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
Store	SEG Online Store

<translate> The transfer function $H\left(s\right)$ of an analog system is the Laplace transform of the impulse-response 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/":): h\left(t\right) . A causal system is one for which 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\left(t\right) is one sided — 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/":): h\left(t\right)=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/":): t<0 . Suppose that we look at the impulse response of a causal system 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/":): t\to \infty . 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} \int\limits^{\infty }_0{|}h\left(t\right){|}\ dt<\infty , \end{align}

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we say that the system is stable.

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

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

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For our purposes, we assume that the parameter a is a real number. The Laplace transform 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\left(s\right)=\int\limits^{\infty }_0{e^{-st}} e^{at}dt=\frac{1}{s-a}. \end{align}

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This integral converges 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/":): {\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 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/":): {\rm |a|<0} . The region of convergence includes the 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/":): i\omega axis (Figure 8a), and the causal 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/":): h\left(t\right) is stable. Case A2 is 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/":): a > 0 . The region of convergence does not include the 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/":): i\omega axis (Figure 8b), and the causal 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/":): h\left(t\right) is unstable.

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

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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\left(t\right)=-e^{at} \mathrm{\;\;\;\; for\;\;\;} t{\rm <0,} \\ h\left(t\right)=0 \mathrm{\;\;\;\; for\;\;\; } t\ge 0. \end{align}

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Its Laplace transform 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\left(s\right)=\int\limits^0_{-\propto }{e^{-st}} \left(-e^{a{\rm t}}\right)dt \;\;\;\;\;\;\; \\ =-\int\limits^{\infty }_0{e^{\left(s-a\right)t}}dt=\frac{1}{s-a}, \end{align}

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which converges 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/":): {\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 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/":): a<0 . The region of convergence includes the 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/":): i\omega axis (Figure 9a), and the anticausal 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/":): h\left(t\right) is unstable. Case B2 is 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/":): a>0 . The region of convergence includes the 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/":): 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.