Z-transform

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	7
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega =-\pi } through Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega =0 } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t=0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\delta }_n=0 } otherwise	1	Everywhere
Delayed impulse	${\delta }_{n-k}$ for fixed k > 0	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z^{ - k} }	$|z|\;>0$
Unit causal step	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_n = 0\;\text{for}\;k < 0} $u_{n}=1\;{\text{for}}\;k\geq \;0$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle nu_n }	${\frac {z}{(z-1)^{2}}}$	$|z|>1$
Causal geometric	$\alpha ^{n}u_{n}$	${\frac {z}{z-\alpha }}={\frac {1}{1-\alpha z^{-}}}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |z|\; > \;\alpha }
Negative anticausal geometric	$-\alpha ^{n}u_{-n-1}$	${\frac {z(z-\cos \theta )}{z^{2}-2\cos \theta \;z+1}}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_n \sin (\theta _n )}	${\frac {z\sin \theta }{z^{2}-2\cos \theta \;z+1}}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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	$u_{n}\alpha ^{n}\sin(\theta n)$	${\frac {Z\alpha \;\sin \theta }{\alpha ^{2}-2\alpha \cos \theta \;Z+Z^{2}}}$	$|Z|\;<\;|\alpha |$

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Fourier transform	Delay: Minimum, mixed, and maximum
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Frequency	Synthetics
Table of Contents (book)

Also in this chapter

• Wavelets
• Fourier transform
• Delay: Minimum, mixed, and maximum
• Two-length wavelets
• Illustrations of spectra
• Delay in general
• Energy
• Autocorrelation
• Canonical representation
• Zero-phase wavelets
• Symmetric wavelets
• Ricker wavelet
• Appendix G: Exercises

External links

find literature about Z-transform