Digital 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

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In one key place, the conventions used in signal processing by geophysicists differ from those used by electrical engineers. At the outset, it is good to point out the differences so as to make the literature more accessible. Geophysicists and electrical engineers have different conventions with respect to the z-transform. Let $h_{0}$ , $h_{1}$ , $h_{2}$ , ... be the impulse response of a causal time-invariant linear filter. The engineering z-transform is

{\begin{aligned}{H}_{\rm {engineenng}}\left(z\right)=h_{0}+h_{1}z^{-1}+h_{2}z^{-2}+\ldots ,\end{aligned}}

(24)

whereas the geophysics Z-transform is

{\begin{aligned}H\left(Z\right)=h_{0}+h_{1}Z+h_{1}Z^{2}+\ldots .\end{aligned}}

(25)

To distinguish between the two, we use a capital Z in the geophysics Z-transform and a lowercase z in the engineering z-transform. Throughout this book, we use mostly the geophysics Z-transform. The two transforms are related thusly: $Z=z^{-1}$ . Whereas the engineering z represents a unit advance operator, the geophysics Z represents a unit-delay operator. The common mathematical terminology for the geophysics Z-transform is generating function.

The transfer function (or system function) of a digital system is defined as the Z-transform of the impulse response. If $h_{k}$ is the impulse response, the transfer function is

{\begin{aligned}H\left(Z\right)=\sum _{k=-\infty }^{\infty }{h_{k}}Z^{k}.\end{aligned}}

(26)

Figure 4. Annular region of convergence of a Laurent series.

The Z-transform establishes a correspondence between the signal $h_{k}$ and the transfer function H(Z). This expression for H(Z) is in the form of a Laurent series (i.e., a power series that can involve negative as well as positive powers of Z). As we know from the theory of functions, the region of convergence of the Laurent series is an annular region (i.e., a ring-shaped region) $r_{1}<|Z|<r_{2}$ , whose inner radius $r_{1}$ and outer radius $r_{2}$ depend on the behavior of the signal as $k\to -\infty$ and as $k\to \infty$ , respectively. Within this ring, the Laurent series has no poles or other singularities (Figure 4). The Z-transform given above is called a two-sided Z-transform because it involves a signal with both negative and positive values of the time index k.

Causal signals are zero for negative values of k, and for such signals, we can use the one-sided Z-transform

{\begin{aligned}F\left(Z\right)=\sum _{k=0}^{\infty }{f_{k}}Z^{k}=Z\left(f_{k}\right).\end{aligned}}

(27)

In this case, the radius of the inner ring is zero, so the region of convergence is the interior ${\rm {|Z|<}}r$ of a circle. The one-sided Z-transform is in the form of a series that involves only nonnegative powers of Z. The symbol Z denotes the parameter in the Z-transform, whereas the symbol Z denotes the one-sided Z-transform itself.

The digital ARMA(p, q) model

{\begin{aligned}y_{k}+{\alpha }_{1}y_{k-1}+\ldots +{\alpha }_{p}y_{k-p}={\beta }_{0}u_{k}\\+{\beta }_{1}u_{k-1}+\ldots +{\beta }_{q}u_{k-q}\end{aligned}}

(28)

represents a recursion relationship between the input signal $u_{k}$ and the output signal $y_{k}$ . If the input signal is causal, then the output signal also must be causal. The recursion relations are

{\begin{array}{l}y_{o}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{0}u_{0}\;\\y_{1}+\alpha _{1}y_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{0}u_{1}+\beta _{1}u_{0}\\y_{2}+\alpha _{1}y_{1}+\alpha _{2}y_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{0}u_{1}+\beta _{1}u_{0}\\...\\y_{k}+\alpha _{1}y_{k-1}+\alpha _{2}y_{k-2}+...+\alpha _{p}y_{k-p}\;\;\;\;\;\;\;\;=\beta _{0}u_{k}+\beta _{1}u_{k-1}+...+\beta _{q}u_{k-q}.\\...\\\end{array}}

(29)

Given the input signal, the first equation determines $y_{0}$ . The next equation can be used to find $y_{1}$ . The third equation then can be solved for $y_{2}$ . By continuing in this way, equation by equation, we can find all values of the output signal. We recall that the impulse response $h_{k}$ is the output resulting from a spike input. Thus we let the input be the Kronecker delta signal ${\delta }_{k}$ . The recursion relations (in the case in which p > q) are

{\begin{array}{l}h_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{0}\\h_{1}+\alpha _{1}h_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{1}\\h_{2}+\alpha _{1}h_{1}\alpha _{2}h_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{2}\\...\\h_{q}+\alpha _{1}h_{q-1}+...+\alpha _{q}h_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{q}\\h_{q+1}+\alpha _{1}h_{q}+...+\alpha _{q}h_{1}+\alpha _{q+1}h_{0}\;\;\;\;\;\;\;\;=0\;\;\;\;\;\;\;\;\;\;\\...\\h_{p}+\alpha _{1}h_{p-1}+...+\alpha _{p}h_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0\\...\\h_{k}+\alpha _{1}h_{k-1}+...+\alpha _{p}h_{k-p}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0\;\;\;{\rm {for}}\;\;k>p.\\\end{array}}

(30)

Solving equation by equation, we can obtain the impulse response $h_{k}$ .

