Appendix N: The energy-delay theorem

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
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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 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 p_n} to an arbitrary 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/":): {\displaystyle x_n} 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 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/":): {\displaystyle \begin{align} y_n= x_n*p_n. \end{align}} (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,


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} \sum^{\infty }_{-\infty }{y^{2}_n}= \sum^{\infty }_{-\infty }{x^{2}_n}. \end{align}} (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


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{array}{l} u_n = x_n \;\;\;{\rm for}\;\;n \le N \\ u_n \; = \;0\;\;\;\;\;{\rm for}\;\;n > N. \\ \end{array}} (N3)

In other words, 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_n} is obtained from 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 x_n} by truncating it for 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 n\le N} . If we pass the truncated input through the all-pass filter, we obtain the output given by


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

The upper limit appearing in the above convolution is allowed because of the causality of all-pass fitters. 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/":): {\displaystyle \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)

for every 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 n\le N} . The total energy of the truncated input 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} \sum^{\infty }_{-{\rm \infty}} {u^{2}_n}= \sum^N_{-\infty }{u^{2}_n}. \end{align}} (N6)

The total energy of the output resulting from the truncated input 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} \sum^{\infty }_{-\infty }{v^{2}_n}= \sum^N_{-\infty }{v^{2}_n}+\sum^{\infty }_{N+1}{v^{2}_n}. \end{align}} (N7)

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


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

Because 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 x_n= u_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 y_n= v_n} for 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 n\le 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/":): {\displaystyle \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/":): {\displaystyle \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[1]).

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/":): {\displaystyle 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/":): {\displaystyle 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/":): {\displaystyle 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/":): {\displaystyle 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/":): {\displaystyle 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/":): {\displaystyle \begin{align} x_n= g_n*z_n. \end{align}} (N11)

If the same 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/":): {\displaystyle 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/":): {\displaystyle 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/":): {\displaystyle \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/":): {\displaystyle \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/":): {\displaystyle \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[2], 1963b[3]). 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.


References

  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 statsionarnykh sluchaynykh protsessov [Properties of the Wold decomposition of stationary stochastic processes (in Russian)]: Teoriya Veroyatnostei i ee Primememiya, Akademiya Nauk SSSR, 7, no. 2, 201–211.
  3. Robinson, E. A., 1963b, Extremal properties of the Wold decomposition: Journal of Mathematical Analysis and Applications, 6, 75–85.

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