Transformada Z

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	7
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

¿En qué se diferencian las definiciones de la transformada Z? Los geofísicos y los ingenieros eléctricos tienen diferentes convenciones con respecto a la transformada z (véase también la discusión en el Capítulo 6). Sea 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_0, h_{{\rm l}}, h_{2}, \dots la respuesta al impulso de un filtro lineal causal invariante en el tiempo. La transformada z de ingeniería (con z minúscula) 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_{\text{engineering}} \;(z) = h_0 + h_1 \;z^{ - 1} + h_2 \;z^{ - 2} + ..., \end{align}

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mientras que la transformada "geofísica Z" (con "Z" mayúscula) es la función generadora

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(Z\right)=h_0+h_{{\rm l}} Z+h_{1}Z^{2}+\dots \end{align}

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Ambos están relacionados 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/":): Z=z^{-1} . Mientras que la "z" de ingeniería representa un operador de avance unitario, la "Z" de geofísica representa un operador de retardo unitario.

En la Tabla 1 se muestran las transformadas "z" de ingeniería de algunas señales comunes.

Al dejar 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=z^{-1} , la Tabla 1 se convierte en la Tabla 2 para las transformadas geofísicas Z correspondientes.

¿Cómo se obtiene la transformada de Fourier a partir de la transformada Z? La transformada de Fourier (convención de ingeniería eléctrica) de una 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_n en términos de frecuencia angular 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/":): \omega 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(\omega \right)=\sum^{\infty }_{n=0 }{h_n}e^{i \omega n}=A\left(\omega \right)e^{-i\phi \left(\omega \right)} \end{align}

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La Transformada de Fourier se obtiene a partir de la transformada z de ingeniería

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(z\right)=h_0+h_{{\rm 1}} z^{-{\rm 1}}+h_{1}z^{-2}+ \dots \end{align}

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por la sustitución 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=e^{i\omega } .

La Transformada de Fourier (convención de ingeniería eléctrica) se obtiene a partir de la transformada Z geofísica

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(Z\right)=h_0+h_{1}Z+h_{1}Z^{2}+\dots \end{align}

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por la sustitución 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=e^{-i\omega } . El lugar geométrico de $ Z=e^{-i\omega } $ es el círculo unitario 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{|}=1 . A medida que la frecuencia angular aumenta desde 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/":): \omega =-\pi pasando 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/":): \omega =0 hasta 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/":): \omega =\pi , el punto $ Z=e^{-i\omega } $ gira alrededor del círculo unitario (en sentido horario) desde Z = +1 pasando por Z = +i hasta Z = -1. La transformada de Fourier representa el valor de la transformada Z en el círculo unitario (Figura 1).

Figure 1.  La transformada de Fourier es igual a los valores de la transformada Z a medida que Z recorre el círculo unitario en el sentido de las agujas del reloj.

Tabla 1. Señales comunes y sus transformadas z de ingeniería eléctrica.

