Analog prediction error

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

Because the same situation is obtained in the analog case as in the digital case, let us merely give examples.

The minimum-delay signal

${\begin{aligned}f\left(t\right)={2}^{\rm {1/2}}e^{-{\rm {t}}}\mathrm {\;} \;{\rm {for}}\;\;t\geq 0\end{aligned}}$

(191)

and the causal nonminimum-delay signal

${\begin{aligned}g\left(t\right)={2}^{\rm {l/2}}e^{-t}\left(1-2t\right)\mathrm {\;} \;{\rm {for}}\;t\geq 0\end{aligned}}$

(192)

have the same magnitude spectrum. The prediction filter for $\varepsilon =1$ is

${\begin{aligned}E\left(s\right)={\frac {{\rm {L}}\left\{f\left(t{\rm {+l}}\right)\right\}}{{\rm {L}}\left\{f\left(t\right)\right\}}}=e^{-1}.\end{aligned}}$

(193)

We recall that L denotes the one-sided Laplace transform (equation 39). First let us consider the minimum-delay signal $f\left(t\right)$. The prediction is obtained by multiplying the signal by $e^{-1}$; that is,

${\begin{aligned}{\hat {f}}\left(t|\varepsilon \right)={\hat {f}}\left(t{\rm {|1}}\right)=f\left(t\right)e^{-1}={2}^{\rm {l/2}}e^{-t}e^{-1}={2}^{\rm {1/2}}e^{-\left(t+1\right)}\mathrm {\;} \;{\rm {for}}\;\;t\geq 0,\end{aligned}}$

(194)

which exactly reproduces the tailgate of the advanced signal $f\left(t{\rm {+\ 1}}\right)$. Thus, the prediction error is given by the front end of the advanced signal; that is,

${\begin{aligned}{\tilde {f}}\left(t|\varepsilon \right)={\tilde {f}}\left(t{\rm {|1}}\right)=f\left(t{\rm {+l}}\right)={2}^{{\rm {1/}}2}e^{-\left(t+1\right)}\mathrm {\;} \;{for}\;-1\leq t<0.\end{aligned}}$

(195)

The prediction-error energy is the front-end energy

${\begin{aligned}\int \limits _{-1}^{0}{f^{2}}\left(t+1\right)dt=\int \limits _{0}^{1}{f^{2}}\left(t\right)dt=\int \limits _{0}^{1}{2}e^{-2t}dt=\int \limits _{0}^{2}{e^{-{\rm {u}}}}du=1-e^{-2}.\end{aligned}}$

(196)

Next let us consider the nonminimum-delay signal $g\left(t\right)$. The prediction is again obtained by multiplying the signal by $e^{-1}$; that is,

${\begin{aligned}{\hat {g}}\left(t|\varepsilon \right)={\hat {g}}\left(t{\rm {|1}}\right)=g\left(t\right)e^{-1}={2}^{\rm {1/2}}e^{-\left(t+1\right)}\left(1-2t\right)\mathrm {\;} \;{\rm {for}}\;t\geq 0.\end{aligned}}$

(197)

This prediction does not exactly reproduce the tailgate of the advanced signal $\left(t{\rm {+\ 1}}\right)$. Thus, the prediction error ${\tilde {g}}\left(t{\rm {|1}}\right)$ is made up of two components. One component is the front-end error

${\begin{aligned}{\tilde {g}}\left(t{\rm {|1}}\right)=g\left(t{\rm {+l}}\right)={2}^{\rm {1/2}}e^{-\left({\rm {t+1}}\right)}\left[1-2\left(t{\rm {+l}}\right)\right]\mathrm {\;} \;{\rm {for}}\;-1\leq t<0,\end{aligned}}$

(198)

and the other component is the tailgate error

${\begin{aligned}{\tilde {g}}(t|1)=g(t+1)-g(t)e^{-1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\\;\;=2^{1/2}e^{-(t+1)}[1-2(t+1)]-2^{1/2}e^{-(t+1)}[1-2t]\;\;\;\;\\\;\;=-2^{3/2}e^{-(t+1)}\;\;\;\;{\rm {for}}\;\;t\geq 0.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\\end{aligned}}$

