Digital prediction/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

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

{\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

{\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,

{\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

{\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

{\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

{\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

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

(105)

where

{\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

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} \mathrm{\;\;\; for\;} k=0, 1, 2,\ldots . \end{align}

(107)

That is, the AR(2) impulse response is a weighted average of two geometric signals with the weights 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/":): A_{2} given above. We want to feed 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 causal prediction 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/":): E\left(Z\right) and obtain as output 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 } . 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

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\right)=Z\left(A_{1}a^k_{1}+A_{2}a^k_{2}\right)=A_{1}{\rm Z}\left(a^k_{1}\right)+A_{2}{\rm Z}\left(a^k_{2}\right) . \end{align}

(108)

The output Z-transform 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} {\rm Z}\left(f_{k+\varepsilon }\right)={\rm Z}\left(A_{1}a^{k+\varepsilon }_{1}+A_{2}a^{k+\varepsilon }_{2}\right)=A_{1}a^{\varepsilon }_{1}{\rm Z}\left(a^k_{1}\right)+A_{2}a^{\varepsilon }_{2}{\rm Z}\left(a^k_{2}\right) . \end{align}

(109)

Thus, the required prediction 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{a^{\varepsilon }_{1}A_{1}{\rm Z}\left(a^k_{1}\right)+a^{\varepsilon }_{2}A_{2}{\rm Z}\left(a^k_{2}\right)}{A_{1}{\rm Z}\left(a^k_{1}\right)+A_{2}{\rm Z}\left(a^k_{2}\right)} , \end{align}

(110)

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} E\left(Z\right)=\frac{a^{\varepsilon }_{1}a_{1}{\left(a_{1}-a_{2}\right)}^{-1}{\left(1-a_{1}Z\right)}^{-1}+a^{\varepsilon }_{2}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{align}

(111)

If we simplify this expression, 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/":): \begin{align} E\left(Z\right)=\frac{a^{\varepsilon +1}_{1}-a^{\varepsilon +1}_{2}} {a_{1}-a_{2}}-a_{1}a_{2}\frac{a^{\varepsilon }_{1}-a^{\varepsilon }_{2}}{a_{1}-a_{2}} Z. \end{align}

(112)

This is the general expression for the prediction system with 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 . Therefore, for unit 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 , the prediction 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_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{align}

(113)

Because the relationship between the AR coefficients 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/":): \alpha _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/":): \alpha _2 and the roots 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/":): a_{2} 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} 1+{\alpha }_{1}Z+{\alpha }_{2}Z^{2}=\left(1-a_{1}Z\right)\left(1-a_{2}Z\right) , \end{align}

(114)

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} \alpha _1 = - (a_1 + a_2 ),\;\;\;\;\alpha _2 = a_1 a_2 . \end{align}

(115)

Thus, the prediction system 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/":): \varepsilon =1 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_1 (Z) = - \alpha _1 - \alpha _2 Z. \end{align}

(116)

If 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 the input (at time index k) to the prediction system (with 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 ), then we define 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) as the output (at the same time index k). The notation 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) should be read as “the predicted value (that is obtained at the present time k) of the future 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/":): f_{k+\varepsilon } (that will not be known until the future time 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 + \varepsilon ).” In other words, the positive integer 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 the prediction distance. In the symbol 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) , the caret stands for predicted value, the subscript k stands for the time at which the predicted value occurs, 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/":): \varepsilon stands for the prediction distance. 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/":): \hat{f}\left(\varepsilon \right) is the prediction of the future 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/":): {\rm value}f_{k+\varepsilon } , the prediction being made at the present time k. Because here 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 , 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/":): \begin{align} \hat{f_k}\left(\varepsilon \right)=\hat{f_k}\left(1\right)=\left(-{\alpha }_{1}-{\alpha }_{2}Z\right)f_k, \end{align}

(117)

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

(118)

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/":): f_k is causal, it is zero for negative k. Thus, this equation gives

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)=-{\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{align}

(119)

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

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

(120)

which generate 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 . Therefore, 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/":): \begin{align} \hat f_{k - 1} (1) = f_k , \end{align}

(121)

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

{\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

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. \end{align}

(123)

As an example, let us find the prediction system that (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/":): \varepsilon =1 ) predicts the impulse response of the AR(p) system 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/":): \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

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

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

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

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

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,

{\begin{aligned}{\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{aligned}}

(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

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 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 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

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

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

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

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,

{\begin{aligned}{\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{aligned}}

(136)

so the above system of equations becomes

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

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

(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

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

{\begin{aligned}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{aligned}}

(140)

The minimum-delay impulse response 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_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

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

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

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 , 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}|< 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

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

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

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

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

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

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

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.

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