# Analog linear time-invariant systems

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A continuous-time signal f(t) is a (real or complex) function that is defined for every real number t. The number t represents continuous time. A continuous-time signal also is called an analog signal. An important type of analog signal is the impulse function ${\displaystyle \delta \left(t\right)}$. This function plays the same role as does the discrete impulse function ${\displaystyle {\delta }_{k}}$, but it is not as easy to define. The impulse function ${\displaystyle \delta \left(t\right)}$, which also is known as the Dirac delta function, can be regarded as a generalized function. A generalized function cannot be defined as an isolated entity but instead must be viewed as the limit of a family of functions. Any family of functions ${\displaystyle {\delta }_{\varepsilon }\left(t\right)}$ with the following properties can be used to define the delta function ${\displaystyle \delta \left(t\right)}$. Here, ${\displaystyle \varepsilon }$ is a parameter that characterizes the functions ${\displaystyle {\delta }_{\varepsilon }\left(t\right)}$. The properties are

 {\displaystyle {\begin{aligned}\int \limits _{-\infty }^{\infty }{{\delta }_{\varepsilon }}\left(t\right)dt{=1,\ }{\mathop {\lim } _{\varepsilon \rightarrow \infty }\ }\int \limits _{\infty }^{-\infty }{{\delta }_{\varepsilon }}\left(t\right)f\left(t\right)dt=f\left(0\right),\end{aligned}}} (17)

where f(t) is any function that is continuous at the origin t = 0. As a result, we can interpret ${\displaystyle \delta \left(t\right)}$ as being a function that satisfies the identity

 {\displaystyle {\begin{aligned}\int \limits _{-\infty }^{\infty }{\delta }\left(t\right)f\left(t\right)dt=f\left(0\right).\end{aligned}}} (18)

All of the formal properties of ${\displaystyle \delta \left(t\right)}$ can be obtained from this identity.

An analog system is a continuous-time system that can be represented by a rule that transforms an analog signal u(t) into another analog signal y(t). The signal u(t) is called the input, and the signal y(t) is called the output. This input-output relationship is shown by

 {\displaystyle {\begin{aligned}y\left(t\right)=S\left(u\left(t\right)\right),\end{aligned}}} (19)

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

 {\displaystyle {\begin{aligned}L\left(au\left(t\right)+bv\left(t\right)\right)=aL\left(u\left(t\right)\right)+bL\left(v\left(t\right)\right)\end{aligned}}} (20)

for any constants a and b and for any signals u(t) and v(t). A system ${\displaystyle y\left(t\right)=S\left(u\left(t\right)\right)}$ is time invariant if

 {\displaystyle {\begin{aligned}y\left(t-t_{0}\right)=S\left(u\left(t-t_{0}\right)\right)\end{aligned}}} (21)

for any real time ${\displaystyle t_{0}.}$.

Let us now give some examples. The delay line ${\displaystyle y\left(t\right)=u\left(t-a\right)}$, where the delay a is constant, is linear and time invariant. The rectifier ${\displaystyle y\left(t\right)=|u\left(t\right)|}$ is nonlinear and time invariant. The amplifier ${\displaystyle y\left(t\right)=t^{2}u\left(t\right)}$ is linear and time varying.

An analog signal is said to be causal if the signal vanishes for time ${\displaystyle t<0}$ or anticausal if the signal vanishes for time ${\displaystyle t\geq 0}$. A noncausal signal is one with an anticausal component. An analog system is causal if a causal input produces a causal output. All physical, real-time systems are causal because they cannot respond to any input that has not yet occurred. From now on, we will be concerned with linear time-invariant systems. Generally, we will mean a causal system unless we specify otherwise.

The impulse-response function represents an important characterization of a linear time-invariant system. The impulse response ${\displaystyle h\left(t\right)}$ is defined as the output of the system to a Dirac delta function input ${\displaystyle \delta \left(t\right)}$; that is, ${\displaystyle h\left(t\right)=L\left(\delta \left(t\right)\right)}$. Because we regard the Dirac delta function as a causal signal, we see that we can distinguish a causal system from a noncausal system by the impulse response. That is, a system is causal if and only if h(t) is one sided (or causal).

As is the case with digital systems, the input-output relationship for an analog system is performed by the convolution operation. However, now the convolution is an integral. If u(t) is the input to an analog filter with impulse response h(t), the output is given by the convolution integral

 {\displaystyle {\begin{aligned}y\left(t\right)=u\left(t\right)*h\left(t\right)=\int \limits _{-\infty }^{\infty }{u}\left(\tau \right)h\left(t-\tau \right)d\tau =\int \limits _{-\infty }^{\infty }{h}\left(\tau \right)u\left(t-\tau \right)d\tau .\end{aligned}}} (22)

The most important class of linear, time-invariant analog systems is the class of ARMA(p,q) systems. Such a system is given by a differential equation (with constant coefficients) of the form

 {\displaystyle {\begin{aligned}y^{\left(p\right)}+{\alpha }_{1}y^{\left(p-1\right)}+\ldots +{\alpha }_{p}y={\beta }_{0}u^{\left(q\right)}+{\beta }_{1}u^{\left(q-1\right)}+\ldots +{\beta }_{q}u.\end{aligned}}} (23)

Here, superscripts in parentheses represent time derivatives; for example, ${\displaystyle y^{\left(n\right)}=d^{n}y/dt^{n}}$. If q = 0, the system reduces to an AR(p) system. On the other hand, if p = 0, the system reduces to an MA(q) system.