Digital prediction error/en

**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	15
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

From the preceding section on prediction, one might get the impression that we have favored minimum-delay signals and have slighted nonminimum-delay ones. However, when we look at the overall prediction problem rather than at the restricted problem given in the preceding section, we will correct this impression.

In the foregoing section, we considered only the prediction of the tailgate. The minimum-delay signal $f_{k}$ and the tailgate of the advanced signal $f_{k+\varepsilon }$ both started at time index $$ k=0 $$ , and there was no concern for any values for $$ k<0 $$ . The signal $f_{k}$ is causal, so it does not have any nonzero values before $$ k=0 $$ , but the advanced signal $f_{k+\varepsilon }$ does have nonzero values in the range $-\varepsilon \leq k<0$ . In fact, the advanced signal starts at $k=-\varepsilon$ , whereas the signal $f_{k}$ starts at $$ k=0 $$ . Because the prediction filter is causal, no output can occur before $$ k=0 $$ , the start of the input signal. Thus, the segment of the advanced signal from $k=-\varepsilon$ to $$ k=0 $$ cannot appear at the output of the filter and thus represents a front-end prediction error. However, as we have seen in the preceding section, perfect prediction of the input signal is obtained at $$ k=0 $$ and beyond, provided that the input signal is minimum delay.

In summary, the minimum-delay signal $f_{k}$ is the input to the causal prediction system. We found the output ${\hat {f_{k}}}\left(\varepsilon \right)$ of the prediction system to be the advanced signal $f_{k+\varepsilon }$ . Because the prediction system is causal, we cannot obtain the front end $f_{k+\varepsilon }$ from $k=-\varepsilon$ to $$ k=-1 $$ as output. However, we can obtain exactly the tailgate $f_{k+\varepsilon }$ from $$ k=0,1,2,..., $$ at the output of the filter. This front end of the advanced signal represents the front-end prediction error, and this error cannot be reduced.

Let us discuss notation. If the input of the prediction system $E\left(z\right)$ is $f_{k}$ , then the actual output is denoted by ${\hat {f}}_{k}(\varepsilon )$ . The desired output is the advanced signal ${\hat {f}}_{k+\varepsilon }$ , where $\varepsilon >0$ . Thus, the actual output ${\hat {f}}_{k}(\varepsilon )$ represents the predicted value of the input signal. Both the input $f_{k}$ and the predicted value ${\hat {f}}_{k}(\varepsilon )$ occur at time index k. That is, the subscript in ${\hat {f}}_{k}(\varepsilon )$ indicates the time of occurrence. The prediction error that occurs at time index k is denoted by

{\begin{aligned}{\tilde {f}}\left(\varepsilon \right)=f_{k+\varepsilon }-{\hat {f}}\left(\varepsilon \right).\end{aligned}}

(152)

The notation ${\tilde {f_{k}}}\left(\varepsilon \right)$ should be read as “the prediction error (that is obtained at the present time k) in predicting the future value $f_{k+\varepsilon }$ (that will not be known until the future time $k+\varepsilon {\rm {)}}$ ).” In the symbol ${\tilde {f_{k}}}\left(\varepsilon \right)$ , the tilde stands for prediction error, the subscript k stands for the time at which the prediction occurs, and $\varepsilon$ stands for the prediction distance. That is, ${\tilde {f_{k}}}\left(\varepsilon \right)$ is the prediction error made in estimating the unknown future value $f_{k+\varepsilon }$ , the prediction being made at the present time k.

Let us now examine the prediction error ${\tilde {f_{k}}}(\varepsilon )$ in more detail. Suppose that a minimum-delay digital signal $f_{k}$ (Table 4, first row below the column headings) is the input to the prediction system

{\begin{aligned}E\left(Z\right)={\frac {{\rm {Z}}\left(f_{k+\varepsilon }\right)}{{\rm {Z}}\left(f_{k}\right)}}={\frac {f_{\varepsilon }+f_{\varepsilon {\rm {+l}}}Z+f_{\varepsilon +2}Z^{2}+\ldots }{f_{0}+f_{1}Z+f_{2}Z^{2}+\ldots }},\end{aligned}}

