Digital prediction

**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

<translate> The prototype minimum-delay digital signal $f_{k}$ is the causal damped geometric signal $f_{k}=a^{k}$ for k = 0, 1, 2, …, where $$ |a|<1 $$ (Table 3, first row below the column headings). We want to let this signal be the input to a causal, linear time-invariant system, which we call the extrapolation (or prediction) system E(Z). For the desired output, we choose the advanced signal (say, for the advance $\varepsilon =2)f_{k+2}=a^{k+2}$ for k = ..., –3, –2, –1, 0, 1, 2, … (Table 3, second row). The front end $a^{0}$ , $a^{1}$ (Table 3, third row) of the advanced signal is anticausal, whereas the tailgate $a^{2},\ a^{3},\ a^{4},\ a^{5},\ a^{6},\ldots$ (Table 3, fourth row) is causal. Because the prediction system is causal, it cannot reach the anticausal front end; thus, it can predict only the causal tailgate. For the tailgate, we demand perfect prediction. We will allow no prediction error whatsoever.

**Table 3.** Minimum-delay signal, advanced signal, front end of advanced signal, and tailgate of advanced signal.
Time k	...	–2	–1	0	1	2	3	4	...
Minimum-delay signal $f_{k}$	...	0	0	1	a	$a^{2}$	$a^{3}$	$a^{4}$	...
Advanced signal $f_{k+2}$	...	1	a	$a^{2}$	$a^{3}$	$a^{4}$	$a^{5}$	$a^{6}$	...
Front end of $f_{k+2}$	...	1	a	0	0	0	0	0	...
Tailgate of $f_{k+2}$	...	0	0	$a^{2}$	$a^{3}$	$a^{4}$	$a^{5}$	$a^{6}$	...

The problem is to find an expression for the causal prediction filter $E\left(Z\right)$ . A naive person would proceed in this way. Both the input and the desired output (i.e., the tailgate) are causal, so we can use the one-sided Z-transform. We recall that the one-sided Z-transform is denoted by Z (equation 27). The one-sided Z-transform of the input is

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{\begin{aligned}{\bf {Z}}(f_{k})=\sum \limits _{k=0}^{\infty }{a^{k}Z^{k}={\frac {1}{1-aZ}},}\end{aligned}}

(99)

whereas the one-sided Z-transform of the tailgate of the desired output is

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{\begin{aligned}{\rm {Z}}\left(f_{k+\varepsilon }\right)=\sum _{k=0}^{\infty }{a^{k+\varepsilon }}Z^{k}=a^{\varepsilon }{\rm {\ }}\sum _{k=0}^{\infty }{a^{k}}Z^{k}={\frac {a^{\varepsilon }}{1-aZ}}.\end{aligned}}

(100)

The prediction distance, or advance ( $\varepsilon$ ), is always a positive integer. The required prediction filter has a transfer function given by the ratio of the output Z-transform over the input Z-transform; that is,

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{\begin{aligned}E\left(Z\right)={\frac {{\rm {Z}}\left(f_{k+\varepsilon }\right)}{{\rm {Z}}\left(f_{k}\right)}}={\frac {a^{\varepsilon }{\left(1-aZ\right)}^{-1}}{{\left(1-aZ\right)}^{-1}}}=a^{\varepsilon }.\end{aligned}}

(101)

This formula is correct, because if we multiply the input signal $a^{k}$ by $a^{\varepsilon }$ , we indeed get the advanced value $a^{k+\varepsilon }$ . A little thought tells us that everywhere the geometric signal has the same shape, so by using a constant attenuation $a^{\varepsilon }$ , we can change the present shape $a^{k}$ into the future shape $a^{k+\varepsilon }$ . (Note: Two curves are said to have the same shape if one is a constant factor times the other.)

The formula $E\left(Z\right)=Z\left(f_{k+\varepsilon }\right){\rm {/Z}}\left(f_{k}\right)$ certainly works in the case in which $f_{k}$ is a geometric signal. Why did we go to the bother of defining minimum delay? One reason is that this formula for $E\left(Z\right)$ works not only for a geometric signal but also for any minimum-delay signal. In fact, we can say this: Perfect prediction in the sense defined above is possible if and only if the input signal is minimum delay. For a minimum-delay signal, $E\left(Z\right)$ gives the causal prediction system for prediction distance $\varepsilon$ . The prediction of a nonminimum-delay signal will be treated in the next section.

