Ztransform
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 Ztransform differ? Geophysicists and electrical engineers have different conventions with respect to the ztransform (see also the discussion in Chapter 6). Let be the impulse response of a causal timeinvariant linear filter. The engineering ztransform (with lowercase z) is
( )
whereas the geophysics Ztransform (with capital Z) is the generating function
( )
The two are related by . Whereas the engineering z represents a unit advance operator, the geophysics Z represents a unit delay operator.
Table 1 gives the engineering ztransforms of some common signals.
By letting , Table 1 becomes Table 2 for the corresponding geophysical Ztransforms.
How is the Fourier transform obtained from the Ztransform? 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 is
( )
The Fourier transform is obtained from the engineering ztransform
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}} ( )
by the substitution .
The Fourier transform (electrical engineering convention) is obtained from the geophysical Ztransform
( )
by the substitution . The locus of is the unit circle . 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 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 Ztransform on the unit circle (Figure 1).
Signal name  Signal  ztransform  Convergence region 

Unit impulse  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  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} }  
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}

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} }} }  
Negative anticausal step  
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 }  
Causal geometric  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  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  
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 )}  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  
Causal geometric sine 
Signal name  Signal  ztransform  Convergence region 

Unit impulse  for
otherwise 
1  Everywhere 
Delayed impulse  for fixed k > 0  
Unit causal step 


Negative anticausal step  
Ramp  
Causal geometric  
Negative anticausal geometric  
Causal cosine  
Causal sine  
Causal geometric cosine  
Causal geometric sine 
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Fourier transform  Delay: Minimum, mixed, and maximum 
Previous chapter  Next chapter 
Frequency  Synthetics 
Also in this chapter
 Wavelets
 Fourier transform
 Delay: Minimum, mixed, and maximum
 Twolength wavelets
 Illustrations of spectra
 Delay in general
 Energy
 Autocorrelation
 Canonical representation
 Zerophase wavelets
 Symmetric wavelets
 Ricker wavelet
 Appendix G: Exercises