Analog transfer functions

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

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

${\begin{aligned}H\left(s\right)=\int \limits _{-\infty }^{\infty }{h}\left(t\right)e^{-st}dt\;\;\;\ \mathrm {where} \;\;\;\;s=\sigma +i\omega .\end{aligned}}$

(38)

The region of convergence is a vertical strip ${\sigma }_{1}<\sigma <{\sigma }_{2}$. Within this strip, H(s) has no poles or other singularities (Figure 5). For causal functions, we use the one-sided Laplace transform

${\begin{aligned}F\left(s\right)=\int \limits _{0}^{\infty }{f}\left(t\right)e^{-st}dt=1\left\{f\left(t\right)\right\}\mathrm {\;} \;\;\;{where}\;\;\;\;\;s=\sigma +i\omega .\end{aligned}}$

(39)

The region of convergence of F(s) is the half-plane $\sigma >{\sigma }_{1}$. The symbol L is used to denote the one-sided Laplace transform.

The analog ARMA system is given by the differential equation

${\begin{aligned}y^{\left(p\right)}+{\alpha }_{1}y^{\left(p-1\right)}+\ldots +{\alpha }_{p}y={\beta }_{\rm {o}}u^{\left(q\right)}+{\beta }_{1}u^{\left(q-1\right)}+\ldots +{\beta }_{q}u,\end{aligned}}$

(40)

where $y^{\left(n\right)}$ denotes $d^{n}y/dt^{n}$ and $u^{\left(n\right)}$ denotes $d^{n}y/dt^{n}$. Let us now assume that the input is causal — that is, $u\left(t\right)=0$ for t < 0 — so that the output also is causal. We also assume that the input and output have zero initial conditions; that is,

${\begin{aligned}u\left(0\right){\rm {=0,\ }}u^{\left(1\right)}\left(0\right){\rm {=0,\ }}u^{\left(2\right)}\left(0\right){\rm {=0,\ldots ,\ }}u^{\left(q-1\right)}\left(0\right)=0\\y\left(0\right){\rm {=0,\ }}y^{\left(1\right)}\left(0\right){\rm {=0,\ }}y^{\left(2\right)}\left(0\right){\rm {=0,\ldots ,\ }}y^{\left(p-1\right)}\left(0\right)=0.\end{aligned}}$

(41)

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

${\begin{aligned}\left(s^{p}+{\alpha }_{1}s^{p-1}+\ldots +{\alpha }_{p}\right){Y}\left(s\right)=\left({\beta }_{0}s^{q}+{\beta }_{1}s^{q-1}+\ldots +{\beta }_{q}\right)U\left(s\right),\end{aligned}}$

(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 $\delta \left(t\right)$, 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

${\begin{aligned}\left(s^{p}+{\alpha }_{1}s^{p-1}+\ldots +{\alpha }_{p}\right)H\left(s\right)\;\;\;\\=\left({\beta }_{0}s^{q}+{\beta }_{1}s^{q-1}+\ldots +{\beta }_{q}\right).\end{aligned}}$

(43)

If we define the polynomials $\alpha \left(s\right)$ and $\beta \left(s\right)$ as

${\begin{aligned}\alpha \left(s\right)=s^{p}+{\alpha }_{1}s^{p-1}+\ldots +{\alpha }_{p}\;\;\;\;\\\beta \left(s\right)={\beta }_{0}s^{q}+{\beta }_{1}s^{q-1}+\ldots +{\beta }_{q},\end{aligned}}$

(44)

Figure 5.  Vertical strip of convergence of a Laplace transform.

then we see that H(s) is

${\begin{aligned}H\left(s\right)={\frac {\beta \left(s\right)}{\alpha \left(s\right)}}.\end{aligned}}$

(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 $\alpha \left(s\right)$ and $\beta \left(s\right)$, we can write H(s) in its factored form as

${\begin{aligned}H\left(s\right)={\frac {{\beta }_{0}\left(s-b_{1}\right)\left(s-b_{2}\right)...\left(s-b_{q}\right)}{\left(s-a_{1}\right)\left(s-a_{2}\right)...\left(s-a_{p}\right)}}.\end{aligned}}$

(46)

The constants $a_{1}$, $a_{2}$, ..., $a_{p}$ are the poles of H(s), and the constants $b_{1}$, $b_{2}$, ..., $b_{q}$ are the zeros. In the case in which the ${\alpha }_{k}$ and $\beta _{k}$ coefficients are real, it follows that all complex $a_{i}$ and $b_{i}$ must occur in complex-conjugate pairs. Because s corresponds to differentiation, it follows that $s^{-1}$ corresponds to integration. Thus, we can write H(s) as

${\begin{aligned}H\left(s\right)={\frac {{\beta }_{0}+{\beta }_{1}s^{-1}+{\beta }_{2}s^{-2}+\ldots +{\beta }_{q}s^{-q}}{1+{\alpha }_{1}s^{-1}+{\alpha }_{2}s^{-2}+\ldots +{\alpha }_{p}s^{-p}}}s^{-\left(p-q\right)}\end{aligned}}$

(47)

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

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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
• 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

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