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: , then the z-transform of the wavelet is


The z-transform may be thought of as , where 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:


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 , which may be factored , which has the roots , , , . (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 in ] 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.


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