(199)

The front-end-error energy is

${\begin{aligned}\int \limits _{-1}^{0}{{\left[{\tilde {g}}\left(t{\rm {|1}}\right)\right]}^{2}}dt=\int \limits _{-1}^{0}{{\left[g\left(t+1\right)\right]}^{2}}dt=\int _{0}^{1}{{\left[g\left(t\right)\right]}^{2}}dt\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\=\int \limits _{0}^{1}{2}e^{-2t}{\left(1-2t\right)}^{2}dt=\int \limits _{0}^{2}{e^{-{\rm {u}}}}{\left(1-u\right)}^{2}du.\end{aligned}}$

(200)

Using tables of integrals, we obtain the front-end-error energy as

${\begin{aligned}\int \limits _{-1}^{0}{{\left[{\tilde {g}}\left(t{\rm {|1}}\right)\right]}^{2}}dt=1-5e^{-2}.\end{aligned}}$

(201)

The tailgate-error energy is

${\begin{aligned}\int \limits _{0}^{\infty }{{\left[{\tilde {g}}\left(t{\rm {|1}}\right)\right]}^{2}}dt=\int \limits _{0}^{\infty }{\rm {8}}e^{-2\left({\rm {t+1}}\right)dt}{\rm {=4}}\int \limits _{2}^{\infty }{e^{-u}}du{\rm {=4}}e^{-2}.\end{aligned}}$

(202)

Thus, the total prediction-error energy is

${\begin{aligned}\int \limits _{-1}^{\infty }{{\left[{\tilde {g}}\left(t{\rm {|1}}\right)\right]}^{2}}dt=1-5e^{-2}{\rm {+4}}e^{-2}=1-e^{-2},\end{aligned}}$

(203)

which is the same as equation 196 for the case of the minimum-delay signal, as we would expect.

Alternatively, ${\hat {g}}\left({\rm {t|1}}\right)$ for $t\geq 0$ can be obtained by passing ${\hat {f}}\left({\rm {t|1}}\right)$ through the all-pass system. Because

${\begin{aligned}F\left(s\right)={\rm {L}}\left\{{2}^{\rm {1/2}}e^{-t}\right\}={2}^{\rm {1/2}}{\left(s+1\right)}^{-1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\G\left(s\right)={\rm {L}}\left\{{2}^{\rm {1/2}}e^{-t}\left(1-2t\right)\right\}={2}^{\rm {l/2}}\left(s-1\right){\left(s+1\right)}^{-2},\end{aligned}}$

(204)

the all-pass system is

${\begin{aligned}P\left(s\right)=G\left(s\right)F^{-1}\left(s\right)=\left(s-1\right){\left(s+1\right)}^{-1}.\end{aligned}}$

(205)

If we expand in partial fractions, we have

${\begin{aligned}P\left(s\right)={\rm {l}}-2{\left(s{\rm {+l}}\right)}^{-1},\end{aligned}}$

(206)

so the all-pass impulse response is the causal function

${\begin{aligned}p\left(t\right)=\delta \left(t\right)-2e^{-t}\mathrm {\;} \;{\rm {for}}\;t\geq 0.\end{aligned}}$

(207)

Thus, the prediction of the nonminimum-delay signal is

${\begin{aligned}{\hat {g}}\left(t{\rm {|1}}\right)=p\left(t\right)*{\hat {f}}\left(t{\rm {|1}}\right)\mathrm {\;} \;{\rm {for}}\;t\geq 0,\end{aligned}}$

(208)

which is

${\begin{aligned}{\hat {g}}\left(t{\rm {|1}}\right)=\left[\delta \left(t\right)-2e^{-t}\right]*\left[{2}^{\rm {1/2}}e^{-\left(f+1\right)}\right]\;\;\;\;\;\;\;\;\;\;\\={2}^{\rm {l/2}}e^{-\left(t{\rm {+l}}\right)}-{2}^{\rm {3/2}}{\rm {\ }}\int _{0}^{t}{e^{-\tau }}e^{-t+\tau -1}d\tau \;\;\;\\={2}^{\rm {1/2}}e^{-\left(t{\rm {+l}}\right)}\left(1-2t\right)\mathrm {\;} {\rm {for}}\;t\geq 0.\;\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}}$

(209)

This is the same result we obtained before (equation 197).

