Z-transform

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
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 be the impulse response of a causal time-invariant linear filter. The engineering z-transform (with lowercase z) is


(16)

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


(17)

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 z-transforms of some common signals.

By letting , 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 in terms of angular frequency is


(18)

The Fourier transform is obtained from the engineering z-transform


(19)

by the substitution .

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


(20)

by the substitution . The locus of is the unit circle . As angular frequency increases from through to , 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 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 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
Table 2. Common signals and their geophysical Z-transforms.
Signal name Signal z-transform 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
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Frequency Synthetics

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