Alternatively, let us multiply the first equation by $Z^{\rm {0}}$ , the second by $Z^{1}$ , the third by ${\rm {Z}}^{2}$ , and so on. We obtain

{\begin{array}{l}h_{0}Z^{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{0}Z^{0}\\h_{1}Z^{1}+\alpha _{0}h_{0}Z^{1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{1}N^{1}\\h_{2}Z^{2}+\alpha _{1}h_{1}Z^{2}+\alpha _{2}h_{0}Z^{2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{2}Z^{2}\\...\\h_{q}Z^{q}+\alpha _{1}h_{q-1}Z^{q}+...+\alpha _{q}h_{0}Z^{q}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\beta _{q}Z^{q}\\h_{q+1}Z^{q+1}+\alpha _{1}h_{q}Z^{q+1}+...+\alpha _{q}h_{1}Z^{q+1}+\alpha _{q+1}h_{0}Z^{q+1}\;\;\;\;\;=0\\...\\h_{p}Z^{p}+\alpha _{1}h_{p-1}Z^{p}+...+\alpha _{p}h_{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0\\...\\h_{k}Z^{k}+\alpha _{1}h_{k-1}Z^{k}+...+\alpha _{p}h_{k-p}Z^{k}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0\;\;\;{\rm {for}}\;k>p.\\\end{array}}

(31)

We now add these equations, column by column, to obtain

{\begin{aligned}H\left(Z\right)+{\alpha }_{1}ZH\left(Z\right)+{\alpha }_{2}Z^{2}H\left(Z\right)+\ldots +{\alpha }_{p}Z^{p}H\left(Z\right)=B\left(Z\right),\end{aligned}}

(32)

where

{\begin{aligned}H\left(Z\right)=\sum _{k=0}^{\infty }{h_{k}}Z^{k},\;\;\;\;\;\;\;\;B\left(Z\right)=\sum _{k=0}^{q}{{\beta }_{k}}Z^{k}.\end{aligned}}

(33)

We recognize H(Z) and B(Z) as the Z-transforms of the impulse response $h_{k}$ and the feedforward coefficients ${\beta }_{k}$ , respectively. If we factor out H(Z), we have

{\begin{aligned}\left(1+{\alpha }_{1}Z+{\alpha }_{2}Z^{2}+\ldots +{\alpha }_{p}Z^{p}\right)H\left(Z\right)=B\left(Z\right).\end{aligned}}

(34)

We recognize the expression in parentheses as the Z-transform A(Z) of the feedback coefficients ${\alpha }_{k}$ ; that is,

{\begin{aligned}1+{\alpha }_{1}Z+{\alpha }_{2}Z^{2}+\ldots +{\alpha }_{p}Z^{p}=A\left(Z\right).\end{aligned}}

(35)

Thus, we have

{\begin{aligned}H\left(Z\right)={\frac {B\left(Z\right)}{A\left(Z\right)}}={\frac {{\beta }_{0}+{\beta }_{1}Z+{\beta }_{2}Z^{2}+\ldots +{\beta }_{q}Z^{q}}{1+{\alpha }_{1}Z+{\alpha }_{2}Z^{2}+\ldots +{\alpha }_{p}Z^{P}}}.\end{aligned}}

(36)

The function H(Z), which is the Z-transform of the impulse response, is the transfer function. The Z-transforms A(Z) and B(Z) are polynomials. We can write H(Z) in factored form as

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

(37)

The constants $a_{{1}^{-1}}$ , $a_{{2}^{-1}}$ , ..., $a_{{\rm {p}}^{-1}}$ are the poles, and the constants $b_{{1}^{-1}}$ , $b_{{2}^{-1}}$ , ..., $b_{{\rm {q}}^{-1}}$ are the zeros of H(Z). In the case in which the ${\alpha }_{k}$ and ${\beta }_{k}$ coefficients are real, all of the complex $a_{i}^{-1}$ and $b_{i}^{-1}$ values occur in complex-conjugate pairs. Because an AR(p) system has only poles, it also is called an all-pole system. Likewise, an MA(q) system is called an all-zero system.