Let us look at the AR(2) system

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{\begin{aligned}y_{k}+{\alpha }_{1}y_{k-1}+{\alpha }_{2}y_{k-2}=u_{k}.\end{aligned}}

(102)

We want to find the prediction system that predicts the impulse response $f_{k}$ . First of all, we know that the impulse response of an AR system is necessarily minimum delay. The impulse response satisfies the difference equation

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{\begin{aligned}f_{k}+{\alpha }_{1}f_{k-1}+{\alpha }_{2}f_{k-2}={\delta }_{k}\mathrm {\;\;for\;} k=0,1,2,\ldots .\end{aligned}}

(103)

The Z-transform of the impulse response is the transfer function

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{\begin{aligned}F\left(Z\right)={\frac {1}{{\rm {l+}}{\alpha }_{1}Z+{\alpha }_{2}Z^{2}}}={\frac {1}{\left(1-a_{1}Z\right)\left(1-a_{2}Z\right)}},\end{aligned}}

(104)

where the poles $a_{1}^{-1}$ and $a_{2}^{-1}$ lie outside the unit circle; that is, $|a_{1}|<1$ and $|a_{2}|<1$ . The impulse response $f_{k}$ can be found by expanding $F\left(Z\right)$ in partial fractions. We have

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{\begin{aligned}F\left(z\right)={\frac {A_{1}}{1-a_{1}Z}}+{\frac {A_{2}}{1-a_{2}Z}},\end{aligned}}

(105)

where

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{\begin{aligned}A_{1}={\frac {a_{1}}{a_{1}-a_{2}}},\;\;\;\;A_{2}={\frac {a_{2}}{a_{2}-a_{1}}}.\end{aligned}}

(106)

Thus, the impulse response is the causal minimum-delay signal

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{\begin{aligned}f_{k}=A_{1}a_{1}^{k}+A_{2}a_{2}^{k}\mathrm {\;\;\;for\;} k=0,1,2,\ldots .\end{aligned}}

(107)

That is, the AR(2) impulse response is a weighted average of two geometric signals with the weights $A_{1}$ and $A_{2}$ given above. We want to feed $f_{k}$ into the causal prediction filter $E\left(Z\right)$ and obtain as output the advanced signal $f_{k+\varepsilon }$ . As we stated above, E(Z) is the ratio of the one-sided Z-transform of the output over the one-sided Z-transform of the input. Of course, we know that the input Z-transform is F(Z). We can obtain symmetry in our expression for E(Z) if we write the input Z-transform F(Z) as

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{\begin{aligned}{\rm {Z}}\left(f_{k}\right)=Z\left(A_{1}a_{1}^{k}+A_{2}a_{2}^{k}\right)=A_{1}{\rm {Z}}\left(a_{1}^{k}\right)+A_{2}{\rm {Z}}\left(a_{2}^{k}\right).\end{aligned}}

(108)

The output Z-transform is

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{\begin{aligned}{\rm {Z}}\left(f_{k+\varepsilon }\right)={\rm {Z}}\left(A_{1}a_{1}^{k+\varepsilon }+A_{2}a_{2}^{k+\varepsilon }\right)=A_{1}a_{1}^{\varepsilon }{\rm {Z}}\left(a_{1}^{k}\right)+A_{2}a_{2}^{\varepsilon }{\rm {Z}}\left(a_{2}^{k}\right).\end{aligned}}

(109)

Thus, the required prediction system is

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{\begin{aligned}E\left(Z\right)={\frac {a_{1}^{\varepsilon }A_{1}{\rm {Z}}\left(a_{1}^{k}\right)+a_{2}^{\varepsilon }A_{2}{\rm {Z}}\left(a_{2}^{k}\right)}{A_{1}{\rm {Z}}\left(a_{1}^{k}\right)+A_{2}{\rm {Z}}\left(a_{2}^{k}\right)}},\end{aligned}}

(110)

which is

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{\begin{aligned}E\left(Z\right)={\frac {a_{1}^{\varepsilon }a_{1}{\left(a_{1}-a_{2}\right)}^{-1}{\left(1-a_{1}Z\right)}^{-1}+a_{2}^{\varepsilon }a_{2}{\left(a_{2}-a_{1}\right)}^{-1}{\left(1-a_{2}Z\right)}^{-1}}{a_{1}{\left(a_{1}-a_{2}\right)}^{-1}{\left(1-a_{1}Z\right)}^{-1}+a_{2}{\left(1-a_{1}\right)}^{-1}{\left(1-a_{2}Z\right)}^{-1}}}.\end{aligned}}