Nombre de la señal	Señal	Transformada z	Región de convergencia
Impulso unitario	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 }_n=1 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/":): t=0 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 }_n=0 en caso contrario	1	En todas partes
Impulso retardado	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 }_{n-k} para k fijo > 0	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^{ - k}	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|\; > 0
Paso causal unitario	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 = 0\;\text{para}\;k < 0 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 = 1\;\text{para}\;k \ge \;0	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/":): \frac{z} {{z - 1}} = \frac{1} {{1 - z^{ - 1} }} Traducciones:Z-transform/55/es	$ |z|\;<0 $
Paso anticausal negativo	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 - 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/":): \frac{z} {{z - 1}} = \frac{1} {{1 - z^{ - 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/":): |z| < 1
Ramp	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/":): nu_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/":): \frac{z} {{(z - 1)^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/":): |z| > 1
Causal geometric	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 ^n u_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/":): \frac{z} {{z - \alpha }} = \frac{1} {{1 - \alpha z^ - }}	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|\; > \;\alpha
Geométrica anticausal negativa	$ -\alpha ^{n}u_{-n-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/":): \frac{{z(z - \cos \theta )}} {{z^2 - 2\cos \theta \;z + 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/":): |z|\; < \;\alpha
Coseno 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/":): u_n \cos (\theta _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/":): \frac{{z(z - \cos \theta )}} {{z^2 - 2\cos \theta \;z + 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/":): |z|\; > 1
Seno 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/":): u_n \sin (\theta _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/":): \frac{{z\sin \theta }} {{z^2 - 2\cos \theta \;z + 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/":): |z|\; > 1
Coseno geométrico 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/":): u_n \alpha ^n \cos (\theta _n )	$ {\frac {z(z-\alpha \cos \theta )}{z^{2}-2\alpha \;\cos \theta \;z+\alpha ^{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/":): |z|\; > \;|\alpha |
Seno geométrico 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/":): u_n \alpha ^n \sin (\theta _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/":): \frac{{z\alpha \sin \theta }}{{z^2 - 2\alpha \cos \theta \;z + \alpha ^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/":): |z|\; > \;|\alpha |

Tabla 2. Señales comunes y sus transformadas Z geofísicas.

Nombre de la señal	Señal	Transformada z	Región de convergencia
Impulso unitario	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 }_n=1 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/":): t=0 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 }_n=0 en caso contrario	1	En todas partes
Impulso retardado	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 }_{n-k} para k fijo > 0	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^k	$ |z|\;>0 $
Paso causal unitario	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 = 0\;\text{para}\;k < 0 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 = 1\;\text{para}\;k \ge \;0	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/":): \frac{1}{{1 - Z}}	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|\; < 1
Paso anticausal negativo	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 - 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/":): \frac{1}{{1 - Z}}	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|\; > 1
Rampa	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/":): nu_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/":): \frac{Z}{{(1 - Z)^2 }}	$ |Z|\;<\;1 $
Geométrica 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/":): \alpha ^n u_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/":): \frac{1} {{1 - \alpha Z}}	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|\; < \;\alpha
Geométrica anticausal negativa	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 ^n u_{ - n - 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/":): \frac{1} {{1 - \alpha Z}}	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|\; > \;\alpha
Coseno 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/":): u_n \cos (\theta 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/":): \frac{{1 - \cos \theta \;Z}} {{1 - 2\;\cos \theta \;Z + Z^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/":): |Z|\; < 1
Seno causal	$ u_{n}\sin(\theta 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/":): \frac{{Z\;\sin \theta }} {{1 - 2\;\cos \theta \;Z + Z^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/":): |Z|\; < 1
Coseno geométrico 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/":): u_n \alpha ^n \cos (\theta 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/":): \frac{{Z(1 - \alpha \cos \theta \;Z)}} {{\alpha ^2 - 2\alpha \;\cos \theta \;Z + Z^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/":): |Z|\; < \;|\alpha |
Seno geométrico 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/":): u_n \alpha ^n \sin (\theta 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/":): \frac{{Z\alpha \;\sin \theta }} {{\alpha ^2 - 2\alpha \cos \theta \;Z + Z^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/":): |Z|\; < \;|\alpha |

Sigue leyendo

Sección previa	Siguiente sección
Transformada de Fourier	Retraso: Mínimo, mixto y máximo
Capítulo previo	Siguiente capítulo
Frecuencia	Sintéticos
Tabla de contenido

También en este capítulo

• Ondículas
• Transformada de Fourier
• Retraso: Mínimo, mixto y máximo
• Ondículas de doble longitude
• Ilustración del espectro
• Retraso en general
• Energía
• Autocorrelación
• Representación canónica
• Ondículas de fase cero
• Ondículas simétricas
• Ondícula de Ricker
• Apéndice G: Ejercicios

Vínculos externos

find literature about Z-transform/es