# 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 $h_{0},h_{\rm {l}},h_{2},\dots$ be the impulse response of a causal time-invariant linear filter. The engineering z-transform (with lowercase z) is

 {\begin{aligned}H_{\text{engineering}}\;(z)=h_{0}+h_{1}\;z^{-1}+h_{2}\;z^{-2}+...,\end{aligned}} (16)

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

 {\begin{aligned}H\left(Z\right)=h_{0}+h_{\rm {l}}Z+h_{1}Z^{2}+\dots \end{aligned}} (17)

The two are related by $Z=z^{-1}$ . 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 $Z=z^{-1}$ , 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 $\omega$ is

 {\begin{aligned}H\left(\omega \right)=\sum _{n=0}^{\infty }{h_{n}}e^{i\omega n}=A\left(\omega \right)e^{-i\phi \left(\omega \right)}\end{aligned}} (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 $z=e^{i\omega }$ .

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

 {\begin{aligned}H\left(Z\right)=h_{0}+h_{1}Z+h_{1}Z^{2}+\dots \end{aligned}} (20)

by the substitution $Z=e^{-i\omega }$ . The locus of $Z=e^{-i\omega }$ is the unit circle ${|}Z{|}=1$ . As angular frequency increases from $\displaystyle \omega =-\pi$ through $\displaystyle \omega =0$ to $\displaystyle \omega =\pi$ , the point $Z=e^{-i\omega }$ 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 ${\delta }_{n}=1$ for $\displaystyle t=0$

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

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

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

1 Everywhere
Delayed impulse ${\delta }_{n-k}$ for fixed k > 0 $z^{k}$ $|z|\;>0$ Unit causal step $u_{n}=0\;{\text{for}}\;k<0$ $u_{n}=1\;{\text{for}}\;k\geq \;0$ ${\frac {1}{1-Z}}$ $|z|\;<1$ Negative anticausal step $-u_{-n-1}$ ${\frac {1}{1-Z}}$ $|z|\;>1$ Ramp $nu_{n}$ ${\frac {Z}{(1-Z)^{2}}}$ $|Z|\;<\;1$ Causal geometric $\alpha ^{n}u_{n}$ ${\frac {1}{1-\alpha Z}}$ $|Z|\;<\;\alpha$ Negative anticausal geometric $-\alpha ^{n}u_{-n-1}$ ${\frac {1}{1-\alpha Z}}$ $|Z|\;>\;\alpha$ Causal cosine $u_{n}\cos(\theta n)$ ${\frac {1-\cos \theta \;Z}{1-2\;\cos \theta \;Z+Z^{2}}}$ $|Z|\;<1$ Causal sine $u_{n}\sin(\theta n)$ ${\frac {Z\;\sin \theta }{1-2\;\cos \theta \;Z+Z^{2}}}$ $|Z|\;<1$ Causal geometric cosine $u_{n}\alpha ^{n}\cos(\theta n)$ ${\frac {Z(1-\alpha \cos \theta \;Z)}{\alpha ^{2}-2\alpha \;\cos \theta \;Z+Z^{2}}}$ $|Z|\;<\;|\alpha |$ Causal geometric sine $u_{n}\alpha ^{n}\sin(\theta n)$ ${\frac {Z\alpha \;\sin \theta }{\alpha ^{2}-2\alpha \cos \theta \;Z+Z^{2}}}$ $|Z|\;<\;|\alpha |$ 