(111)

If we simplify this expression, we obtain

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{\begin{aligned}E\left(Z\right)={\frac {a_{1}^{\varepsilon +1}-a_{2}^{\varepsilon +1}}{a_{1}-a_{2}}}-a_{1}a_{2}{\frac {a_{1}^{\varepsilon }-a_{2}^{\varepsilon }}{a_{1}-a_{2}}}Z.\end{aligned}}

(112)

This is the general expression for the prediction system with prediction distance $\varepsilon$ . Therefore, for unit prediction distance $\varepsilon =1$ , the prediction system is

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{\begin{aligned}E_{1}(Z)={\frac {a_{1}^{2}-a_{2}^{2}}{a_{1}-a_{2}}}-a_{1}a_{2}Z=a_{1}+a_{2}-a_{1}a_{2}Z.\end{aligned}}

(113)

Because the relationship between the AR coefficients $\alpha _{1}$ and $\alpha _{2}$ and the roots $a_{1}$ and $a_{2}$ is

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{\begin{aligned}1+{\alpha }_{1}Z+{\alpha }_{2}Z^{2}=\left(1-a_{1}Z\right)\left(1-a_{2}Z\right),\end{aligned}}

(114)

we have

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{\begin{aligned}\alpha _{1}=-(a_{1}+a_{2}),\;\;\;\;\alpha _{2}=a_{1}a_{2}.\end{aligned}}

(115)

Thus, the prediction system for $\varepsilon =1$ is

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{\begin{aligned}E_{1}(Z)=-\alpha _{1}-\alpha _{2}Z.\end{aligned}}

(116)

If $f_{k}$ is the input (at time index k) to the prediction system (with prediction distance $\varepsilon$ ), then we define ${\hat {f_{k}}}\left(\varepsilon \right)$ as the output (at the same time index k). The notation ${\hat {f_{k}}}\left(\varepsilon \right)$ should be read as “the predicted value (that is obtained at the present time k) of the future value $f_{k+\varepsilon }$ (that will not be known until the future time $k+\varepsilon$ ).” In other words, the positive integer $\varepsilon$ is the prediction distance. In the symbol ${\hat {f_{k}}}\left(\varepsilon \right)$ , the caret stands for predicted value, the subscript k stands for the time at which the predicted value occurs, and $\varepsilon$ stands for the prediction distance. That is, ${\hat {f}}\left(\varepsilon \right)$ is the prediction of the future value ${\rm {value}}f_{k+\varepsilon }$ , the prediction being made at the present time k. Because here $\varepsilon =1$ , we can write

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{\begin{aligned}{\hat {f_{k}}}\left(\varepsilon \right)={\hat {f_{k}}}\left(1\right)=\left(-{\alpha }_{1}-{\alpha }_{2}Z\right)f_{k},\end{aligned}}

(117)

which is

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{\begin{aligned}{\hat {f_{k}}}\left(1\right)=-{\alpha }_{1}f_{k}-{\alpha }_{2}f_{k-1}.\end{aligned}}

(118)

Because $f_{k}$ is causal, it is zero for negative k. Thus, this equation gives

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{\begin{aligned}{\hat {f}}\left(1\right)=-{\alpha }_{1}f_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\{\hat {f}}\left(1\right)=-{\alpha }_{1}f_{1}-{\alpha }_{2}f_{0}\;\;\\{\hat {f}}\left(1\right)=-{\alpha }_{1}f_{2}-{\alpha }_{2}f_{1}.\\\ldots .\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}}

(119)

With the initial condition $f_{0}={\delta }_{0}=1$ , the above equations 119 are equivalent to the equations

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{\begin{aligned}f_{k}+{\alpha }_{1}f_{k-1}+{\alpha }_{2}f_{k-2}={\delta }_{k}\mathrm {\;\;for\;} k=0,1,2,\ldots ,\end{aligned}}

(120)

which generate the impulse response $f_{k}$ . Therefore, it follows that

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{\begin{aligned}{\hat {f}}_{k-1}(1)=f_{k},\end{aligned}}

(121)

which shows that the prediction system perfectly predicts the tailgate $f_{1}$ , $f_{2},f_{3}$ , ... of the impulse response. In a nutshell, we can find the prediction system with $\varepsilon =1$ for the impulse response of an AR(2) system by inspection. If the AR(2) system is

