# Causality and stability of analog systems

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 15 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The transfer function ${\displaystyle H\left(s\right)}$ of an analog system is the Laplace transform of the impulse-response function ${\displaystyle h\left(t\right)}$. A causal system is one for which ${\displaystyle h\left(t\right)}$ is one sided — that is, ${\displaystyle h\left(t\right)=0}$ for ${\displaystyle t<0}$. Suppose that we look at the impulse response of a causal system as ${\displaystyle t\to \infty }$. If

 {\displaystyle {\begin{aligned}\int \limits _{0}^{\infty }{|}h\left(t\right){|}\ dt<\infty ,\end{aligned}}} (62)

we say that the system is stable.

In Example A, the prototype causal analog signal is the causal exponential

 {\displaystyle {\begin{aligned}h\left(t\right)=0\mathrm {\;\;\;for\ } t{\rm {<0,}}\;\;\;h\left(t\right)=e^{at}\mathrm {\;\;\;for\ } t{\rm {>0.}}\end{aligned}}} (63)

For our purposes, we assume that the parameter a is a real number. The Laplace transform is

 {\displaystyle {\begin{aligned}H\left(s\right)=\int \limits _{0}^{\infty }{e^{-st}}e^{at}dt={\frac {1}{s-a}}.\end{aligned}}} (64)
Figure 8.  (a) For a stable causal analog signal, the region of convergence is a right half-plane that includes the ${\displaystyle i\omega }$ axis. (b) For an unstable causal analog signal, the region of convergence is a right half-plane that does not include the ${\displaystyle i\omega }$ axis.

This integral converges for ${\displaystyle {\rm {\ Re\ }}\left(s\right)>a}$. That is, the region of convergence is the half-plane to the right of the point a. Case A1 is for ${\displaystyle {\rm {|a|<0}}}$. The region of convergence includes the ${\displaystyle i\omega }$ axis (Figure 8a), and the causal signal ${\displaystyle h\left(t\right)}$ is stable. Case A2 is for ${\displaystyle a>0}$. The region of convergence does not include the ${\displaystyle i\omega }$ axis (Figure 8b), and the causal signal ${\displaystyle h\left(t\right)}$ is unstable.

In Example B, the prototype anticausal analog signal is the anticausal exponential

 {\displaystyle {\begin{aligned}h\left(t\right)=-e^{at}\mathrm {\;\;\;\;for\;\;\;} t{\rm {<0,}}\\h\left(t\right)=0\mathrm {\;\;\;\;for\;\;\;} t\geq 0.\end{aligned}}} (65)

Its Laplace transform is

 {\displaystyle {\begin{aligned}H\left(s\right)=\int \limits _{-\propto }^{0}{e^{-st}}\left(-e^{a{\rm {t}}}\right)dt\;\;\;\;\;\;\;\\=-\int \limits _{0}^{\infty }{e^{\left(s-a\right)t}}dt={\frac {1}{s-a}},\end{aligned}}} (66)
Figure 9.  (a) For an unstable anticausal analog signal, the region of convergence is a left half-plane that does not include the ${\displaystyle i\omega }$ axis. (b) For a stable anticausal analog signal, the region of convergence is a left half-plane that includes the '${\displaystyle i\omega }$ axis.

which converges for ${\displaystyle {\rm {\ Re\ }}\left(s\right). That is, the region of convergence is the half-plane to the left of point a. Case B1 is for ${\displaystyle a<0}$. The region of convergence includes the ${\displaystyle i\omega }$ axis (Figure 9a), and the anticausal signal ${\displaystyle h\left(t\right)}$ is unstable. Case B2 is for ${\displaystyle a>0}$. The region of convergence includes the ${\displaystyle i\omega }$ axis (Figure 9b), and the anticausal signal ${\displaystyle h\left(t\right)}$ is stable. As is the case with digital signals, we usually would choose stability at the expense of causality in the case of analog signals.

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