Digital linear time-invariant systems

**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> A discrete-time 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/":): x_k is a signal of (real or complex) numbers defined for every integer k. The index k represents the set of discrete, equally spaced time points. A discrete time signal often is called a digital signal. It is customary to denote time spacing between the discrete time points 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/":): \Delta t . Actual time t is related to the index k by the equation 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/":): t=k\Delta t . Usually, time is measured in seconds.

An important type of digital signal is the impulse function. The discrete impulse function 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/":): {\delta }_k is defined as the signal made up entirely of zeros except for a value of one at time index 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=0 . 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/":): {\delta }_k is 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} {\delta }_k=1 \mathrm{\;\;\;\; for\;\;\;\; } k=0\\ {\delta }_k=0 \mathrm{\;\;\;\; for\;\;\;\; } k\ne 0. \end{align}

(1)

The impulse function 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/":): {\delta }_k is also called the Kronecker delta function. A more popular name for the delta function is the unit spike.

A digital system is a discrete time system that can be represented by a rule that transforms a 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/":): u_k into another 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/":): y_k . The 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/":): u_k is called the input, and the 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/":): y_k is the output. This relationship is indicated by the notation

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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=S\left(u_k\right) , \end{align}

(2)

where S denotes the digital system. A system L is linear if

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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} L\left(au_k+bv_k\right)=aL\left(u_k\right)+bL\left(v_k\right). \end{align}

(3)

for any constants a and b and for any digital signals 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/":): u_k 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/":): v_k . In particular, 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/":): y_k is the output of a linear system for input 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/":): u_k , then 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/":): ay_k is the output for input 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/":): au_k . That is, if we amplify the input of a linear system by a constant factor, the output also is amplified by this same constant factor. In addition, 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/":): y_k 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/":): z_k are the respective outputs of a linear system for the inputs 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/":): u_k 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/":): v_k , then 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/":): y_k+z_k is the output for input 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/":): u_k+v_k . That is, if we add the inputs of a linear system, the outputs also are added.

A digital system $y_{k}=S\left(u_{k}\right)$ is said to be time invariant 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/":): y_{k-n}=S\left(u_{k-n}\right) for any integer n. That is, a system is time invariant if a shift in the input produces the same shift in the output.

Let us give some examples. We begin with the delay line 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/":): y_k=u_{k-n} , where n is a constant. The delay line 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/":): y_k=u_{k-n} is linear and time invariant. The rectifier 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/":): y_k=|u_k| is nonlinear and time invariant. The amplifier 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/":): y_k=k^{2}u_k is linear and time varying. It is time varying because (1) a shift in the input 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/":): u_k produces the 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/":): L\left(u_{k-n}\right)=k^{2}u_{k-n;}\left(2\right) a shift in the 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/":): y_k=k^{2}u_k produces 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/":): y_{k-n}={\left(k-n\right)}^{2}u_{k-n} ; and (3) the two outputs are not the same.

A 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/":): u_k is one-sided, or causal, 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/":): u_k=0 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<0 . A 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/":): u_k is anticausal 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/":): u_k=0 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\ge 0 . Finally, a 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/":): u_k is noncausal if it has an anticausal component.

A system is said to be causal if a causal input yields a causal output. That is, a causal system is a system with this property: 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/":): u_k=0 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<0 , then 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/":): y_k=0 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<0 . If the system represents a physical phenomenon operating in real time, then the system must be causal. However, if the system represents an analysis of past recordings (such as a seismic record) so that k represents nominal time (i.e., time marks on the record), then the system need not be causal.

A linear time-invariant system is characterized by its impulse response. 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/":): h_k is defined as the output resulting from a spike input 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/":): {\delta }_k ; that 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} h_k=L\left({\delta }_k\right) . \end{align}

(4)

Because the impulse 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/":): {\delta }_k is causal, we see that a causal system has a causal impulse response, whereas a noncausal system has a noncausal impulse response. Let us now express, in terms 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/":): h_k , the 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/":): y_k of a linear time-invariant system that has an arbitrary input $u_{k}$ . Because of time invariance, the output resulting from the input 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/":): {\delta }_{k-n} 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/":): h_{k-n} for any n; that 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} L\left({\delta }_{k-n}\right)=h_{k-n}. \end{align}

(5)

Because of linearity, the output resulting from the input 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/":): u_n{\delta }_{k-n} 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/":): u_nh_{k-n} . The input 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/":): u_k can be written as

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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} u_k=\sum^{\infty }_{n=-\infty }{u_n}{\delta }_{k-n}. \end{align}

(6)

The output resulting from this input 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} y_k=L\left(u_k\right)=L\left(\sum^{\infty }_{n=-\infty }{u_n}{\delta }_{k-n}\right)=\sum^{\infty }_{n=-\propto }{u_n}L\left({\delta }_{k-n}\right) , \end{align}