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{\begin{aligned}y_{k}+{\alpha }_{1}y_{k-1}+{\alpha }_{2}y_{k-2},\end{aligned}}

(122)

then the prediction system is the MA(1) system given by

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{\begin{aligned}E_{1}\left(Z\right)=-{\alpha }_{1}-{\alpha }_{2}Z.\end{aligned}}

(123)

As an example, let us find the prediction system that (for $\varepsilon =1$ ) predicts the impulse response of the AR(p) system given by

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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_k+{\alpha }_{1}y_{k-1}+\ldots +{\alpha }_py_{k-p}=u_k. \end{align}

(124)

By inspection, the prediction system is

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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_{1}\left(Z\right)=-{\alpha }_{1}-{\alpha }_{2}Z-\ldots -{\alpha }_pZ^p. \end{align}

(125)

Let us look at the minimum-delay MA(1) system

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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\left(Z\right)={\rm Z} \left(f_k\right)=1-bZ\;\;\; (\text{where}\ |b{\rm |<l}). \end{align}

(126)

By inspection, we see that the causal impulse response is

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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={\delta }_k-b{\delta }_{k-1}\;\;\; \text{for}\ k=0, 1, 2,..., \end{align}

(127)

which in longhand is

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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_0{\rm =1\ ,\ }f_{1}=-b, \ f_{2}{\rm \ =0,\ }f_{3}{\rm \ =0,\ldots} . \end{align}

(128)

Thus

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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} {\rm Z}\left(f_{k+1}\right)=f_{1}+f_{2}Z+f_{3}Z^{2}{\rm +\ldots\ =}-b\ \\ {\rm Z}\left(f_{k+2}\right)=f_{2}+f_{3}Z+f_{4}Z^{2}{\rm +\ldots\ =0},\;\;\; \end{align}

(129)

and in general,

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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} {\rm Z}\left(f_{k+\varepsilon }\right)=f_{\varepsilon }+f_{\varepsilon +1}Z+f_{\varepsilon +2}Z^{2}+\dots {\rm =\ 0\;\; for\;}\varepsilon {\rm >l}. \end{align}

(130)

Thus, the prediction system 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 is

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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_{\varepsilon }\left(Z\right)=\frac{{\rm Z}\left(f_{k+\varepsilon }\right)}{{\rm Z}\left(f_k\right)}= \frac{-b}{1-bZ}\;\; \text{for}\; \varepsilon =1 \;\; {\rm and} =0 \;\; {\rm for}\; \varepsilon\; {\rm 2}, 3, 4,... . \end{align}

(131)

So if the prediction distance $\varepsilon$ is greater than one, only the trivial 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}\left(\varepsilon \right)= 0 can be obtained. The one-step prediction system is

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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_{1}\left(z\right)=\frac{-b}{1-bZ}=-b\left(1+bZ+b^{2}Z^{2}+b^{3}Z^{3}+\dots \right) . \end{align}

(132)

Therefore, the prediction is

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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)=\left(-b-b^{2}Z-b^{3}Z^{2}-b^{4}Z^{3}{\rm +\ldots }\right){\rm \ }f_k , \end{align}

(133)

which is

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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)=-bf_k-b^{2}f_{k-1}-b^{3}f_{k-2}-{\rm \ }b^{4}f_{k-3}-\ldots . \end{align}

(134)

Because f(k) is causal, we have the system of equations

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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)=-bf_0 \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\; \\ \hat{f}\left(1\right)=-bf_{1}-b^{2}f_0\ \;\;\;\;\;\;\;\;\; \\ \hat{f}\left(1\right)=-bf_{2}-b^{2}f_{1}-b^{3}f_0 \ \\ ... \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\; \end{align}

(135)

Because the impulse response 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, perfect prediction is obtained. Thus,

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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_0}\left(1\right)=f_{1}\ , \hat{f}\left(1\right)=f_{2}\ , \hat{f_2}\left(1\right)=f_{3},\ldots, \end{align}

(136)

so the above system of equations becomes

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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_{1}=-bf_0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ f_{2}=-bf_{1}-b^{2}f_0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ f_{3}=-bf_{2}-b^{2}f_{1}-b^{3}f_0\;\;\;\;\;\; \\ ... \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{align}

(137)

With the initial condition 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= 1 , we find

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{\begin{aligned}f_{1}=-b\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\f_{2}=-b\left(-b\right)-b^{2}=0\;\;\;\;\;\;\;\;\;\\f_{3}=-b^{2}\left(-b\right)-b^{3}=0\;\;\;\;\;\;\;\\...\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{aligned}}

(138)

which indeed is the impulse response of the given MA(1) system.

