Appendix N: The energy-delay theorem/en

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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

Suppose that the response of an all-pass filter $ p_{n} $ to an arbitrary input $ x_{n} $ is $ y_{n} $:

$ {\begin{aligned}y_{n}=x_{n}*p_{n}.\end{aligned}} $

(N1)

Because the amplitude spectrum of an all-pass filter is unity, the energy of the input is equal to the energy of the output; that is,

$ {\begin{aligned}\sum _{-\infty }^{\infty }{y_{n}^{2}}=\sum _{-\infty }^{\infty }{x_{n}^{2}}.\end{aligned}} $

(N2)

In other words, the output of an all-pass filter has the same total energy as does the input. Now let us define the truncated input as

$ {\begin{array}{l}u_{n}=x_{n}\;\;\;{\rm {for}}\;\;n\leq N\\u_{n}\;=\;0\;\;\;\;\;{\rm {for}}\;\;n>N.\\\end{array}} $

(N3)

In other words, $ u_{n} $ is obtained from $ x_{n} $ by truncating it for $ n\leq N $. If we pass the truncated input through the all-pass filter, we obtain the output given by

$ {\begin{aligned}v_{n}=\sum _{{j=}-\infty }^{n}{p_{j}}u_{n-j}.\end{aligned}} $

(N4)

The upper limit appearing in the above convolution is allowed because of the causality of all-pass fitters. We conclude that

$ {\begin{aligned}v_{n}=\sum _{j=-\propto }^{N}{p_{j}}u_{n-j}=\sum _{j=-\propto }^{N}{p_{j}}x_{n-j}=y_{n}\end{aligned}} $

(N5)

for every $ n\leq N $. The total energy of the truncated input is

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

(N6)

The total energy of the output resulting from the truncated input is

$ {\begin{aligned}\sum _{-\infty }^{\infty }{v_{n}^{2}}=\sum _{-\infty }^{N}{v_{n}^{2}}+\sum _{N+1}^{\infty }{v_{n}^{2}}.\end{aligned}} $

(N7)

Equations N-6 and N-7 for total energy are equal, so we have

$ {\begin{aligned}\sum _{-\propto }^{N}{u_{n}^{2}}=\sum _{-\infty }^{N}{y_{n}^{2}}+\sum _{N+1}^{\infty }{v_{n}^{2}},\end{aligned}} $

(N8)

Because $ x_{n}=u_{n} $ and $ y_{n}=v_{n} $ for $ n\leq N $, 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/":): \begin{align} \sum^N_{-\propto }{x^{2}_n}= \sum^N_{-\propto }{y^{2}_n}+\sum^{\infty }_{N+1}{v^{2}_n}, \end{align}

(N9)

From equation N-9, we conclude 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/":): \begin{align} \sum^N_{-\infty }{X^{2}_n}\ge \sum^N_{-\propto }{y^{2}_n}. \end{align}

(N10)

Equation N-10 is the first result. It says that the partial energy (i.e., the energy buildup) of the input is greater than the partial energy (i.e., the energy buildup) of the output of an all-pass filter (Robinson, 1962).

Now we are ready to establish the energy-delay theorem. The canonical representation says that any causal stable filter 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 can be expressed as a minimum-delay filter 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 in cascade with an all-pass filter 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 . If the input 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 is applied to the minimum-delay filter 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 , the output 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/":): \begin{align} x_n= g_n*z_n. \end{align}

(N11)

If the same input $ z_{n} $ is applied to the causal filter 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 , the output 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/":): \begin{align} y_n= h_n*z_n= \left[g_{n}*z_n\right]*p_n. \end{align}

(N12)

Equations N-11 and N-12 show 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/":): \begin{align} y_n= x_n*p_n. \end{align}

(N13)

From the first result, 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/":): \begin{align} \sum^N_{-\infty }{x^{2}_n}\ge \sum^N_{-\propto }{y^{2}_n}. \end{align}

(N14)

This equation is the energy-delay theorem (Robinson, 1963a, 1963b). It says that if the same input is applied to both a minimum-delay filter and any other stable causal filter with the same amplitude spectrum, then the partial energy of the output of the minimum-delay filter is greater than or equal to the partial energy of the output of the other filter. In other words, a minimum-delay filter produces less energy delay than does any other causal filter with the same amplitude spectrum. This theorem is the reason for the use of the term minimum delay, referring to the fact that a minimum-delay filter delays the energy the least.

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Also in this chapter

• Phase rotation
• Alternate approaches to phase rotation
• Band-limiting root approximation

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