Dictionary:Z-transform

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A transform useful for representing time series and calculating the effects of various operations. If the sample values of a wavelet at successive times are: ${\displaystyle x_{t}=[x_{0},x_{1},x_{2},x_{3}...,x_{n}]}$, then the z-transform of the wavelet is

${\displaystyle x_{z}=x_{0}+x_{1}z+x_{2}z^{2}+x_{3}z^{3}+...+x_{n}z^{n}.}$

The z-transform may be thought of as ${\displaystyle z=e^{-i\omega t}}$, where ${\displaystyle \omega ={\text{angular frequency}}}$ this allows one to relate it to the Fourier transform. The z-transform technique is an easy way of converting from the time domain into a form which can be treated as in the frequency domain. Convolution can be accomplished by merely multiplying the z-transforms of the waveforms being convolved, and the inverse of a filter can be found by finding the reciprocal of the filter's z-transform. The z-transform polynomial can be factored and expressed as the product of doublets of the form:

${\displaystyle f(z)=(z-a)(z-b)(z-c)...(z-n).}$

The magnitudes of the roots or zeros for which this expression vanishes, i.e., z=a, z=b, etc. (which may be complex), indicate whether the doublets are minimum or maximum phase. Values greater than unity are said to "lie outside the unit circle."

FIG. Z-3.z-plane. (a) The wavelet (10, -2, -1, 2 1) has the z-transform ${\displaystyle 10-2z-z^{2}+2z^{3}+z^{4}}$, which may be factored ${\displaystyle (2+j+z)(2-j+z)(-1-j+z)(-1+j+z)}$, which has the roots ${\displaystyle (-2-j)}$, ${\displaystyle (-2+j)}$, ${\displaystyle (1-j)}$, ${\displaystyle (1+j)}$. (b) A plot of these roots in the z-plane is shown. Since all roots lie outside a circle of radius 1 (the unit circle), the wavelet is minimum phase.

If all the roots lie outside the unit circle, the function is minimum phase; if all are inside, it is maximum phase. Values for which an expression becomes infinite [such as ${\displaystyle r}$ in ${\displaystyle {\frac {1}{z-r}}}$] are called poles or singularities. Filters are sometimes designed in the z-plane.[1][2] Sometimes the opposite convention is used, i.e., successive sample values are multiplied by successively higher negative powers of z and then the criteria for minimum and maximum phase with respect to the unit circle is reversed.

References

1. Robinson, E. A., and Treitel, S., 1964, Principles of digital filtering: Geophysics, 29, 395–404.
2. Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press, pgs. 292, 549-550.