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