# Analog transfer functions

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

**(**)

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

**(**)

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

**(**)

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,

**(**)

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

**(**)

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

**(**)

If we define the polynomials and as

**(**)

then we see that *H*(*s*) is

**(**)

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

**(**)

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

**(**)

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

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## Also in this chapter

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