Likewise, ${\tilde {g}}\left(t{\rm {|1}}\right)$ can be obtained by passing ${\tilde {f}}\left(t{\rm {|1}}\right)$ through $p\left(t\right)$. The error ${\tilde {f}}\left(t{\rm {|1}}\right)$ is nonzero only in the range $-\varepsilon \leq t<0$, whereas the error ${\tilde {g}}\left(t{\rm {|1}}\right)$ is nonzero in the range $-\varepsilon \leq t<\infty$. First we will evaluate ${\tilde {g}}\left(t{\rm {|1}}\right)$ in the subrange $-\varepsilon \leq t<0$ and then in the subrange $0\leq t<\infty$. We have

${\begin{aligned}{\tilde {g}}\left(t{\rm {|1}}\right)={\tilde {f}}\left(t{\rm {|l}}\right)*p\left(t\right){\rm {=\ }}\int \limits _{-1}^{t}{{2}^{\rm {1/2}}}{\rm {\ }}e^{-\left(\tau {\rm {+l}}\right)}\left[\delta \left(t-\tau \right)-2e^{-\left(t\tau \right)}\right]d\tau \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\={2}^{\rm {1/2}}e^{-\left(t+1\right)}\left(1-2\right)=-{2}^{\rm {l/2}}e^{-\left(t{\rm {+l}}\right)}\mathrm {\;} \;{\rm {for}}\;-\varepsilon \leq t<0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}}$

(210)

Figure 12.  (a) Minimum-delay wavelet. (b) Nonminimum-delay wavelet. The desired output (which is the given wavelet advanced by one time unit) is shown in (c) for the minimum-delay wavelet and in (d) for the nonminimum-delay wavelet. The actual output (which is the predicted wavelet) is shown in (e) for the minimum-delay wavelet and in (f) for the nonminimum-delay wavelet. The prediction error is shown in (g) for the minimum-delay wavelet and in (h) for the nonminimum-delay wavelet. (Note: Each curve on the right can be obtained from the corresponding curve on the left by means of the same all-pass filter.)

whereas

${\begin{aligned}{\tilde {g}}(t|1)=\int \limits _{-1}^{0}{2^{1/2}e^{-(\tau +1)}[\delta (t-\tau )-2e^{-(t-\tau )}]\;d\tau \;\;\;{\rm {for}}\;\;{\rm {0}}\;\leq t<\infty .}\end{aligned}}$

(211)

In this last integral, the spike of the delta function does not lie within the integration range $-1\leq \tau <0$, so the contribution resulting from the delta function is zero. Thus,

${\begin{aligned}{\tilde {g}}(t|1)=\int \limits _{-1}^{0}{2^{1/2}e^{-(\tau +1)}(-2)e^{-(t{\rm {-}}\tau )}}d\tau =-2^{3/2}e^{-(t+1)}\;\;\;{\rm {for}}\;{\rm {0}}\leq {\rm {t<}}\infty {\rm {.}}\end{aligned}}$

(212)

We see that we have obtained the same prediction error that we did before (equation 199).

The examples we have just treated are illustrated in Figure 12, which is similar to Figures 9.81 and 9.82 in Robinson (1962). The minimum-delay case is on the left. Note how the prediction system clearly splits the desired output (i.e., the advanced input) into two parts so that one part is the prediction error (anticausal) and the other part is the prediction (causal). The nonminimum-delay case is on the right. Each curve on the right can be obtained from the corresponding curve on the left by means of the all-pass filter.

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Also in this chapter

• Introduction
• Digital linear time-invariant systems
• Analog linear time-invariant systems
• Digital transfer functions
• Analog transfer functions
• Causality and stability of digital systems
• Causality and stability of analog systems
• Frequency response of a digital system
• Digital prediction
• Digital prediction error

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