# Z-transform

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 7 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

How do the definitions of the Z-transform differ? Geophysicists and electrical engineers have different conventions with respect to the z-transform (see also the discussion in Chapter 6). Let $\displaystyle h_0, h_{{\rm l}}, h_{2}, \dots$ be the impulse response of a causal time-invariant linear filter. The engineering z-transform (with lowercase z) is

 \displaystyle \begin{align} H_{\text{engineering}} \;(z) = h_0 + h_1 \;z^{ - 1} + h_2 \;z^{ - 2} + ..., \end{align} (16)

whereas the geophysics Z-transform (with capital Z) is the generating function

 \displaystyle \begin{align} H\left(Z\right)=h_0+h_{{\rm l}} Z+h_{1}Z^{2}+\dots \end{align} (17)

The two are related by $\displaystyle Z=z^{-1}$ . Whereas the engineering z represents a unit advance operator, the geophysics Z represents a unit delay operator.

Table 1 gives the engineering z-transforms of some common signals.

By letting $\displaystyle Z=z^{-1}$ , Table 1 becomes Table 2 for the corresponding geophysical Z-transforms.

How is the Fourier transform obtained from the Z-transform? The Fourier transform (electrical engineering convention) of a causal signal $\displaystyle h_n$ in terms of angular frequency $\displaystyle \omega$ is

 \displaystyle \begin{align} H\left(\omega \right)=\sum^{\infty }_{n=0 }{h_n}e^{i \omega n}=A\left(\omega \right)e^{-i\phi \left(\omega \right)} \end{align} (18)

The Fourier transform is obtained from the engineering z-transform

 \displaystyle \begin{align} H\left(z\right)=h_0+h_{{\rm 1}} z^{-{\rm 1}}+h_{1}z^{-2}+ \dots \end{align} (19)

by the substitution $\displaystyle z=e^{i\omega }$ .

The Fourier transform (electrical engineering convention) is obtained from the geophysical Z-transform

 \displaystyle \begin{align} H\left(Z\right)=h_0+h_{1}Z+h_{1}Z^{2}+\dots \end{align} (20)

by the substitution $\displaystyle Z=e^{-i\omega }$ . The locus of $\displaystyle Z=e^{-i\omega }$ is the unit circle $\displaystyle {|}Z{|}=1$ . As angular frequency increases from $\displaystyle \omega =-\pi$ through $\displaystyle \omega =0$ to $\displaystyle \omega =\pi$ , the point $\displaystyle Z=e^{-i\omega }$ goes around the unit circle (in a clockwise direction) from Z = +1 through Z = +i to Z = -1. The Fourier transform represents the value of the Z-transform on the unit circle (Figure 1).

Figure 1.  The Fourier transform is equal to the values of the Z-transform as Z traverses the unit circle in the clockwise direction.
Table 1. Common signals and their electrical engineering z-transforms.
Signal name Signal z-transform Convergence region
Unit impulse $\displaystyle {\delta }_n=1$ for $\displaystyle t=0$

$\displaystyle {\delta }_n=0$ otherwise

1 Everywhere
Delayed impulse $\displaystyle {\delta }_{n-k}$ for fixed k > 0 $\displaystyle z^{ - k}$ $\displaystyle |z|\; > 0$
Unit causal step $\displaystyle u_n = 0\;\text{for}\;k < 0$