(153)

where Z denotes the one-sided Z-transform (equation 27). Note that $F\left(Z\right)=Z\left(f_{k}\right)$ . The advanced signal is $f_{k+\varepsilon }$ (Table 4, second row). The actual output ${\hat {f_{k}}}\left(\varepsilon \right)$ has the Z-transform given by

{\begin{aligned}F\left(Z\right)E\left(Z\right)=Z\left(f_{k}\right){\rm {\ }}{\frac {{\rm {Z}}\left(f_{k+\varepsilon }\right)}{{\rm {Z}}\left(f_{k}\right)}}=Z\left(f_{k+\varepsilon }\right).\end{aligned}}

(154)

Thus, the predicted value ${\hat {f_{k}}}\left(\varepsilon \right)$ (Table 4, third row) is the causal tailgate of the advanced input $f_{k+\varepsilon }$ ; that is, the predicted values are …, ${\hat {f}}_{-2}(\varepsilon )=0,\;{\hat {f}}_{-1}(\varepsilon )=0,\;{\hat {f}}_{0}(\varepsilon )=f_{\varepsilon },{\hat {f}}_{1}(\varepsilon )=f_{\varepsilon +1},{\hat {f}}_{\varepsilon +1},{\hat {f}}(\varepsilon )=f_{\varepsilon +2},\ldots .$ Hence, the prediction error ${\widetilde {f_{k}}}\left(\varepsilon \right)$ (Table 3, fourth row) is the anticausal front end of the advanced input $f_{k+\varepsilon }$ ; that is, the prediction error is $...{\tilde {f}}_{-\varepsilon -1}(\varepsilon )=0$ , ${\tilde {f}}_{-\varepsilon }(\varepsilon )=f_{0},{\tilde {f}}_{-\varepsilon +1}(\varepsilon )=f_{1},...,\;{\tilde {f}}_{-1}(\varepsilon )=f_{\varepsilon -1},\;{\tilde {f}}_{0}(\varepsilon )=0,{\tilde {f}}_{1}(\varepsilon )=0,\;{\tilde {f}}_{2}(\varepsilon )=0$ . Therefore, the error energy is the front-end energy of the minimum-delay input; that is,

{\begin{aligned}\sum _{k=-\varepsilon }^{-1}{f_{k+\varepsilon }^{2}}=f_{0}^{2}+f_{1}^{2}+\ldots +f_{\varepsilon -1}^{2}.\end{aligned}}

(155)

**Table 4.** Minimum-delay signal, advanced signal, predicted values, and prediction error.
Time k	$\varepsilon$	$\varepsilon +1$	...	–2	–1	0	1	2	...
Minimum-delay signal	0	0	...	0	0	$f_{0}$	$f_{1}$	$f_{2}$	...
Advanced signal	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/":): f_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/":): f_{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/":): f_{\varepsilon -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/":): f_{\varepsilon -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/":): f_{\varepsilon }	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/":): f_{\varepsilon +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/":): f_{\varepsilon +2}	...
Predicted values	0	0	...	0	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/":): f_{\varepsilon }	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/":): f_{\varepsilon +1}	$f_{\varepsilon +2}$	...
Prediction error	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/":): f_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/":): f_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/":): f_{\varepsilon -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/":): f_{\varepsilon -1}	0	0	0	...

This is the minimum possible error energy that can be obtained by a causal linear time-invariant prediction system because this error cannot be reduced.

Next let us convert the minimum-delay signal 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/":): f_k into the nonminimum-delay signal 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_k by passing 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/":): f_k through a nontrivial all-pass system 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_k ; 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/":): \begin{align} g_k\ =P_k*f_k. \end{align}

(156)

We also pass the predicted value ${\hat {f_{k}}}\left(\varepsilon \right)$ , as well as the prediction error 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/":): \tilde{f_k}\left(\varepsilon \right) , through the same all-pass system; that is, 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/":): \begin{align} \hat{g}\left(\varepsilon \right)\ =P_k*\hat{f}\left(\varepsilon \right) \ \\ \tilde{g}\left(\varepsilon \right)\ =P_k*\tilde{f}\left(\varepsilon \right) . \end{align}