Let us look next at the digital minimum-delay ARMA(1,1) system

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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_k-ay_{k-1}=x_k-bx_{k-1} , \end{align}

(139)

so 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/":): |a|< 1 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/":): |b|< 1 . The transfer function is

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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\left(z\right)={\rm Z}\left\{f_k\right\}=\frac{1-bZ}{1-aZ}=1+\frac{\left(a-b\right)Z}{1-aZ}. \end{align}

(140)

The minimum-delay impulse response is

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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_0{\rm =1,\ }f_{1}=a-b, \ f_{2}=\left(a-b\right)a, \ f_{3}=\left(a-b\right)a^{2}{\rm \ ,\ldots ,\ }f_{k+\varepsilon }=\left(a-b\right)a^{k+\varepsilon -1},\ldots, \end{align}

(141)

so

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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} {\rm Z}\left(f_{k+\varepsilon }\right)=\left(a-b\right)a^{\varepsilon -1}\sum^{\infty }_{k=0}{a^k}Z^k=\left(a-b\right)a^{\varepsilon -1}{\left(1-aZ\right)}^{-1}. \end{align}

(142)

Thus, the prediction system is

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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{1-aZ}{1-bZ}{\rm \ }\frac{a-b}{1-aZ}{\rm \ }a^{\varepsilon -1}=\frac{a-b}{1-bZ}a^{\varepsilon -1}, \end{align}

(143)

which is an AR(1) system.

Next let us consider the minimum-delay ARMA(2,1) system

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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\left(Z\right)={\rm Z}\left(f_k\right)=\frac{1-bZ}{\left(1-a_{1}Z\right)\left(1-a_{2}Z\right)}, \end{align}

(144)

where 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/":): |b|< 1 , $|a_{1}|<1$ , 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/":): |a_{2}|< 1 . The partial fraction expansion is

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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\left(Z\right)=\frac{A-{1}} {1-a_{1}Z}+\frac{A_{2}}{1-a_{2}Z}, \end{align}

(145)

where

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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} A_1 = (a_1 - b)(a_1 - a_2 )^{ - 1} ,\;\;\;\;\;A_2 = (a_2 - b)(a_2 - a_1 )^{ - 1} . \end{align}

(146)

The expansion of the fractions gives

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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(z) = A_1 \sum\limits_{k = 0}^\infty {a_1^k Z^k + A_2 \sum\limits_{k = 0}^\infty {a_2^k Z^k ,} } \end{align}

(147)

so the impulse response is

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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_{1}a^k_{1}+A_{2}a^k_{2}. \end{align}

(148)

The advanced impulse response is

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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 }=A_{1}a^{k+\varepsilon }_{1}+A_{2}a^{k+\varepsilon }_{2}. \end{align}

(149)

Thus, the prediction system is

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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} E(Z) = \frac{{{\bf Z}(f_{k + \varepsilon } )}} {{{\bf Z}(f_k )}} = \frac{{a_1^\varepsilon A_1 {\bf Z}(a_1^k ) + a_2^\varepsilon A_2 {\bf Z}(a_2^k )}}{{A_1 {\bf Z}(a_1^k ) + A_2 {\bf Z}(a_2^k )}}, \\ \;\;\;\;\;\;\;\; = \frac{{a_1^\varepsilon (a_1 - b)(a_1 - a_2 )^{ - 1} (1 - a_1 Z)^{ - 1} + a_2^\varepsilon (a_2 - b)(a_2 - a_1 )^{ - 1} (1 - a_2 Z)^{ - 1} }}{{(a_1 - b)(a_1 - a_2 )^{ - 1} (1 - a_1 Z)^{ - 1} + (a_2 - b)(a_2 - a_1 )^{ - 1} (1 - a_2 Z)^{ - 1} }} \\ \end{array} \end{align}

(150)

which, simplified, is

</translate> <translate>

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{a^{\varepsilon }_{1}\left(a_{1}-b\right)\left(1-a_{2}Z\right)-a^{\varepsilon }_{2}\left(a_{2}-b\right)\left(1-a_{1}Z\right)}{\left(a_{1}-a_{2}\right)\left(1-bZ\right)}. \end{align}

(151)

We see that the prediction system is an ARMA(1,1) system.