$\displaystyle u_n = 1\;\text{for}\;k \ge \;0$

$\displaystyle \frac{z} {{z - 1}} = \frac{1} {{1 - z^{ - 1} }}$ $\displaystyle |z|\; < 0$
Negative anticausal step $\displaystyle - u_{ - n - 1}$ $\displaystyle \frac{z} {{z - 1}} = \frac{1} {{1 - z^{ - 1} }}$ $\displaystyle |z| < 1$
Ramp $\displaystyle nu_n$ $\displaystyle \frac{z} {{(z - 1)^2 }}$ $\displaystyle |z| > 1$
Causal geometric $\displaystyle \alpha ^n u_n$ $\displaystyle \frac{z} {{z - \alpha }} = \frac{1} {{1 - \alpha z^ - }}$ $\displaystyle |z|\; > \;\alpha$
Negative anticausal geometric $\displaystyle - \alpha ^n u_{ - n - 1}$ $\displaystyle \frac{{z(z - \cos \theta )}} {{z^2 - 2\cos \theta \;z + 1}}$ $\displaystyle |z|\; < \;\alpha$
Causal cosine $\displaystyle u_n \cos (\theta _n )$ $\displaystyle \frac{{z(z - \cos \theta )}} {{z^2 - 2\cos \theta \;z + 1}}$ $\displaystyle |z|\; > 1$
Causal sine $\displaystyle u_n \sin (\theta _n )$ $\displaystyle \frac{{z\sin \theta }} {{z^2 - 2\cos \theta \;z + 1}}$ $\displaystyle |z|\; > 1$
Causal geometric cosine $\displaystyle u_n \alpha ^n \cos (\theta _n )$ $\displaystyle \frac{{z(z - \alpha \cos \theta )}} {{z^2 - 2\alpha \;\cos \theta \;z + \alpha ^2 }}$ $\displaystyle |z|\; > \;|\alpha |$
Causal geometric sine $\displaystyle u_n \alpha ^n \sin (\theta _n )$ ${\displaystyle {\frac {z\alpha \sin \theta }{z^{2}-2\alpha \cos \theta \;z+\alpha ^{2}}}}$ ${\displaystyle |z|\;>\;|\alpha |}$
Table 2. Common signals and their geophysical Z-transforms.
Signal name Signal z-transform Convergence region
Unit impulse ${\displaystyle {\delta }_{n}=1}$ for ${\displaystyle t=0}$

$\displaystyle {\delta }_n=0$ otherwise

1 Everywhere
Delayed impulse $\displaystyle {\delta }_{n-k}$ for fixed k > 0 $\displaystyle z^k$ $\displaystyle |z|\; > 0$
Unit causal step $\displaystyle u_n = 0\;\text{for}\;k < 0$

$\displaystyle u_n = 1\;\text{for}\;k \ge \;0$

$\displaystyle \frac{1}{{1 - Z}}$ $\displaystyle |z|\; < 1$
Negative anticausal step $\displaystyle - u_{ - n - 1}$ $\displaystyle \frac{1}{{1 - Z}}$ $\displaystyle |z|\; > 1$
Ramp $\displaystyle nu_n$ $\displaystyle \frac{Z}{{(1 - Z)^2 }}$ $\displaystyle |Z|\; < \;1$
Causal geometric $\displaystyle \alpha ^n u_n$ $\displaystyle \frac{1} {{1 - \alpha Z}}$ $\displaystyle |Z|\; < \;\alpha$
Negative anticausal geometric $\displaystyle - \alpha ^n u_{ - n - 1}$ $\displaystyle \frac{1} {{1 - \alpha Z}}$ $\displaystyle |Z|\; > \;\alpha$
Causal cosine $\displaystyle u_n \cos (\theta n)$ $\displaystyle \frac{{1 - \cos \theta \;Z}} {{1 - 2\;\cos \theta \;Z + Z^2 }}$ $\displaystyle |Z|\; < 1$
Causal sine $\displaystyle u_n \sin (\theta n)$ $\displaystyle \frac{{Z\;\sin \theta }} {{1 - 2\;\cos \theta \;Z + Z^2 }}$ $\displaystyle |Z|\; < 1$
Causal geometric cosine $\displaystyle u_n \alpha ^n \cos (\theta n)$ $\displaystyle \frac{{Z(1 - \alpha \cos \theta \;Z)}} {{\alpha ^2 - 2\alpha \;\cos \theta \;Z + Z^2 }}$ $\displaystyle |Z|\; < \;|\alpha |$
Causal geometric sine $\displaystyle u_n \alpha ^n \sin (\theta n)$ $\displaystyle \frac{{Z\alpha \;\sin \theta }} {{\alpha ^2 - 2\alpha \cos \theta \;Z + Z^2 }}$ $\displaystyle |Z|\; < \;|\alpha |$