Causality and stability of analog systems

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
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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 transfer function of an analog system is the Laplace transform of the impulse-response function . A causal system is one for which is one sided — that is, for . Suppose that we look at the impulse response of a causal system as . If


(62)

we say that the system is stable.

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


(63)

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


(64)
Figure 8.  (a) For a stable causal analog signal, the region of convergence is a right half-plane that includes the axis. (b) For an unstable causal analog signal, the region of convergence is a right half-plane that does not include the axis.

This integral converges for . That is, the region of convergence is the half-plane to the right of the point a. Case A1 is for . The region of convergence includes the axis (Figure 8a), and the causal signal is stable. Case A2 is for . The region of convergence does not include the axis (Figure 8b), and the causal signal is unstable.

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


(65)

Its Laplace transform is


(66)
Figure 9.  (a) For an unstable anticausal analog signal, the region of convergence is a left half-plane that does not include the axis. (b) For a stable anticausal analog signal, the region of convergence is a left half-plane that includes the ' axis.

which converges for . That is, the region of convergence is the half-plane to the left of point a. Case B1 is for . The region of convergence includes the axis (Figure 9a), and the anticausal signal is unstable. Case B2 is for . The region of convergence includes the axis (Figure 9b), and the anticausal signal 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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