# Appendix N: The energy-delay theorem

Other languages:
English • ‎español
Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 13 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

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

 {\displaystyle {\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,

 {\displaystyle {\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

 ${\displaystyle {\begin{array}{l}u_{n}=x_{n}\;\;\;{\rm {for}}\;\;n\leq N\\u_{n}\;=\;0\;\;\;\;\;{\rm {for}}\;\;n>N.\\\end{array}}}$ (N3)

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

 {\displaystyle {\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

 {\displaystyle {\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 ${\displaystyle n\leq N}$. The total energy of the truncated input is

 {\displaystyle {\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

 {\displaystyle {\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

 {\displaystyle {\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 ${\displaystyle x_{n}=u_{n}}$ and ${\displaystyle y_{n}=v_{n}}$ for ${\displaystyle n\leq N}$, we obtain

 {\displaystyle {\begin{aligned}\sum _{-\propto }^{N}{x_{n}^{2}}=\sum _{-\propto }^{N}{y_{n}^{2}}+\sum _{N+1}^{\infty }{v_{n}^{2}},\end{aligned}}} (N9)

From equation N-9, we conclude that

 {\displaystyle {\begin{aligned}\sum _{-\infty }^{N}{X_{n}^{2}}\geq \sum _{-\propto }^{N}{y_{n}^{2}}.\end{aligned}}} (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 ${\displaystyle h_{n}}$ can be expressed as a minimum-delay filter ${\displaystyle g_{n}}$ in cascade with an all-pass filter ${\displaystyle p_{n}}$. If the input ${\displaystyle z_{n}}$ is applied to the minimum-delay filter ${\displaystyle g_{n}}$, the output is

 {\displaystyle {\begin{aligned}x_{n}=g_{n}*z_{n}.\end{aligned}}} (N11)

If the same input ${\displaystyle z_{n}}$ is applied to the causal filter ${\displaystyle h_{n}}$, the output is

 {\displaystyle {\begin{aligned}y_{n}=h_{n}*z_{n}=\left[g_{n}*z_{n}\right]*p_{n}.\end{aligned}}} (N12)

Equations N-11 and N-12 show that

 {\displaystyle {\begin{aligned}y_{n}=x_{n}*p_{n}.\end{aligned}}} (N13)

From the first result, it follows that

 {\displaystyle {\begin{aligned}\sum _{-\infty }^{N}{x_{n}^{2}}\geq \sum _{-\propto }^{N}{y_{n}^{2}}.\end{aligned}}} (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.