(157)

Note 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/":): \hat{f_k}\left(\varepsilon \right) 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/":): \tilde{f_k}\left(\varepsilon \right) are, respectively, the causal tailgate and the anticausal front end of the advanced signal 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/":): f_{k+\varepsilon } . That is, the identity “desired output equals actual output plus error,” or

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} f_{k+\varepsilon }\ =\hat{f}\left(\varepsilon \right)+\tilde{f}\left(\varepsilon \right) , \end{align}

(158)

represents a clean split of 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/":): f_{k+\varepsilon } into its causal and anticausal parts. This clean split happens only in the case of a minimum-delay signal 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/":): f_k . Such a clean split does not happen 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/":): g_k . That is, in the identity

{\begin{aligned}g_{k+\varepsilon }={\hat {g}}_{k}(\varepsilon )+{\tilde {g}}_{k}(\varepsilon ),\end{aligned}}

(159)

the actual output 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/":): \hat{g}\left(\varepsilon \right) is causal, but the prediction error 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/":): \tilde{g}\left(\varepsilon \right) has both a causal component and an anticausal component. The anticausal component, of course, is the anticausal front end of the advanced signal 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_{k+\varepsilon } . This anticausal front end cannot be reached by a causal prediction system, so it must represent prediction error.

However, the causal tailgate of the advanced signal cannot be predicted perfectly, and as a result, the prediction error 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/":): \tilde{g}\left(\varepsilon \right) has a causal component also. However, the all-pass energy-delay theorem (equation 88) tells us that both prediction errors — that is, both 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/":): \tilde{f_k}\left(\varepsilon \right) 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/":): \tilde{g}\left(\varepsilon \right)- have the same energy.

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^{-1}_{k=-\varepsilon }{{\left[\tilde{f}\left(\varepsilon \right)\right]}^{2}} =\sum^{-1}_{k=-\varepsilon }{{\left[\tilde{g}\left(\varepsilon \right)\right]}^{2}}+\sum^{\infty }_{k=0}{{\left[\tilde{g}\left(\varepsilon \right)\right]}^{2}} \end{align}

(160)

Thus, we have the following prediction-system theorem (Robinson, 1963, theorem 10 (g), p. 206). Let 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_k be a stable causal signal with canonical representation 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_k=p_k*f_k, , where $p_{k}$ is all-pass 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/":): f_k is minimum delay. Then we can obtain the predicted value 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/":): \hat{g}\left(\varepsilon \right) for prediction distance 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/":): \varepsilon >0 by passing 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_k into the prediction system

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} E\left(Z\right)=\frac{{\rm Z}\left(f_{k+\varepsilon }\right)}{{\rm Z}\left(f_k\right)}. \end{align}

(161)

The predicted value is equal to the output of the all-pass system with the causal tailgate of 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/":): f_{k+\varepsilon } as input; 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/":): \begin{align} \hat{g}\left(\varepsilon \right)=\sum^{\infty }_{n=0}{f_{{\rm t+}{\mathcal E}} }p_{k-n}. \end{align}

(162)

The prediction error

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} \tilde{g}\left(\varepsilon \right)=g_{k+\varepsilon }-\hat{g}\left(\varepsilon \right) \end{align}

(163)

is equal to the output of the all-pass system with the input being the anticausal front end of $f_{k+\varepsilon }$ ; 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/":): \begin{align} \tilde{g}\left(\varepsilon \right)=\sum^{-1}_{n=-\varepsilon }{f_{n+\varepsilon P_{k-n}} }. \end{align}

(164)

The prediction-error energy is equal to the front-end energy of the minimum-delay signal; 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/":): \begin{align} \sum\limits_{k = - \varepsilon }^\infty {[\tilde g_k (\varepsilon )]^2 = \sum\limits_{k = - \varepsilon }^{ - 1} {[f_{k + \varepsilon } ]^2 = [f_0 ]^2 + [f_1 ]^2 + ... + [f_{\varepsilon - 1} )]^2 .} } \end{align}

(165)

