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

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

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we say that the system is stable.

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

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

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

${\begin{aligned}H\left(s\right)=\int \limits _{0}^{\infty }{e^{-st}}e^{at}dt={\frac {1}{s-a}}.\end{aligned}}$

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Figure 8.  (a) For a stable causal analog signal, the region of convergence is a right half-plane that includes the $i\omega$ axis. (b) For an unstable causal analog signal, the region of convergence is a right half-plane that does not include the $i\omega$ axis.

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

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

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

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Its Laplace transform is

${\begin{aligned}H\left(s\right)=\int \limits _{-\propto }^{0}{e^{-st}}\left(-e^{a{\rm {t}}}\right)dt\;\;\;\;\;\;\;\\=-\int \limits _{0}^{\infty }{e^{\left(s-a\right)t}}dt={\frac {1}{s-a}},\end{aligned}}$

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Figure 9.  (a) For an unstable anticausal analog signal, the region of convergence is a left half-plane that does not include the $i\omega$ axis. (b) For a stable anticausal analog signal, the region of convergence is a left half-plane that includes the '$i\omega$ axis.

which converges for ${\rm {\ Re\ }}\left(s\right)<a$. That is, the region of convergence is the half-plane to the left of point a. Case B1 is for $a<0$. The region of convergence includes the $i\omega$ axis (Figure 9a), and the anticausal signal $h\left(t\right)$ is unstable. Case B2 is for $a>0$. The region of convergence includes the $i\omega$ axis (Figure 9b), and the anticausal signal $h\left(t\right)$ 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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Table of Contents (book)

Also in this chapter

• Introduction
• Digital linear time-invariant systems
• Analog linear time-invariant systems
• Digital transfer functions
• Analog transfer functions
• Causality and stability of digital systems
• Frequency response of a digital system
• Digital prediction
• Digital prediction error
• Analog prediction error

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