# Canonical representation

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 |

What is the canonical representation of a causal wavelet? Let be the minimum-phase wavelet with the same amplitude spectrum as that of the causal wavelet *w*. Then is called the minimum-phase counterpart of *w*. The *canonical representation' states that any causal wavelet *w* can be represented as the convolution of its minimum-phase counterpart and a causal all-pass wavelet *p*; that is,*

**(**)

Because the inverse of a minimum-phase wavelet is minimum phase, it follows that is minimum phase and hence is causal. From the canonical representation, we see that the inverse wavelet is given by

**(**)

Two cases can occur. In the first case, the causal wavelet *w* is itself minimum phase. Then the causal all-pass wavelet is trivial, so the inverse wavelet is simply . In this case, the inverse wavelet is minimum phase and causal.

In the second case, the causal wavelet *w* is not minimum phase. Then the causal all-pass wavelet *p* is not trivial and therefore its inverse is a one-sided noncausal wavelet. In this case, the inverse wavelet is a two-sided noncausal wavelet.

## Continue reading

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Autocorrelation | Zero-phase wavelets |

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Frequency | Synthetics |

## Also in this chapter

- Wavelets
- Fourier transform
- Z-transform
- Delay: Minimum, mixed, and maximum
- Two-length wavelets
- Illustrations of spectra
- Delay in general
- Energy
- Autocorrelation
- Zero-phase wavelets
- Symmetric wavelets
- Ricker wavelet
- Appendix G: Exercises