As an example of the theorem, let us predict the stable causal signal

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} g_k\ =-a^{k+1}+ka^{k-1}\left(1-a^{2}\right) (\text{where}\ \textit{a}\; \text{is real and}\ k{ =0,1,2,}\ \dots ) \end{align}

(166)

for a prediction distance 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/":): \varepsilon = 1 . First, we find its Z-transform:

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} G\left(Z\right)=-\sum^{\infty }_{k=0}{a^{k+1}} Z^k+\sum^{\infty }_{k=1}{k}a^{k-1}\left(1-a^{2}\right)Z^k \ \\ =-a\sum^{\infty }_{k=0}{a^k}Z^k+\left(1-a^{2}\right)Z\sum^{\infty }_{k=1}{k}a^{k-1}Z^{\left(k-1\right)}\ \\ =-a\sum^{\infty }_{k=0}{a^k}Z^k+\left(1-a^{2}\right)Z\sum^{\infty }_{i=0}{\left(i+1\right)}a^iZ^j\ \\ =\frac{-a}{1-aZ}+\frac{\left(1-a^{2}\right)Z}{{\left(1-aZ\right)}^{2}}=\frac{-a+Z}{{\left(1-aZ\right)}^{2}}. \end{align}

(167)

Thus, we can write 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\left(Z\right) in the form

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} G\left(Z\right)= \frac{-a +Z 1}{1-aZ 1-aZ}, \end{align}

(168)

where the first factor is the all-pass system 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\left(Z\right) and the second factor is the minimum-delay system 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/":): F\left(Z\right) . Thus, the minimum-delay signal $f_{k}$ 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} f_k =a^k\;\;\; \text{(where}\ k=0,1,2, \dots ). \end{align}

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Therefore, the least-error-energy-predicting system 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} E\left(Z\right)=\frac{{\rm Z}\left(a^{k+1}\right)}{{\rm Z}\left(a^k\right)}=a, \end{align}

(170)

which is merely the constant a. Thus, the predicted value of 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_{k+1} is obtained by multiplying 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_k by the constant a; 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/":): \begin{align} \hat{g}\left(1\right)=ag_k=-a^{k+2}+k\; a^k\left(1-a^{2}\right) \text{(where}\ k=0,1,2, \dots ). \end{align}

(171)

The prediction error 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} \tilde{g}\left(1\right)=g_{k+1}\ \;\;\;\;\;\;\;\;\;\;\; \text{for}\ k=-1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ \tilde{g}\left(1\right)=g_{k+1}-ag_k\ \;\;\;\; \text{for}\ k=0, 1, 2,... . \;\;\;\;\;\;\;\;\;\;\; \end{align}

(172)

The two components are the front-end prediction error

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} g_{-1}\left(1\right)=g_0=-a\ \;\;\; \text{for}\ k=-1 \end{align}

(173)

and the tailgate prediction error

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} \tilde{g}\left(1\right)=\left[-a^{k+2}+\left(k+1\right)a^{\left(k+1\right)-1}\left(1-a^{2}\right)\right]-a\left[-a^{k+1}+ka^{k-1}\left(1-a^{2}\right)\right] \ \\ =\left(1-a^{2}\right)a^k\ \text{for}\ k=0, 1, 2,... . \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{align}

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The error energy is the sum of

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} \begin{array}{l} {\rm Front - end}\;{\rm error}\;{\rm energy}\; = ( - a)^2 = a^2 \\ {\rm Tailgate}\;{\rm error}\;{\rm energy}\;\;\;\,\;\;\; = \;(1 - a^2 )^2 [1 + a^2 + a^4 + ...] = (1 - a^2 )^2 \frac{1}{{1 - a^2 }} = 1 - a^2 . \\ \end{array} \end{align}

(175)

Thus, the total prediction-error energy is $a^{2}+\left({\rm {1\ }}-a^{2}\right)$ , which is unity.

