# 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{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

 {\displaystyle {\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 ${\displaystyle Z=z^{-1}}$. 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{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{aligned}H\left(z\right)=h_{0}+h_{\rm {1}}z^{-{\rm {1}}}+h_{1}z^{-2}+\dots \end{aligned}}} (19)

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

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

 {\displaystyle {\begin{aligned}H\left(Z\right)=h_{0}+h_{1}Z+h_{1}Z^{2}+\dots \end{aligned}}} (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 ${\displaystyle u_{n}=0\;{\text{for}}\;k<0}$

${\displaystyle u_{n}=1\;{\text{for}}\;k\geq \;0}$

${\displaystyle {\frac {z}{z-1}}={\frac {1}{1-z^{-1}}}}$ ${\displaystyle |z|\;<0}$
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 ${\displaystyle u_{n}=0\;{\text{for}}\;k<0}$

${\displaystyle u_{n}=1\;{\text{for}}\;k\geq \;0}$

${\displaystyle {\frac {1}{1-Z}}}$ ${\displaystyle |z|\;<1}$
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 ${\displaystyle u_{n}\alpha ^{n}\sin(\theta n)}$ ${\displaystyle {\frac {Z\alpha \;\sin \theta }{\alpha ^{2}-2\alpha \cos \theta \;Z+Z^{2}}}}$ ${\displaystyle |Z|\;<\;|\alpha |}$