Analog transfer functions

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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 (or system function) of an analog system is defined as the Laplace transform of the impulse response. If h(t) is the impulse response, then the transfer function is the Laplace transform


(38)

The region of convergence is a vertical strip . Within this strip, H(s) has no poles or other singularities (Figure 5). For causal functions, we use the one-sided Laplace transform


(39)

The region of convergence of F(s) is the half-plane . The symbol L is used to denote the one-sided Laplace transform.

The analog ARMA system is given by the differential equation


(40)

where denotes and denotes . Let us now assume that the input is causal — that is, for t < 0 — so that the output also is causal. We also assume that the input and output have zero initial conditions; that is,


(41)

If we take the (one-sided) Laplace transform of the above differential equation, we obtain


(42)

where U(s) is the Laplace transform of the input and Y(s) is the Laplace transform of the output. In the case in which the input is the Dirac delta function , which has a Laplace transform equal to 1, then the output is the impulse response function h(t), with the Laplace transform H(s). In such a case, the above equation becomes


(43)

If we define the polynomials and as


(44)
Figure 5.  Vertical strip of convergence of a Laplace transform.

then we see that H(s) is


(45)

The function H(s), which is the Laplace transform of the impulse response h(t), is the transfer function. If we factor the polynomials and , we can write H(s) in its factored form as


(46)

The constants , , ..., are the poles of H(s), and the constants , , ..., are the zeros. In the case in which the and coefficients are real, it follows that all complex and must occur in complex-conjugate pairs. Because s corresponds to differentiation, it follows that corresponds to integration. Thus, we can write H(s) as


(47)

when we want to implement the system by means of integrating circuits.


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