As a check, let us verify the results of the foregoing example by using another approach. We now want to predict the minimum-delay signal 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/":): f_k and pass the result through the all-pass filter to predict 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_k . Again, we let 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/":): \varepsilon = 1 . The prediction of the minimum-delay signal 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} \hat{f}\left(1\right)=af_k\ \text{(for}\ k=0,1,2, \dots ). \end{align}

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The prediction error 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} \tilde{f}\left(1\right)=f_{k{\rm +l}} \mathrm \;\;\;\;\;\;\;\;\;\; {\rm for}\; k=-1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ \overline{f}\left(1\right)=f_{k+1}-af_k \mathrm \;\;\;{\rm for}\; k=0, 1, 2, \ldots. \;\;\;\;\;\;\;\;\;\;\;\;\; \end{align}

(177)

Respectively, the two components are the front-end prediction error (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/":): k=-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/":): \begin{align} \tilde f_{ - 1} (1) = f_0 = a^0 = 1 \end{align}

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and the tailgate prediction error (for k = 0, 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/":): \begin{align} \tilde{f_k}\left(1\right)=a^{k+1}-a\left(a^k\right)=0. \end{align}

(179)

This result confirms our knowledge that the tailgate prediction error of a minimum-delay signal is zero, and the entire prediction error is the front-end prediction error. The prediction 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/":): \hat g_k (1) of the signal ${\rm {g}}_{k}$ can be obtained by passing the prediction 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/":): \hat{f_k}\left(1\right) through the all-pass filter; 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/":): \begin{align} \hat{g}\left(1\right)=\hat{f_k}\left(1\right)*p_k \end{align}

(180)

or

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} \hat{g_k}\left(1\right)=a\left(a^k\right)*p_k. \end{align}

(181)

From equation 168, the all-pass system 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} P\left(Z\right)=\frac{-a+Z}{1-aZ} , \end{align}

(182)

which can be expanded 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/":): \begin{align} P\left(Z\right)=-a+\frac{\left(1-a^{2}\right)Z}{1-aZ} , \end{align}

(183)

which 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} P\left(Z\right)=-a+\left(1-a^{2}\right)Z\sum^{\infty }_{i=0}{a^i}Z^i=-a+\left(1-a^{2}\right)\sum^{\infty }_{k=1}{a^{k-1}} Z^k. \end{align}

(184)

Thus, the all-pass signal 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} p_k = - a\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm for}\;k = 0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ p_k = (1 - a^2 )a^{k - 1} \;\;\;{\rm for}\,\,k = 1,2,3,...\,. \\ \end{align}

(185)

Therefore, from equations 181 and 185, the prediction 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} \hat{g}\left(1\right)=a^{k{\rm +l}} *p_k=-a^{k+2}+ka^k\left(1-a^{2}\right) \mathrm{for} k=0, 1, 2,\ldots . \end{align}

(186)

Likewise, the prediction error ${\tilde {g}}\left(1\right)$ is obtained by passing the prediction error 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/":): \tilde{f_k}\left(1\right) through the all-pass operator; 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/":): \begin{align} \tilde{g}\left(1\right)=\tilde{f}\left(1\right)*p_k. \end{align}

(187)

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/":): \tilde{f_k}\left(1\right) consists of only the front-end component

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} \tilde{f}\left(1\right)=a^0{\rm \ }{\delta }_{k+1}={\delta }_{k+1} , \end{align}

(188)

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/":): \begin{align} \tilde{g_k}\left(1\right)=a^0{\rm \ }{\delta }_{k+1}*P_k, \end{align}

(189)

which 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} \tilde{g}\left(1\right)=p_{k{\rm +l}} . \end{align}

(190)

In this case, we see that the prediction error is given by the advanced all-pass signal. Thus, the front-end prediction error 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/":): \tilde{g}-1\left(1\right)=p_0=-a \mathrm \;{for}\; k=-1 , which is the same as equation 173. The tailgate prediction error 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/":): \tilde{g}\left(1\right)=\left({\rm 1\ }-a^{2}\right)a^k \mathrm\; {\rm for}\; k=0, 1 , 2, \ldots, , which is the same as equation 174. The total prediction-error energy is equal to the front-end prediction-error energy of the minimum-delay signal; 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/":): {\left[\tilde{f}\left(1\right)\right]}^{2}={\rm 1.} .

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