(7)

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} y_k=\sum^{\infty }_{n=-\infty }{u_n}h_{k-n}. \end{align}

(8)

The expression on the right is a convolution. It represents the convolution 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/":): u_k 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/":): h_k and is denoted 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/":): u_k*h_k . Convolution is commutative, so we can write the input-output relationship as

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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=u_k*h_k=\sum^{\infty }_{n=-\infty }{u_n}h_{k-n}=\sum^{\infty }_{n=-\infty }{h_n}u_{k-n}. \end{align}

(9)

The most important class of linear time-invariant digital systems is the class represented by a finite-difference equation (with constant coefficients) of the form

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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}={\beta }_0u_k+{\beta }_{1}u_{k-1}+\ldots +{\beta }_qy_{k-q}. \end{align}

(10)

This difference equation represents an ARMA(p,q) system. In the case of q = 0 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/":): \beta_0=1 , it reduces to the AR(p) 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+{\alpha }_{1}y_{k-1}+\ldots +{\alpha }_py_{k-p}=u_k. \end{align}

(11)

On the other hand, if p = 0, the ARMA(p,q) system reduces to the MA(q) 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={\beta }_0u_k+{\beta }_{1}u_{k-1}+\ldots +{\beta }_qu_{k-q}. \end{align}

(12)

Most of our work will deal with the three difference equations 10, 11, and 12. Let us now make some remarks about these three equations. In the form in which they are written above, we want index k to represent the present instant of time, 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/":): u_k would be the present input 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/":): y_k the present output. As a result, the other input values 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/":): u_{k-1} , $u_{k-2},\ldots ,u_{k-q}$ represent past values, and the other output values 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/":): y_{k - 1} ,\;y_{k - 2} ,\; \ldots ,y_{k - p} represent past values. Thus, the above difference equations, so interpreted, involve only present and past values of the input and output and no future values. Under this interpretation, the systems represented by equations 10, 11, and 12 are causal — that is, they can be implemented in real time. Conceptually, the MA system is the easiest one to visualize.

We now want to introduce the backward-shift (or unit-delay) operator Z, which is defined 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/":): Zx_k=x_{k-1} for any 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/":): x_k . In terms of the backward-shift operator, the MA(q) equation 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} y_k={\beta }_0u_k+{\beta }_{1}Zu_k+{\beta }_{2}Z^{2}u_k+\ldots +{\beta }_qZ^qu_k \end{align}

(13)

or

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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=}\left({\beta }_0+{\beta }_{1}Z+{\beta }_{2}Z^{2}+\ldots +{\beta }_qZ^q\right)u_k. \end{align}

(14)

A diagram for this system is shown in Figure 1. In such diagrams, a branch point is indicated by a solid circle. At a branch point, the signal that goes out on each branch is the same as the signal that enters the branch point. A small open circle represents a summation point.

**Figure 1.** Digital MA(q) (moving-average) system.

At a summation point, all the incoming signals are added to produce the outgoing signal.

In Figure 1, the forward direction (i.e., the direction from input to output) is from left to right. Because all of the arrows associated with the constant multipliers 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/":): {\beta }_0,{\beta }_{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/":): {\beta }_{2}...,\beta_{q} are in the forward direction, an MA system is seen to be a feedforward system. In a feedforward system, the present 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/":): y_k depends on the present and past values 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/":): u_k , 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/":): u_{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/":): u_{k-2} , ... of the input. In the case of an MA system, only a finite number q of past values are involved.

Next, let us construct a diagram for an AR(p) system, which we can write as

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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{array}{l} (1 + \alpha _1 Z + \alpha _2 Z^2 + ... \\ \;\;\;\; + \alpha _p Z^p )y_k = u_k . \\ \end{array}

(15)

**Figure 2.** Digital AR(p) (autoregressive) system.

A diagram for this system is shown in Figure 2. Because all the arrows associated with the multipliers 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} , ${\alpha }_{2},...,p$ are in the backward direction, an AR system is seen to be a (pure) feedback system. In a feedback system, the present input 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/":): u_k can be expressed in terms of the present and past values 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/":): y_k , 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/":): y_{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/":): y_{k-2} , ... of the output. In the case of an AR system, only a finite number pof past values are involved.

Finally, let us construct a diagram of the ARMA(p, q) 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} \left(1+{\alpha }_{1}Z+{\alpha }_{2}Z^{2}+\dots +{\alpha }_pZ^p\right)y_k=\left({\beta }_0+{\beta }_{1}Z+{\beta }_{2}Z^{2}+\dots +{\beta }_qZ^q\right)u_k. \end{align}

(16)

An ARMA(p, q) system can be regarded as the combination in series of an AR(p) system and an MA(q) system. A diagram for an ARMA system is shown in Figure 3.