario. de Análisis y Aplicaciones Matemáticas, 6, 75–85.

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